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ɔP	[	2h3Vv'mЗ6ٚ}/}tt>w)ǔ9999?999999*999999999999999999999999999999)`wל#_[gCG;Os+kS/Ûۛǝ7ϛ˜))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))--------------------------------------------------------------------------------------------------------------------------xD0,G,D0,G,D0,G,D0,G,D0,G,D0,G,Math::Pari::Not a matrix where matrix expectedin, ...gne_gne_expMath::Pari::PARIcolMath::Pari::interface5Math::Pari::interface25Math::Pari::interface45Math::Pari::have_highlevelgp_readvec_stream: reaching %ld entries
unknown code in readobjmatrix(0,%ld)read failed%s written by an incompatible version of GPPolGGDGbnrisconductor(a1,{a2},{a3}): returns 1 if the modulus is the conductor of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc), and 0 otherwise. Slightly faster than bnrconductor if this is the only desired resultcore(n,{flag=0}): unique (positive of negative) squarefree integer d dividing n such that n/d is a square. If (optional) flag is non-null, output the two-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecotan(x): cotangent of xderivellak(e,n): computes the n-th Fourier coefficient of the L-function of the elliptic curve eellidentifyforsubgroupkroneckermatcompanionmateigen(x): eigenvectors of the matrix x given as columns of a matrixmattransposeminpolynfdisc(x,{flag=0},{p}): discriminant of the number field defined by the polynomial x using round 4. Optional args flag and p are as in nfbasisnfeltpowmodprnfnewprecpolcyclo(n,{v=x}): n-th cyclotomic polynomial (in variable v)lGDnpolinterpolate(xa,{ya},{x},{&e}): polynomial interpolation at x according to data vectors xa, ya (ie return P such that P(xa[i]) = ya[i] for all i). If ya is omitter, return P such that P(i) = xa[i]. If present, e will contain an error estimate on the returned valueprimesprintpquadunit(x): fundamental unit of the quadratic field of discriminant x where x must be positivernfcharpoly(nf,T,alpha,{var=x}): characteristic polynomial of alpha over nf, where alpha belongs to the algebra defined by T over nf. Returns a polynomial in variable var (x by default)rnfisfree(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notsqrt(x): square root of xsqrtnwrite1(filename,a): write the string expression a (same output as print1) to filenameznlog(x,g): g as output by znprimroot (modulo a prime). Return smallest non-negative n such that g^n = xdependent vectors in lllfpin precision %ld
fincke_pohsttime [max,t12,loop,reds,fin] = [%ld, %ld, %ld, %ld, %ld]
insufficient precision for generators, not givencompleting bnf (building cycgen)
*** Ideal no %ld: [%Z, %Z, %Z, %Z]
gmul2nincorrect denominator in initgaloisborne: %ZA4GaloisConj: %ld hop sur %ld iterations
too large negative arg %ld in zetagpolylognon integral index in prodinf1non integral index in sumposgdeflateLLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld
subresallnon polynomial or series type substituted in a seriesgfloor2ninvalid vector in qfevalincorrect variable in caradjmatrixqz when the first 2 dets are zerohnfall[2], li = %ldN0 in QuickPol: %ld 
polrelnumN0 = %ld
RecCoeff3: no solution found!
invalid flagimpossible assignment I-->S\vskip\smallskipamount$\displaystyle{\tt\%#1} = #2$expected character: ']'regulafor this traceTerminal does not define justify_textCan't find Gnuplot drawing engine DLL,
	set GNUPLOT_DRAW_DLL environment variable to the name of the DLL,
	or install Perl module Term::Gnuplot, e.g., by running
		perl -MCPAN -e "install Term::Gnuplot"
	With Term::Gnuplot, if you don't have root access, consult
		perldoc -q "my own"
	alternatively, you can use an uninstalled version of Term::Gnuplot
	by running GP/PARI as
		env PERL5OPT=-Mblib=/directory/of/build/of/Term-Gnuplot gp
Entering compute_data()

plotinitvLGGD0,L,plotlinetype(w,type): change the type of following lines in rectwindow w. type -2 corresponds to frames, -1 to axes, larger values may correspond to something else. w=-1 changes highlevel plottingplotscale(w,x1,x2,y1,y2): scale the coordinates in rectwindow w so that x goes from x1 to x2 and y from y1 to y2 (y2<y1 is allowed)too many iterations in roots2() ( laguer() ):
SubCyclo: conductor:%ld
wrong type in galoissubcyclogauss. i=%lddetint. k=%ldFpM_invimageincorrect modpr format  fastnu: G is computed
Rg_to_FlFpXQX_gcdmodulargcd: trial division failedQXQ_inv: mod %ld (bound 2^%ld)IFAC: untested integer declared primeincorrect dimensions in initrect%ld %ld p
%ld %ld m %ld %ld l
L(6) = PSL(2,5) = A_5(6)incorrect galois automorphism in galoisapplynegative valuation in laplacegasin[rnfkummer] bnfinit(Kz)Step 4
Step 18
M* = %Z
(lower bound for regulator) M = %Z
weipellnumMPQS: sieving interval = [%ld, %ld]
MPQS:   (%ld of these from partial relations)
 
\\ MATRIX READ BY MPQS
FREL=MPQS: splitting N after %ld kernel vector%s

[Non trivial conditional class group.
  *** May miss solutions of the norm equation  errdelta = %Z
Unrecognized action in a templateorderstuunused characters: %sbreak not allowed here (reading function args)skipidentifier (unknown code)invalid coefficients in rootsalgdep(x,n)=algebraic relations up to degree n of xbilhell(e,z1,z2)=canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsGD0,G,D0,G,D3,L,factpol2(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0) using root findingidealaddmultone(nf,list)=when the sum of the ideals in the number field K defined by nf and given in the vector list is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1ideallistarch(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, without generatorsindsort(x)=indirect sorting of the vector xisisom(x,y)=tests whether the number field defined by the polynomial x is isomorphic to the one defined by y; 0 if not, otherwise all the isomorphismsispspispsp(x)=true(1) if x is a strong pseudoprime, false(0) if notisqrtlll(x)=lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lllgen(x)=lll reduction of the vectors forming the matrix x with polynomial coefficients (gives the unimodular transformation matrix)mod(x,y)=creates the integer x modulo y on the PARI stackmodreverse(x)=reverse polymod of the polymod x, if it existsnumer(x)=numerator of xpolintpolredabsnoredrhorealrhorealnodsigma(x)=sum of the divisors of xsignat(x)=signature of the symmetric matrix xsinh(x)=hyperbolic sine of xsumalt2(X=a,expr)=Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at asuminf(X=a,expr)=infinite sum (X goes from a to infinity) of real or complex expression exprtruncweipellzell(e,z)=In the complex case, lattice point corresponding to the point z on the elliptic curve ezidealstarinit(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*ratlift: bmax must be > 0, found
	bmax=%Z
Got the type 0x%x instead of CV=0x%x or GV=0x%x in %s, %iarg1, arg2, arg3=0gcmp1_flexible_intn = 02.030518Math::Pari::allocatememUnsupported code '%.1s' in signature '%s' of a PARI functiont_FRAC*** hashcode = %lu
\frac{%s is set (%s), but is not a directorymat(%ld,%ld) = I/O: closing file %s (code %d) 
skipping directory %s  *** %s: %sbernvecbittest(x,n): gives bit number n (coefficient of 2^n) of the integer x. Negative numbers behave as if modulo big power of 2bnfdecodemodulebnfsignunit(bnf): matrix of signs of the real embeddings of the system of fundamental units found by bnfinitbnrrootnumbercomponent(x,s): the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]. For list objects such as nf, bnf, bnr or ell, it is much easier to use member functions starting with "."GD0,L,dirmul(x,y): multiplication of the Dirichlet series x by the Dirichlet series yellapelleisnum(om,k,{flag=0}): om=[om1,om2] being a 2-component vector giving a basis of a lattice L and k an even positive integer, computes the numerical value of the Eisenstein series of weight k. When flag is non-zero and k=4 or 6, this gives g2 or g3 with the correct normalizationelletaellgenerators(E): if E is an elliptic curve as output by ellinit(), return the generators of the Mordell-Weil group associated to the curve. This function depends on the curve being referenced in the elldata database.elllseriesGPidealpowideleprincipalincgam(s,x,{y}): incomplete gamma function. y is optional and is the precomputed value of gamma(s)isprime(x,{flag=0}): true(1) if x is a (proven) prime number, false(0) if not. If flag is 0 or omitted, use a combination of algorithms. If flag is 1, the primality is certified by the Pocklington-Lehmer Test. If flag is 2, the primality is certified using the APRCL test.iGGlex(x,y): compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)listsortmatkermatrixqz(x,p): if p>=0, transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwise. If p=-1, finds a basis of the intersection with Z^n of the lattice spanned by the columns of x. If p=-2, finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmaxmodreverse(x): reverse polymod of the polymod x, if it existsnfeltdiveuc(nf,a,b): gives algebraic integer q such that a-bq is smallnfeltmulnfeltvalnormnorml2(x): square of the L2-norm of the vector xnumtoperm(n,k): permutation number k (mod n!) of n letters (n C-integer)padicprecpolcompositumpoldiscreduced(f): vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fpolrootspadic(x,p,r): p-adic roots of the polynomial x to precision rpolsylvestermatrix(x,y): forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowsprecprime(x): largest pseudoprime <= x, 0 if x<=1primepi(x): the prime counting function pi(x) = #{p <= x, p prime}.qfbredqfrep(x,B,{flag=0}): vector of (half) the number of vectors of norms from 1 to B for the integral and definite quadratic form x. Binary digits of flag mean 1: count vectors of even norm from 1 to 2B, 2: return a t_VECSMALL instead of a t_VECreturn({x=0}): return from current subroutine with result xsumalt(X=a,expr,{flag=0}): Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialstaylorvectorsmallEntering LLL
bitsize bound: %ld
bitsize Norm: %ld
incorrect big number fieldregulator is zero.
KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld
rnfidealhermiteSecond arg. must be integer in galoisconj4GaloisAnalysis:Nbtest=%ld,p=%ld,o=%ld,n_o=%d,best p=%ld,ord=%ld,k=%ld
partial factorization is not meaningful hereQ_contentnot the same variables in sylvestermatrixreduceddiscsmithginvmodqfi_unitassmatmatrixqz2hnf[1]. i=%ldsqred2nb lines > nb columns in hnfmodinconsistent datanot a polynomial of degree 2 in quadhilbertpowsubFBquadto find factor %ZTR_POL, i = %ld/%ldplotpointsplotpointsizereducible modulus in factornfSubcyclo: val=%ld
subcyclo_cyclicIncomplete curve name in ellsearchNo such elliptic curveincorrect mask in vecextractSolving the triangular system
ZM_invFpM_kerincorrect polynomial in rnf function  entering Nilordrelative basis computed
1/0 exponent in FpXQ_sqrtnpolint_triv2 (i = %ld)bound for resultant: 2^%ld
...
ECM: time = %6ld ms, B1 phase done, SQUFOF: blacklisting a = %ld on second cycle
	But it wasn't a pure power.
ECM: stack tight, using heap space
IFAC: trying Shanks' SQUFOF, will fail silently if input
      is too large for it.
IFAC: incorporating set of %ld factor(s)
checkrnfplease apply bnfinit firstmissing units in %sgalois of degree higher than 11nsiso0Partitions of %ld (%ld)
read_objectcaching Bernoulli numbers 2*%ld to 2*%ld, prec = %ld
non-positive integer argument in cxgammaGamma not defined for non-integral p-adic numberdegree overflow in pow_monomevaluation overflow in sqrtn[rnfkummer] Selmer groupnot an integral curve in elllocalredpoint not on elliptic curveLPRELMPQS: relations file truncated?!
 0MPQS: full relations file %s than expectedinvalid polynomial in thue (need deg>2)C (bitsize) : %d
x <= %Z
not a vector of ideals in idealaddmultooneisrealapprrefine_Fapell2(e,p)=computes a_p for the elliptic curve e using Jacobi symbolsGD0.1,G,D0.1,G,D5,G,conductorofchar(bnr,chi)=conductor of the character chi on the ray class group bnrideallistunitarchgenimageisinclfastkerlaplacemathell(e,x)=gives the height matrix for vector of points x on elliptic curve e using theta functionsGGVVInfval(nf,a,pr)=valuation of element a at the prime prreduceddiscrootmod(x,p)=roots mod p of the polynomial xrounderror(x)=maximum error found in rounding xseries(x,v)=convert x (usually a vector) into a power series with variable v, starting with the constant coefficientsetsearch(x,y)=looks if y belongs to the set x. Returns 0 if it is not, otherwise returns the index j such that y==x[j]simplify(x)=simplify the object x as much as possiblesmith(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vectorsubellthetanullk(q,k)=k'th derivative at z=0 of theta(q,z)entering zlog, arg1, arg2, arg3, arg4gand_ggt_gsubv5.14.0Math::Pari::listPariArray index %i out of rangeLV=GGEpD0,L,D0,L, %2d: %s
)
[]
\pmatrix{
 %ld unnamed objects read. Returning then in a vectormod(\PARIbreak I/O: opening file %s (mode %s)
%s not written for a %ld bit architecture.use pari_warn for warningsgcdMod(x,y): creates 'x modulo y'.Pi=Pi(): the constant pi, with current precisionbesseljbesseln(nu,x): N-bessel function of index nu and argument xbitand(x,y): bitwise "and" of two integers x and y. Negative numbers behave as if modulo big power of 2bnfnarrow(bnf): given a big number field as output by bnfinit, gives as a 3-component vector the structure of the narrow class groupbnfreg(P,{tech=[]}): compute the regulator of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnrconductorofcharcoshcosh(x): hyperbolic cosine of xdilog(x): dilogarithm of xelladd(e,z1,z2): sum of the points z1 and z2 on elliptic curve eforstepforstep(X=a,b,s,seq): the sequence is evaluated, X going from a to b in steps of s (can be a vector of steps)galoisexportgetrand(): current value of random number seedidealaddtooneideallist(nf,bound,{flag=4}): vector of vectors L of all idealstar of all ideals of norm<=bound. If (optional) flag is present, its binary digits are toggles meaning 1: give generators; 2: add units; 4: give only the ideals and not the bid.intformal(x,{y}): formal integration of x with respect to the main variable of y, or to the main variable of x if y is omittedintnuminitgen(t,a,b,ph,{m=0},{flag=0}): initialize tables for  integrations from a to b using abcissas ph(t) and weights ph'(t). Note that  there is no equal sign after the variable name t since t always goes from  -infty to +infty, but it is ph(t) which goes from a to b, and this is not  checked. If flag = 1 or 2, multiply the reserved table length by 4^flag, to  avoid corresponding error.isfundamentalispower(x,{k},{&n}): true (1) if x is a k-th power, false (0) if not. If n is given and a k-th root was computed in the process, put that in n. If k is omitted, return the maximal k >= 2 such that x = n^k is a perfect power, or 0 if no such k exist.lengthlexlistputmathessnfsnfpermtonum(vect): ordinal (between 1 and n!) of permutation vectpolrecipqfminim(x,{bound},{maxnum},{flag=0}): number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0. flag is optional, and can be 0: default; 1: returns the first minimal vector found (ignore maxnum); 2: as 0 but uses a more robust, slower implementation, valid for non integral quadratic formsquadgenrnfidealreltoabs(rnf,x): transforms the ideal x from relative to absolute representationGGD2,L,setminusuntilweber(x,{flag=0}): One of Weber's f function of x. flag is optional, and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x) such that (j=(f^24-16)^3/f^24), 1: function f1(x)=eta(x/2)/eta(x) such that (j=(f1^24+16)^3/f2^24), 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x) such that (j=(f2^24+16)^3/f2^24)vsDGznstar(n): 3-component vector v, giving the structure of (Z/nZ)^*. v[1] is the order (i.e. eulerphi(n)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorsModified P: %Z
Initialization time = %ld
smallvectors looking for norm < %Z
factor base
(more relations needed: %ld)
/conjugate %ld: %Z
galoisconj4FlxqX_safegcdpadic polylogarithmintegral from infty to infty or from -infty to -inftynfhilbert not soluble at finite place: %Z
polsym_gensylvestermatrixmatratliftnfgcdfor this block of tracesLLL_cmbf: chk_factors failedgevalnegative discriminant in %smatrixqz_auxallhnfmodlength (lg) overflowboldfgispower for non-rational argumentsquadhilbert needs a fundamental discriminantremaining modular factor(s): %ld
panic: gnuplotTerminal does not define pointsizeggvalnegationgexpocouldn't open dynamic symbol table of processplotkillplotrecthplotrecthrawpsplothraw   %3ld fact. of degree %3ld
non-positive precision in factorpadicSubcyclo: borne=%Z
mattodiagonalFqM_ker  f = %Z,
  a = %Z
, parameters: %Z^%ld
  f = %ZmodulargcdLucasModC(6) = 6 = 3[x]2S6F_42(7) = 7:6coefget_red_G: starting LLL, prec = %ld (%ld + %ld)
        all integer roots are double roots
indefinite invariant polynomial in gpoly() %ld^%ld
$$$$$ Tschirnhaus transformation of degree %ld: $$$$$
 )
negative degree in legendrenot a set in setintersectpsi of power seriesgsqrtnp-adic argument out of range in gsinStep 5
Computing Newton sums: not an Abelian extension in rnfnormgroup?MPQS: first sorting at %ld%%, then every %3.1f%% / %3.1f%%

MPQS: giving up.
MPQS: Gauss done: kernel has rank %ld, taking gcds...
MPQS: no solutions found from linear system solver5th powernot a tnf in thuec1 = %Z
c2 = %Z
Indice <= %Z
  c8  = %Z
[1]: nfhermitemod, i = %ldincorrect vector length in idealrederror in parse_option_stringcan't kill thatonly functions can be aliased.%s = break not allowed after #algdep2asin(x)=inverse sine of xbuchrealchptell(x,y)=change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]comprealraw(x,y)=Gaussian composition without reduction of the binary quadratic forms x and y of positive discriminantdiscrayabs(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayabslistarchall(bnf,bound)=gives list of discrayabscond of all modules up to norm bound with all possible archimedean places arch in reverse lexicographic order, in a longvector formatfactpol(x,l,hint)=factorization over Z of the polynomial x up to degree l (complete if l=0) using Hensel lift, knowing that the degree of each factor is a multiple of hintgaloisconj(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field, using p-adics, LLL on integral basis (not always complete)galoisconj1gotohell2idealdivexact(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nf when the quotient is known to be an integral idealideallistunitarchimagecompl(x)=vector of column indices not corresponding to the indices given by the function imageincgam4lllgramint(x)=lll reduction of the lattice whose gram matrix is the integral matrix x (gives the unimodular transformation matrix)localred(e,p)=e being an ellliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_pGGGGplseriesell(e,s,N,A)=L-series at s of the elliptic curve e, where |N| is the conductor, sign(N) the sign of the functional equation, and A a cut-off point close to 1newtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime pnfdiveucpoly(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientpolyrevprecrounderrorsetrand(n)=reset the seed of the random number generator to nsignunit(bnf)=matrix of signs of the real embeddings of the system of fundamental units found by buchinitsmallinitellsmithclean(z)=if z=[u,v,d] as output by smith2, removes from u,v,d the rows and columns corresponding to entries equal to 1 in dsmithpol(x)=Smith normal form (i.e. elementary divisors) of the matrix x with polynomial coefficients, expressed as a vectorthreetotwo2vecsort(x,k)=sorts the vector of vector (or matrix) x according to the value of its k-th componentzeta(s)=Riemann zeta function at sFp_shanks, k = %ldIdeallistfloorr (precision loss in truncation)PARI: %.*s%*s%sgcmpglt_gdiv_gle`%s' is not a Pari function name%ld%sour data takes %ld words out of %ld words on stack
Math::Pari::interface_flexible_voidMath::Pari::interface_flexible_longMath::Pari::interface30V=GGIV=GGID%020ld\cr}
den = real = #\right) %s.gpunexpected endianness in %smallocing NULL object in newblocleaving recover()
  ***   %s %s, please report %s %s %s %s.  [hint] you can increase GP stack with allocatemem()
0x%p:	0x%lx	%lu
besselh1(nu,x): H^1-bessel function of index nu and argument xbnfisintnormbnfisnorm(bnf,x,{flag=1}): Tries to tell whether x (in Q) is the norm of some fractional y (in bnf). Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-unit, with S a certain list of primes (in bnf) containing (among others) all primes dividing x. If bnf is known to be Galois, set flag=0 (in this case, x is a norm iff b=1). If flag is non zero the program adds to S all the primes : dividing flag if flag<0, or less than flag if flag>0. The answer is guaranteed (i.e x norm iff b=1) under GRH, if S contains all primes less than 12.log(disc(Bnf))^2, where Bnf is the Galois closure of bnfbnfsignunitbnrinitdireulerGGDnelltaniyamagammaideallog(nf,x,bid): if bid is a big ideal, as given by idealstar(nf,I,1) or idealstar(nf,I,2), gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)idealminintcircintformallGDGD&lindep(x,{flag=0}): Z-linear dependencies between components of x. flag is optional, and can be 0: default, PSLQ; -1: using Hastad et al; -2: returns a non-trivial kernel vector (not integral in general); positive, and in that case should be between 0.5 and 1.0 times the accuracy in decimal digits of x, using a standard LLLmatdiagonal(x): creates the diagonal matrix whose diagonal entries are the entries of the vector xnfeltdiv(nf,a,b): element a/b in nfnfhnfmodnfsubfields(nf,{d=0}): find all subfields of degree d of number field nf (all subfields if d is null or omitted). Result is a vector of subfields, each being given by [g,h], where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nfpoldisc(x,{v}): discriminant of the polynomial x, with respect to main variable if v is omitted, with respect to v otherwisepolzagier(n,m): Zagier's polynomials of index n,mprecision(x,{n}): change the precision of x to be n (n C-integer). If n is omitted, output real precision of object xprodeuler(X=a,b,expr): Euler product (X runs over the primes between a and b) of real or complex expressionrandomrnfhnfbasisrnfisfreernfnormgrouproundlLsignsizedigit(x): maximum number of decimal digits minus one of (the coefficients of) xsubgrouplistsumnumalt(X=a,sig,s,{tab},{flag=0}): numerical summation of (-1)^X s from X = ceiling(a) to +infinity. Note that the (-1)^X must not be included. sig is either a scalar or a two-component vector coded as in intnum, and the  scalar part is larger than all the real parts of the poles of s. Uses intnum, hence tab is as in intnum. If flag is nonzero, assumes that the function to  be summed satisfies conj(f(z))=f(conj(z)), and then up to twice faster.sumpos(X=a,expr,{flag=0}): sum of positive series expr, the formal variable X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialsvss*znorderlllint[2], kmax = %ld
...LLL reducing precision to %ld
Matrix to be reduced:
%Z
lindep2lllgramallgen    Norm(P) > Zimmert bound
    #%ld in factor base
small norm relations
#### Computing check
please apply rnfequation(,,1)S4GaloisConj:sigma=%Z
GaloisConj: %d hops on %Z tests
czetanot a vector in forvecnot a vector of two-component vectors in forvecnon integral index in prodinfprime_loop_initLLL_cmbf [no factor]gisirreducible
LLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)
nextSousResultant j = %ld/%ldnot a t_QFR in powrealrawquadhilbertreal
Trying modulus = %Z and subgroup = %Z
non quadratic residue in gsqrt(raw)new_galois_format[%ld,%ld] polynomial variable must have highest priority in nffactormodPrime ideal chosen: %Z
Z = %Z
delta[%ld] = %Z
d-1 test failed
p = 1 in Z_lvalremgcvtopno more room in this listcan't find symbol '%s' in dynamic symbol table of processplotcolor(w,c): in rectwindow w, set default color to c. Possible values for c are 1=black, 2=blue, 3=sienna, 4=red, 5=cornsilk, 6=grey, 7=gainsboroughpsdrawnot a group in forsubgroupisdiagonalincorrect object in diagonalrowred j=%ldnewtonsums  (eq,er) = (%ld,%ld)
ECM: time = %6ld ms,	ellfacteur giving up.
SQUFOF: ...found nothing on the ambiguous cycle
	after %ld steps there, time = %ld ms
SQUFOF: squfof_ambig returned %ld
`*where' out of bounds in ifac_sort_oneE(4) = 2[x]2false nf in nf_get_signs = 0 is a pole (gzetakall)xbest = %Z
polredabs (precision problem)      Working with polynomial #%ld:
lim, nn: [%ld, %ld], la = %lf
[apell1] giant steps, i = %ldlseriesellinitellnot a prime in localred[2]: mpqs_solve_linear_system  Baker = %Z
not a module in nfsmithaprcl test fails! this is highly improbableno more variables availableb8biddiffr2identifier (unknown code)not a suitable VECSMALL componentcurve not defined over Rparametersabs(x)=absolute value (or modulus) of xbilhellconcat(x,y)=concatenation of x and ycoredisc2det(x)=determinant of the matrix xdiscrayabscondfloor(x)=floor of x=largest integer<=xfpnidealinv2ideallistunit(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, without generatorsidmatimagecomplincgam2(s,x)=incomplete gamma function (for debugging only)V=GGID3,L,pisprincipalraygen(bnf,x)=bnf being output by buchrayinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorkerint(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkerint2lllgramkerimpolylog(m,x)=m-th polylogarithm of xpolylogd(m,x)=D_m~-modified m-th polylog of xpolzagrayclassnolist(bnf,liste)=if listes is as output by idealisunit or similar, gives list of corresponding ray class numbersrhoreal(x)=single reduction step of the binary quadratic form x of positive discriminantrnfdiscf(nf,pol)=given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2srgcd(x,y)=polynomial gcd of x and y using the subresultant algorithmtaylor(x,y)=taylor expansion of x with respect to the main variable of ytwototwowfalgtobasis_ia not invertible in ff_PHlog_FpInternal error in sv2pari!_gbitneg_gneg_logtag%li items moved off stack, onStack=%ld, offStack=%lda, b, c=0Math::Pari::PARIMath::Pari::interface14Math::Pari::interface23$$;$Math::Pari::interface28$$$$;$Access to elements of not-a-vector%s not implemented on this architecturet_MATembedded braces (in parser)Qfb(#<%d>
  current stack size: %lu (%.3f Mbytes)
Eulervrrallocatemem({s=0}): allocates a new stack of s bytes. doubles the stack if s is omittedasin(x): inverse sine of xbesselibezoutres(x,y): gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbitxor(x,y): bitwise "exclusive or" of two integers x and y. Negative numbers behave as if modulo big power of 2bnrL1bnrclassnocorediscelllseries(e,s,{A=1}): L-series at s of the elliptic curve e, where A a cut-off point close to 1GD&ellrootnoforidealappridealintersectidealmin(nf,ix,{vdir}): minimum of the ideal ix in the direction vdir in the number field nfidealprimedec(nf,p): prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a. Z_K, e,f as usual, a as vector of components on the integral basis, b Lenstra's constantintfuncinit(X=a,b,s,{flag=0},{m=0}): initialize tables for integrations  from a to b using a weight s(X). Essential for integral transforms such as intmellininv, intlaplaceinv and intfourier, since it avoids recomputing all the time the same quantities. Must then be used with intmellininvshort (for intmellininv) and directly with intnum and not with the corresponding  integral transforms for the others. See help for intnum for coding of a  and b, and m is as in intnuminit. If flag is nonzero, assumes that  s(-X)=conj(s(X)), which is twice faster.lpkillmatalgtobasismatintersect(x,y): intersection of the vector spaces whose bases are the columns of x and ymatsupplementmattranspose(x): x~=transpose of xnextnfeltmodnfgaloisapplynfisidealDGGpolredpolresultantpolsubcycloLGD0,L,pprintp1qfjacobi(x): eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xqfperfection(a): rank of matrix of xx~ for x minimal vectors of a gram matrix aquadclassunitrnfdedekind(nf,T,pr): relative Dedekind criterion over nf, applied to the order defined by a root of irreducible polynomial T, modulo the prime ideal pr. Returns [flag,basis,val], where basis is a pseudo-basis of the enlarged order, flag is 1 iff this order is pr-maximal, and val is the valuation in pr of the order discriminantrnfeltabstorelrnfeltup(rnf,x): expresses x (belonging to the base field) on the relative fieldserconvolsin(x): sine of xsinhsubst(x,y,z): in expression x, replace the variable y by the expression ztracevaluationznlog %ldgramtm1 = %ZlllintpartialallpslqL2R1 = %ld, R2 = %ld
D = %Z
small_norm (precision too low)BOUND = %.4g
frobenius powerGaloisConj:I will try %Z permutations
veceint1k < 0 in thetanullkneed exponential decrease in intinvmellinshortboth nonzero real and imag. part in coding, real ignorednegative number of components in vectorlast factor still to be checked
Root boundcan't deflate this power series (d = %ld): %Zfactor of general polynomialQ_divmuli_to_intThnf[2]. i=%lddep = %Z
(on with colors)prettyprinteryesforbidden or incompatible types in hilFp_sqrtngisprimeintegral part not significant in sfcontbuchquadincorrect parameters in quadclassunitsubFBquad (%ld elt.)polynomial variable must have highest priority in nffactorTerminal size directive without ','plotplotmove(w,x,y): move cursor to position x,y in rectwindow wrootpadicgauss_pivot_ker. k=%ld, n=%ld with parameters: %Z^%ld
QXQ_inv: char 0 check failedECM: time = %6ld ms, entering B2 phase, p = %lu
ECM: finishing curves %ld...%ld
SQUFOF: entering main loop with forms
	(1, %ld, %ld) and (1, %ld, %ld)
	of discriminants
	%Z and %Z, respectively
IFAC: checking for odd power
IFAC: trying Lenstra-Montgomery ECM
%9.*gA3S4you found a counter-example to a conjecture, please report!argument too large in ggammasum from 0 to N-1large Minkowski bound: certification will be VERY longpartition functiongoodsingular curve in ellinitMPQS: starting main loop
MPQS: kN has %ld decimal digits
,<%lu>MPQS: sizing out of tune, FB size or tolerance
	too large* real root no %ld/%ld
Semirat. reduction: B0 -> %Z x <= %Z
LLL_First_Pass successful !!
nfhermitenot a valid identifierfthis should be an integerall_roots: restarting, i = %ld, e = %ld
agm(x,y)=arithmetic-geometric mean of x and yGLLpanelldiscfeint1(x)=exponential integral E1(x)factor(x)=factorization of xfactorpadic(x,p,r)=p-adic factorization of the polynomial x to precision r, using the round 4 algorithmforstep(X=a,b,s,seq)=the sequence is evaluated, X going from a to b in steps of sglobalredideallistunitarch(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitinitalgred(x)=x being a nonconstant irreducible polynomial, finds (using polred) a simpler polynomial pol defining the same number field, and gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual), r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]isinclkbessel(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)norm(x)=norm of xo(a^b)=O(a^b)=p-adic or power series zero with precision given by bphi(x)=Euler's totient function of xsetintersect(x,y)=intersection of the sets x and ysize(x)=maximum number of decimal digits minus one of (the coefficients of) xsqred(x)=square reduction of the (symmetric) matrix x ( returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)weberf2(x)=Weber's f2 function of x (j=(f2^24+16)^3/f2^24)module too large in Fp_shanksD0,G,Same iterator for a double loopgle_gbitxor_gltval$$Math::Pari::interface18Math::Pari::interface32Math::Pari::pari_pprintMath::Pari::changevalue$Math::Pari::initmem not defined!LGD0,L[pipe:] '%s' failed[&=%016lx] mallocing NULL objectbot=0x%lx	top=0x%lx
Piallocatemembernvec(x): Vector of rational Bernoulli numbers B_0, B_2,...up to B_(2x)bnfinit(P,{flag=0},{tech=[]}): compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units, 3: small bnfinit, which can be converted to a big one using bnfmake. See manual for details about techbnfisprincipal(bnf,x,{flag=1}): bnf being output by bnfinit (with flag<=2), gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. flag is optional, whose meaning is: 0: output only v; 1: default; 2: output only v, precision being doubled until the result is obtained; 3: as 2 but output generatorsGDGD0,L,pbnrclassbnrisprincipal(bnr,x,{flag=1}): bnr being output by bnrinit, gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. If (optional) flag is set to 0, output only vcharpolychinese(x,{y}): x,y being both intmods (or polmods) computes z in the same residue classes as x and ycontfracpnqn(x): [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the continued fraction xelladdGGD2,L,pelllocalred(e,p): e being an elliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the Kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_pfactorintidealprimedecissquarelistput(list,x,{n}): sets n-th element of list equal to x. If n is omitted or greater than the current list length, just append xmathnfmatrank(x): rank of the matrix xmin(x,y): minimum of x and ynfeltpowmodpr(nf,a,k,pr): element a^k modulo pr in nf, where pr is in modpr format (see nfmodprinit)numbpart(x): number of partitions of xpolredord(x): reduction of the polynomial x, staying in the same orderprimepiprodqfperfectionrnfidealdown(rnf,x): finds the intersection of the ideal x with the base fieldrnfisnorminitrnfpolredabsshiftmulsubstvec(x,v,w): in expression x, make a best effort to replace the variables v1,...,vn by the expression w1,...,wnzetakinitzncoppersmith
Checking LLL basis
Increasing dim, delta = %d t = %d
yMonomorphismLift: true early solution.
f=%Z
 borne=%Z
 l-borne=%Z
S4GaloisConj:Testing %d/3:%d/2:%d/2:%d/4:%Z:%Z
tab[q^-s] from 1 to N-1cannot give a power series result in k/n bessel functionnon-positive valuation in etax = 0 in Fourierqrom3: iteration %ld: %Z
prod: remaining objects %ld
forbidden substitution in a scalar typezero argument in O()main variable must have higher priority in gtosergreal/gimagincompatible field degrees in conjvec[2]: smithall, i = %ldvaluation (valp) overflowzero polynomialbad argument for an elliptic curve related function   datadir = "%s"
comment>    %s = 1 (on)
primitive root mod %Z does not existnegative argument in factorial functionclassnonfsqffarch/
Subfields of degree %ld: %Z
embedding = %Z
installvrrD"",r,D"",s,vV=GGIDGDGpplotboxvLGGbitwise negationzero polynomial in FpXQ_pow. %Z not primeQpXQ_to_ZXYexact type in subgrouplistfor this typeFpM_gauss. i=%ldFlxqM_kerResult for prime %Z is:
%Z
not a factorisation in nfbasisZY_ZXY_resultant_all: LERS needs lambda for one roundifac_startIFAC: main loop: repeated new factor
	%Z
	%Z
ifac_crack [Z_issquarerem miss]ifac_sort_onecomposite equals prime in ifac_sort_oneGaloisIndex: Using hash value u=%ld
 chk_gen_init: skipfirst = %ld
not a set in setminusa transcendental function%Z should divide valuation (= %ld) in sqrtnser_powStep 3
beta LLL-reduced mod U^l = %Z
old method: y = %Z, M0 = %Z
Searching minimum of T2-form on units:
k not a positive even integer in elleisnumcut-off point must be positive in lseriesellinitell for 2-adic numbersCM_ellpow
MPQS: passing the %3.1f%% sort point
 and combining\\ KERNEL COMPUTED BY MPQS
KERNEL=shorternot a correct ideal list in %sget_archUnrecognized id '%s' in a stringified flagclgpzkstthis function uses a killed variablebuchcertifybuchrayinitgendecodefactorhilbpiideallistzstargenlindep2(x,dec)=Z-linear dependencies between components of x using LLL, where dec should be about one half the number of decimal digits of precisionlogagmmatrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xminimreduceddisc(f)=vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial frootsof1(nf)=number of roots of unity and primitive root of unity in the number field nfvecindexsortname, v = 99Math::Pari::STOREMath::Pari::interface199$;$Math::Pari::interface87Math::Pari::_2boolMath::Pari::DESTROYbernzone%3ld:%3ld ]

 / closenew bloc, size %6lu (no %ld): %08lx
-->Polrev(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termVecsmallVecsmall({x=[]}): transforms the object x into a VECSMALL. Empty vector if x is omittedacosh(x): inverse hyperbolic cosine of xbnfclassunit(P,{flag=0},{tech=[]}): compute the class group, regulator of the number field defined by the polynomial P, and also the fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units. See manual for details about tech. P may also be a non-zero integer, and is then considered as the discriminant of a quadratic orderlGDGDGV=GGEDGellpointtoz(e,P): lattice point z corresponding to the point P on the elliptic curve eellsigma(om,z,{flag=0}): om=[om1,om2], value of the Weierstrass sigma function of the lattice generated by om at z if flag = 0 (default). If flag = 1, arbitrary determination of the logarithm of sigma. If flag = 2 or 3, same but using the product expansion instead of theta serieselltaniyama(e): modular parametrization of elliptic curve efactornffibonacci(x): fibonacci number of index x (x C-integer)hilbertidealaddtoone(nf,x,{y}): if y is omitted, when the sum of the ideals in the number field K defined by nf and given in the vector x is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1. Otherwise, x and y are ideals, and if they sum up to 1, find one element in each of them such that the sum is 1idealred(nf,x,{vdir=0}): LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealvalintmellininvnfisideal(nf,x): true(1) if x is an ideal in the number field nf, false(0) if notprecprimeprinttex(a): outputs a in TeX formatqfbnucompqflllgramqflllgram(x,{flag=0}): LLL reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix). flag is optional and can be 0: default,1: lllgramint algorithm for integer matrices, 4: lllgramkerim giving the kernel and the LLL reduced image, 5: lllgramkerimgen same when the matrix has polynomial coefficients, 8: lllgramgen, same as qflllgram when the coefficients are polynomialsrnfdet(nf,order): given a pseudomatrix, compute its pseudodeterminantrnfeltreltoabs(rnf,x): transforms the element x from relative to absolute representationrnfidealabstorel(rnf,x): transforms the ideal x from absolute to relative representationrnfkummerserlaplace(x): replaces the power series sum of a_n*x^n/n! by sum of a_n*x^n. For the reverse operation, use serconvol(x,exp(X))lGGD0,L,znstar(jid=%ld,jdir=%ld)incompatible variables in gredMonomorphismLift: false early solution.
GaloisConj:Entree Init Test
S4GaloisConj:pj=%Z
Combinatorics too hard : would need %Z tests!
 I'll skip it but you will get a partial result...q >= 1 in thetazero argument in a k/n bessel functioncan't factor %Zsrgcd: dr = %ld
invalid quadratic form in qfevalgtoserdiscriminant not congruent to 0,1 mod 4 in %sdifferent discriminants in qfb_compnot a t_REAL in 4th component of a t_QFRhnf_special    matb cleaned up (using Id block)
hnfspec [%ld x %ld] --> [%ld x %ld]non coprime ideals in hnfmergeCompute %sstark (computation impossible)(use old functions, don't ignore case)parisizenoneGPHELPreal_relations#### Tentative regulator: %Z
gnuplot-like plotting environment not loaded yetCan't resolve 'get_term_ftable' function from Gnuplot drawing engine '%s': %sGroup(())bad index in listinsertplotrbox(w,dx,dy): if the cursor is at (x1,y1), draw a box with diagonal (x1,y1)-(x1+dx,y1+dy) in rectwindow w (cursor does not move)FpX_factor_2subcycloshallowtranshnf_invimageIndexPartial: factor %Z^%ld Trying lambda = %ld
ifac_sumdivmulti-curves cannot be plot recursivelyS1galois (bug3)galoisapplyget_bnfpolmult. tableget_red_Gchk_gen_init: generator %Z
Galoisbig: reduced polynomial #1 = %Z

*** Entering isin_%ld_G_H_(%ld,%ld)
gpsi0 argument in cotan[rnfkummer] candidate list[1]: discrayabslistarchneither bnf nor bnr in conductor or discrayXcan't evaluate ellzeta at a pole%lu is not prime, use ellakMPQS: Ran out of primes for A, giving up.
MPQS: chose Q_%ld(x) = %Z x^2 %c %Z x + C
MPQS: chose primes for Ainvalid polynomial in thue (need n>2)B0 -> %Z
not a matrix in nfsmithStep4: q-values (# = %ld, largest = %ld): [install] updating '%s' prototype; module not reloadedbreak not allowed after !break not allowed here (expanding string)global variable not allowedadj(x)=adjoint matrix of xatan(x)=inverse tangent of xatanh(x)=inverse hyperbolic tangent of xbasisbasis2(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 2 algorithmbin(x,y)=binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xcfcvtoi(x)=truncation of x, without taking into account loss of integer part precisiondetint(x)=some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with hermitemodgaloisconjforce(nf)=list of conjugates of a root of the polynomial x=nf[1] in the Galois number field nf, using p-adics, LLL on integral basis. Guaranteed to be complete if the field is Galois, otherwise there is an infinite loopideallistunitgenisinclfast(nf1,nf2)=tests whether the number nf1 is isomorphic to a subfield of nf2 or not. If it gives a non-zero result, this proves that this is the case. However if it gives zero, nf1 may still be isomorphic to a subfield of nf2 so you have to use the much slower isincl to be sureisirreducible(x)=true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantisoncurve(e,x)=true(1) if x is on elliptic curve e, false(0) if notisset(x)=true(1) if x is a set (row vector with strictly increasing entries), false(0) if notlexsortmat(x)=transforms any GEN x into a matrixnupowpowell(e,x,n)=n times the point x on elliptic curve e (n in Z)rnfequation2(nf,pol)=given a pol with coefficients in nf, gives [apol,th], where apol is the absolute equation of the number field defined by pol and th expresses the root of nf[1] in terms of the root of apolrnfisfree(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notsortthreetotwo2(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]g, eltgadd_gge_gmodMath::Pari::sv2pariMath::Pari::int_set_term_ftableCorrupted data: should be variable Occupation of the PARI stack : %6.2f percent
%016lx  Strtex({str}*): translates its (string) arguments to TeX format and returns the resulting stringacosbestapprbnfissunitbnfsunit(bnf,S): compute the fundamental S-units of the number field bnf output by bnfinit, S being a list of prime ideals. res[1] contains the S-units, res[5] the S-classgroup. See manual for detailsbnrclass(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this module. flag is optional, and can be 0: default, 1: compute data necessary for working in the ray class group, for example with functions such as bnrisprincipal or bnrdisc, without computing the generators of the ray class group, or 2: with the generators. When flag=1 or 2, the fifth component is the ray class group structure obtained when flag=0bnrdisclistconcatconcat(x,{y}): concatenation of x and y, which can be scalars, vectors or matrices, or lists (in this last case, both x and y have to be lists). If y is omitted, x has to be a list or row vector and its elements are concatenatedGGGpellheightvGVIgaloisfixedfield(gal,perm,{flag},{v=y}): gal being a galois field as output by galoisinit and perm an element of gal.group or a vector of such elements, return [P,x] such that P is a polynomial defining the fixed field of gal[1] by the subgroup generated by perm, and x is a root of P in gal expressed as a polmod in gal.pol. If flag is 1 return only P. If flag is 2 return [P,x,F] where F is the factorization of gal.pol over the field defined by P, where the variable v stands for a root of Pgaloisisabelian(gal,{flag=0}): gal being as output by galoisinit, return 0 if gal is not abelian, the HNF matrix of gal over gal.gen if flag=0, 1 if flag is 1, and the SNF of gal is flag=2idealmul(nf,x,y,{flag=0}): product of the two ideals x and y in the number field nf. If (optional) flag is non-nul, reduce the resultidealtwoelt(nf,x,{a}): two-element representation of an ideal x in the number field nf. If (optional) a is non-zero, first element will be equal to alGGGincgamc(s,x): complementary incomplete gamma functionVGGED0,L,D0,L,pmatdetmatisdiagonal(x): true(1) if x is a diagonal matrix, false(0) otherwisematmultodiagonal(x,y): product of matrices x and y, knowing that the result will be a diagonal matrix. Much faster than general multiplication in that casematpascalmatrixmatsolve(M,B): gaussian solution of MX=B (M matrix, B column vector)nffactornffactormod(nf,pol,pr): factorize polynomial pol modulo prime ideal pr in number field nfpolinterpolatepolylog(m,x,{flag=0}): m-th polylogarithm of x. flag is optional, and can be 0: default, 1: D_m~-modified m-th polylog of x, 2: D_m-modified m-th polylog of x, 3: P_m-modified m-th polylog of xqfbclassnoqfbhclassno(x): Hurwitz-Kronecker class number of x>0qfsignquadpolytan(x): tangent of xtaylor(x,y): taylor expansion of x with respect to the main variable of yteichmuller(x): teichmuller character of p-adic number xvecextract(x,y,{z}): extraction of the components of the matrix or vector x according to y and z. If z is omitted, y designs columns, otherwise y corresponds to rows and z to columns. y and z can be vectors (of indices), strings (indicating ranges as in "1..10") or masks (integers whose binary representation indicates the indices to extract, from left to right 1, 2, 4, 8, etc.)writezetak(nfz,s,{flag=0}): Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by zetakinit (NOT by nfinit) flag is optional, and can be 0: default, compute zetak, or non-zero: compute the lambdak function, i.e. with the gamma factorslllintfirst LLL: prec = %ld
minim0, rank>=%ldzero ideal in isprincipalelement is not in the base field in rnfelementdownGaloisConj:increase prec of p-adic roots of %ld.
GaloisConj:Testing S4 first
GaloisConj:Orbite:%Z
GaloisConj:Fini!
A4GaloisConj:%ld hop sur %d iterations max
Combinatorics too hard : would need %Z tests!
I will skip it, but it may induce an infinite loopFlx_to_FlvzpsolublenfRgX_extgcd, dr = %ldnot a squarefree polynomial in sturmnot a variable in substvecincorrect object in O()negative definite t_QFIjacobimore rows than columns in matrixqzhnflll	character no: %ld (%ld/%ld)
get_sep: argument too long (< %ld chars)arguments must be positive integers%c%ld.%ldechoPARIbreak\hskip 0pt plus \hsize\relax\discretionary{}{}{}}   path = "%s"
for this exponent, GSmin = %Z
Time reduction: %ld
-perl -MDynaLoader -we %cpackage DynaLoader; print mod2fname([qw(Term Gnuplot)]) if defined &mod2fname%coverflow in calc_blockgabsgaffect (gen_0)gaffect (pol_1/pol_x)plotrmove(w,dx,dy): move cursor to position (dx,dy) relative to the present position in the rectwindow wpolfnfFqX_factorSubcyclo: %ld orbits with %ld elements each
eigendet. col = %ldget_normrnfordmaxsnextpr: %lu should have been prime but isn't
ellfacteursIFAC: trying Pollard-Brent rho method
polynomial not in Z[X] in %sS3A_4(6) = [2^2]3negative index in vecsortglngammagpow: 0 to a forbidden powergexppolred(compositum) = %Z
incorrect subgroup in %sincompatible p-adic numbers in initellnorm too large in CMlocalredMPQS: factoring number of %ld decimal digits
MPQS: size of factor base = %ld
MPQS: found %ld full relations
MPQS:            %4.3g full relations per 100 polynomials
manyMPQS: FB [-1,2  c15 = %Z
not a correct ideal list in nfsmithnfhermitemodStep6: testing potential divisors
Non-numeric argument of an action in a templatecycdiscexpr%s[1-%ld]char2cyclo(n)=n-th cyclotomic polynomialdilog(x)=dilogarithm of xGD0,G,D0,G,D0,L,factcantor(x,p)=factorization mod p of the polynomial x using Cantor-Zassenhausfactoredbasisfactorpadic2galoisconj1(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field nf, using complex numbers, LLL on integral basis (not always complete)hclassno(x)=Hurwitz-Kronecker class number of x>0hilbert(n)=Hilbert matrix of order n (n C-integer)initalgred2(P)=P being a nonconstant irreducible polynomial, gives a two-element vector [nf,mod(a,pol)], where nf is as output by initalgred and mod(a,pol) is a polymod equal to mod(x,P) and pol=nf[1]initzeta(x)=compute number field information necessary to use zetak, where x is an irreducible polynomialisisomfast(nf1,nf2)=tests whether the number fields nf1 and nf2 are isomorphic or not. If it gives a non-zero result, this proves that they are isomorphic. However if it gives zero, nf1 and nf2 may still be isomorphic so you have to use the much slower isisom to be surekbessel2lllkerim(x)=kernel and lll reduction of the vectors forming the integral matrix xlllkerimgen(x)=kernel and lll reduction of the vectors forming the matrix x with polynomial coefficientsmathellmatrix(m,n,X,Y,expr)=mXn matrix of expression expr, the row variable X going  from 1 to m and the column variable Y going from 1 to nmatsize(x)=number of rows and columns of the vector/matrix x as a 2-vectormu(x)=Moebius function of xnfdivnfmodnumdiv(x)=number of divisors of xpadicprec(x,p)=absolute p-adic precision of object xpointell(e,z)=coordinates of point on the curve e corresponding to the complex number zpolredabsnored(x)=a smallest generating polynomial of the number field for the T2 norm on the roots without initial polredpolylogdpolylogdoldsumpos2(X=a,expr)=sum of positive series expr, the formal variable X starting at a, using Zagier's polynomialsunit(x)=fundamental unit of the quadratic field of discriminant x where x must be positiveratilft: amax must be >= 0, found
	amax=%Z
arg1, arg2=0_gcmp0_absMath::Pari::pari2num_;@Math::Pari::interface11Math::Pari::interface209Math::Pari::interface33Math::Pari::pari_version_expMath::Pari::__detach_stackXSUB call through interface with a NULL code%c[%ld;%ld;%ldmt_QFRNULLList([mat(deletedoubling stack size; new stack = %lu (%.3f Mbytes)I=I(): square root of -1VecvD0,L,bernfrac(x): Bernoulli number B_x, as a rational numberbesselnbestappr(x,k): gives the best approximation to the real x with denominator less or equal to kbezout(x,y): gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dbnfclgpbnfregbnrclassno(bnf,x): ray class number of the module x for the big number field bnf. Faster than bnrclass if only the ray class number is wantedGDGDGDGcontentcontfrac(x,{b},{lmax}): continued fraction expansion of x (x rational,real or rational function). b and lmax are both optional, where b is the vector of numerators of the continued fraction, and lmax is a bound for the number of terms in the continued fraction expansiondefault({opt},{v}): returns the current value of the current default opt. If v is present, set opt to v first. If no argument is given, print a list of all defaults as well as their values.divisorseint1(x,{n}): exponential integral E1(x). If n is present, computes the vector of the first n values of the exponential integral E1(n.x) (x > 0)elleisnumellheight(e,x,{flag=2}): canonical height of point x on elliptic curve E defined by the vector e. flag is optional and selects the algorithm used to compute the archimedean local height. Its meaning is 0: use theta-functions, 1: use Tate's method, 2: use Mestre's AGMellordinate(e,x): y-coordinates corresponding to x-ordinate x on elliptic curve eellztopoint(e,z): coordinates of point P on the curve e corresponding to the complex number zidealnorm(nf,x): norm of the ideal x in the number field nfimag(x): imaginary part of xintmellininvshort(sig,x,tab): numerical integration on the  line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). Compulsory table tab has been  precomputed using the command intfuncinit(t=[[-1],sig[2]],[[1],sig[2]],s)  (with possibly its two optional additional parameters), where sig[2] = 1 if not given. Orders of magnitude faster than intmellininv.intnumintnuminit(a,b,{m=0}): initialize tables for integrations from a to b. See help for intnum for coding of a and b. Possible types: compact interval, semi-compact (one extremity at + or - infinity) or R, and very slowly, slowly or exponentially decreasing, or sine or cosine oscillating at infinities,listcreate(n): creates an empty list of maximum length nmatdiagonalmatmuldiagonalnextprimenfhilbert(nf,a,b,{p}): if p is omitted, global Hilbert symbol (a,b) in nf, that is 1 if X^2-aY^2-bZ^2 has a non-trivial solution (X,Y,Z) in nf, -1 otherwise. Otherwise compute the local symbol modulo the prime ideal pnfisinclnumeratorpolrootsqfbpowraw(x,n): n-th power without reduction of the binary quadratic form xrnfidealabstorelsetsearch(x,y,{flag=0}): looks if y belongs to the set x. If flag is 0 or omitted, returns 0 if it is not, otherwise returns the index j such that y==x[j]. If flag is non-zero, return 0 if y belongs to x, otherwise the index j where it should be insertedsqrtintGnPthetavaluation(x,p): valuation of x with respect to pzetalllfp giving upincrementalGSgenbase change matrix =
red_mod_units
#### Tentative class number: %Z
++ LV[%ld] = %Z  nb. fact./nb. small norm = %ld/%ld = %.3f
main variable must be of higher priority in rnfinitalgFixedField: Weight: %Z
gzetagpolylogz0 argument in nfhilbertpplease apply rnfisnorminit firstnot a positive exponent in polhenselliftnot a factorisation in factorbackgdivmodgsubstinvalid base change matrix in qf_base_changesquare discriminant in %sd must be positive    after phase2:
hnflll (reducing), i = %ldhnfall[3], j = %ldTrying to find another modulus...diff(CHI) = %Z* Root Number: cond. no %ld/%ld (%ld chars)
unknown function or error in formal parametersvariable name expectedobsolete functionconstant polynomial   realprecision = %ld significant digits(use old functions, ignore case)   %s = 0 (off)
fundunitinsufficient precision for p = 2 in hilbertlist of numerators too short in sfcontf2 %ldPp = %Z, lift to p^%ld
cvtop2normalizegsigneplotdrawfactmodSubCyclo: testing %ld^%ld
Subcyclo: complex=%ld
    mu'= det, col %ld / %ldimpossible concatenation: %s %Z . %s %ZFqM_gauss_pivotmodpr initialized for integers only!PL: N-1 factored!
Rho: time = %6ld ms,	%3ld rounds, back to normal mode
IFAC: main loop: repeated old factor
	%Z
IFAC: after main loop: repeated old factor
	%Z
not a vector in rectdrawgaloisfalse nf in nf_get_r1
initzeta: N0 = %Z
galois in degree > 11not a series in laplaceglogincorrect character length in KerCharanell for n >= %lunot a complex multiplication in powelltorsell (bug2) 1 1    B0 -> %Z
looking for a fundamental unit of norm -1
aprcl: e(t) too smallnumeric id in a stringified flagareac4; or ] expectedO(a^b)=o(a^b)=p-adic or power series zero with precision given by bbuchnarrow(bnf)=given a big number field as output by buchinitxx, gives as a 3-component vector the structure of the narrow class groupGnconductor(bnr,subgroup)=conductor of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdegree(x)=degree of the polynomial or rational function x. -1 if equal 0, 0 if non-zero scalardiscrayabslistarchdiscrayabslistarch(bnf,arch,bound)=gives list of discrayabscond of all modules up to norm bound with archimedean places arch, in a longvector formaterfc(x)=complementary error functionimage(x)=basis of the image of the matrix xisidealkbessel2(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)krolll1lllgenomega(x)=number of unrepeated prime divisors of xpascalpolredabsallprodinf(X=a,expr)=infinite product (X goes from a to infinity) of real or complex expressionquadgen(x)=standard generator of quadratic order of discriminant xseriessupplement(x)=supplement the columns of the matrix x to an invertible matrixveclexsortzidealstarinitzsigneratlift: must have 0 <= x < m, found
	x=%Z
	m=%Z
Math::PariAttempt to ask Perl to free PARI function not installed from Perlgcmp_1_gbitand_geqMath::Pari::sv2parimatMath::Pari::interface299$$$$$;$$Math::Pari::reset_on_reloadlGGvV=GGvLGGG\cr
 I/O: can't remove file %s%s is set (%s), but is not writeable(%c,lgefint=%ld): %s .%ld.%ldGP_DATA_DIRtypeerinvmoderList({x=[]}): transforms the vector or list x into a list. Empty list if x is omittedGGGDGpargbesseljhGLGD-1,L,ellchangepointvs*exp(x): exponential of xfactor(x,{lim}): factorization of x. lim is optional and can be set whenever x is of (possibly recursive) rational type. If lim is set return partial factorization, using primes up to lim (up to primelimit if lim=0)galoisfixedfieldidealinvideallistarch(nf,list,arch): list is a vector of vectors of of bid's as output by ideallist. Return a vector of vectors with the same number of components as the original list. The leaves give information about moduli whose finite part is as in original list, in the same order, and archimedean part is now arch. The information contained is of the same kind as was present in the input.matimagecompl(x): vector of column indices not corresponding to the indices given by the function matimagematrankmatsizenfdetintnfhnf(nf,x): if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfisisom(x,y): as nfisincl but tests whether x is isomorphic to ynfsnf(nf,x): if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xGGGLpolisirreducible(x): true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantprintp1(a): outputs a (in beautified format) without ending with newlineqfbprimeform(x,p): returns the prime form of discriminant x, whose first coefficient is pqfbred(x,{flag=0},{D},{isqrtD},{sqrtD}): reduction of the binary quadratic form x. All other args. are optional. D, isqrtD and sqrtD, if present, supply the values of the discriminant, floor(sqrt(D)) and sqrt(D) respectively. If D<0, its value is not used and all references to Shanks's distance hereafter are meaningless. flag can be any of 0: default, uses Shanks's distance function d; 1: use d, do a single reduction step; 2: do not use d; 3: do not use d, single reduction step.quadregulator(x): regulator of the real quadratic field of discriminant xrealreturnrnfidealmul(rnf,x,y): relative version of idealmul, where rnf is a relative numberfieldsetrandGnGtrapvecmaxvecsortlllint[1], kmax = %ldcount_max = %ld
lllfp[1], kmax = %ldzero polynomial forbidden*sorting...
KCZ = %ld, KC = %ld, n = %ld
Galois group almost certainly not weakly super solvableGaloisConj: Bound %Z
%d%% Entering veceint1:
szetaargument must belong to upper half-planeinfinities of the same sign in intnuminitgennormalizing a polynomial with 0 leading termmain variable must be of higher priority in rnfisnorminitmissing case in gdivexact\a log/atan appears in intformalgrndtoiinvalid vector in qfbevalzero discriminant in Qfbnot a t_QFI in nucompIt's not a square...
RecCoeff (cf = %ld, B = %Z)
not a definite matrix in lllgram(no backward compatibility)   prompt%s = "%s"
large exponent in Mod(a,N)^n: reduce n mod phi(N)bestappr0classno2sfcont2smith/class groupfilename of Term::Gnuplot does not end in `.pm': `%s'comparisongaffect (gpi)rootsoldsubcyclo_rootsnb subgroup = %ld
trying to concat elements of an empty vectorTreating p^k = %Z^%ld
this combination of flags in rnfpolredabsnewtonsums: result doesn't fit in cache
non invertible polynomial in FpXQ_invsnextpr: integer wraparound after prime %lu
flag PLOT_PARAMETRIC ignoredpolynomial not in Z[X,Y] in %sfalse nf in nf_get_r2matrix Mtschirnnot vectors in polinterpolatemain variable in kummer must not be xbug%d in kummerLbnrclassno  z1 = %Z
akellreduction mod SL2 (reduce_z)MPQS: found %3.1f%% of the required relations
,%lu...] Wait a second --
MPQS: computing logarithm approximations for p_i in FB
principalidealnot a factorization in idealapprfactcodifftoo many parameters in user-defined function calladdell(e,z1,z2)=sum of the points z1 and z2 on elliptic curve ebasistoalgbuchgenforcefucf(x)=continued fraction expansion of x (x rational,real or rational function)char2(x,y)=characteristic polynomial of the matrix x expressed with variable y, using the Hessenberg form. Can be much faster or much slower than char, depending on the base ringcompo(x,s)=the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]content(x)=gcd of all the components of x, when this makes senseconvol(x,y)=convolution (or Hadamard product) of two power seriesdivsumfactoredpolred2(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives elements and minimal polynomials)hermitepermhermiteperm(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Batut's algorithmincgam4(s,x,y)=incomplete gamma function where y=gamma(s) is precomputedV=GGID0,L,pisprincipalgenkerint1powellpowrealraw(x,n)=n-th power without reduction of the binary quadratic form x of positive discriminantprec(x,n)=change the precision of x to be n (n C-integer)random()=random integer between 0 and 2^31-1reverseset(x)=convert x into a set, i.e. a row vector with strictly increasing coefficientsshift(x,n)=shift x left n bits if n>=0, right -n bits if n<0shiftmul(x,n)=multiply x by 2^n (n>=0 or n<0)trunc(x)=truncation of x;when x is a power series,take away the O(X^)vvector(n,X,expr)=column vector with n components of expression expr (X ranges from 1 to n)zidealstarinitgenCan't install Perl function with prototype `%s'gequalggegmulMath::Pari::interface3GGGGDGDGD---- (type RETURN to continue) ----O(}{st(%c,expo=%ld):Mod(write failedno such error number: %ld %s: %s
,errpilegdiverGGD0,L,Strexpand({str}*): concatenates its (string) argument into a single string, performing tilde expansionalgdepbernrealGGD0,L,pcomponentellap(e,p,{flag=0}): computes a_p for the elliptic curve e using Shanks-Mestre's method. flag is optional and can be set to 0 (default) or 1 (use Jacobi symbols)ellsearch(N): if N is an integer, it is taken as a conductor else if N is a string, it can be a curve name ("11a1"), a isogeny class ("11a") or a conductor ("11"). Return all curves in the elldata database that match the  property.fordivgaloissubfieldsgetheapgetheap(): 2-component vector giving the current number of objects in the heap and the space they occupyidealdiv(nf,x,y,{flag=0}): quotient x/y of two ideals x and y in HNF in the number field nf. If (optional) flag is non-null, the quotient is supposed to be an integral ideal (slightly faster)idealfactoridealhnfidealstar(nf,I,{flag=1}): gives the structure of (Z_K/I)^*. flag is optional, and can be 0: simply gives the structure as a 3-component vector v such that v[1] is the order (i.e. eulerphi(I)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generators. If flag=1 (default), gives idealstarinit, i.e. a 6-component vector [I,v,fa,f2,U,V] where v is as above without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*. Finally if flag=2, same as with flag=1 except that the generators are also givenintnumrombintnumsteplngammaLDGnext({n=1}): interrupt execution of current instruction sequence, and start another iteration from the n-th innermost enclosing loopspoltchebiqfbnucomp(x,y,l): composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedqfbnupowreal(x): real part of xrnfidealhnf(rnf,x): relative version of idealhnf, where rnf is a relative numberfieldsigma(x,{k=1}): sum of the k-th powers of the divisors of x. k is optional and if omitted is assumed to be equal to 1sumnumaltk = negative bound in zncoppersmithexpected shvector bitsize: %ld
inconsistent primes in plindepnegative polynomial degree in algdeptime for ct = %ld : %ld
*%ld %ld 
#### Looking for random relations
non-monic polynomial in galoisconj4galoisconj _may_ hang up for this polynomialtwistpartialzeta (1), j = %ld/%ldbad argument for modular functioninfinities of different type in intnuminitgenuseless flag in rnfisnorm: the extension is Galoisroots must be bracketed in solvenot a polynomial in polhenselliftsubresexta log appears in intformalzero discriminant in %scornacchiahess, m = %ldEntering hnffinal:
Warning:accuracy problemsinteger too largelogfiledefault: incorrect value for %s [0:off / 1:on]associationnegative integer in sqrtint
rel = Terminal does not define put_textCan't load Gnuplot drawing engine from '%s': %sThis runtime link with gnuplot-shim does not implement midlevel start/end functionschanging f(x): p divides disc(g)
candidate = %Z
empty vector in vecmaxplothsizesLGD0,L,Berlekamp_kerapprpadicFqX_split_Trager: choosing k = %ld
Matrix of wrong dimensions in galoissubcycloroots_to_polinfinite group in forsubgroup  countsub = %ld
gaddmatnegative size in fill_scalcolgaussmodulonot a pseudo-matrix in rnfdetcompositum  entering Dedekind Basis with parameters p=%Z
  leaving Decomp: f1 = %Z
f2 = %Z
e = %Z
de= %Z
not a pseudo-matrix in %sFFInit: using subcyclo(%ld, %ld)
squfof [caller of] (n or 3n is a square)	But it nevertheless wasn't a %ld%s power.
	checking modulo %ld
factoring 0 in ifac_startIFAC: new partial factorization structure (%ld slots)
IFAC: main loop: another factor was divisible by
	%Z
IFAC: cracking composite
	%Z
          a(n)partitions( %ld ) is meaninglessincorrect precision in transcgpow: 0 to a non positive exponentnot a factorisation in decodemodule
MPQS: split N whilst combining, time = %ld ms
MPQS: Gauss elimination will require more than
	128MBy of memoryMPQS (relation is a nonsquare)MPQS panickingnon-monic polynomial in thuex2 -> %Z
* Checking for small solutions
not a suitable vector of elements in idealchinesebreak not allowedcan't derive thisunknown function '%s', expected '=' instead ofacos(x)=inverse cosine of xbezout(x,y)=gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,pchar1(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using Lagrange interpolationchinese(x,y)=x,y being integers modulo mx and my,finds z such that z is congruent to x mod mx and y mod mychptellcompocomprealrawdiscrayabscond(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if fmodule is not the conductordiscrayabslist(bnf,listes)=if listes is a 2-component vector as output by ideallistunit or similar, gives list of corresponding discrayabscondgetrand()=current value of random number seedincgam3(s,x)=complementary incomplete gamma functionisprincipalraygenkaramul(x,y,k)=THIS FUNCTION HAS BEEN SUPPRESSEDlabelmin(x,y)=minimum of x and ynfreduce(nf,a,id)=gives r such that a-r is the ideal id and r is smallGGGD&V=GID1,L,pregula(x)=regulator of the real quadratic field of discriminant xrnfbasis(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setsmallfactsubcyclo(p,d)=finds an equation for the d-th degree subfield of Q(zeta_p), where p must be a prime powervec(x)=transforms the object x into a vector. Used mainly if x is a polynomial or a power seriesvecmin(x)=minimum of the elements of the vector/matrix xzideallognegative power in element_powid_mod_pnot a matrix in matbasistoalgSomething very wrong  in %s, %iarg0, arg00, arg1=0, arg2=0, arg3=0gdivround_cosname, valnewsize = 0Math::Pari::pari2nvMath::Pari::interface1Math::Pari::interface16[;]
I/O: removed file %s
^%ldunknown user duplicate indeterminates in reordertalkerO(a^b): p-adic or power series zero with precision given by bStrtexacos(x): inverse cosine of xcotandirdivellwpexpGLDnforvec(x=v,seq,{flag=0}): v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1 if flag is zero or omitted. If flag = 1 (resp. flag = 2), restrict to increasing (resp. strictly increasing) sequencesgaloisexport(gal,{flag}): gal being a galois field as output by galoisinit, output a string representing the underlying permutation group in GAP notation (default) or Magma notation (flag = 1)GGD0,L,Dngaloisidentify(gal): gal being a galois field as output by galoisinit, output the isomorphism class of the underlying abstract group as a two-components vector [o,i], where o is the group order, and i is the group index in the GAP4 small group libraryidealhnf(nf,a,{b}): hermite normal form of the ideal a in the number field nf, whatever form a may have. If called as idealhnf(nf,a,b), the ideal is given as aZ_K+bZ_K in the number field K defined by nfidealmulidealredidealtwoeltV=GGEDGpintlaplaceinv(X=sig,x,s,{tab}): numerical integration on the line real(z) = sig of s(z)exp(xz)dz/(2*I*Pi), i.e. inverse Laplace transform of s at x. tab is as in intnum.intnum(X=a,b,s,{tab}): numerical integration of s from a to b with  respect to X. a (and similarly b) is coded as follows. It can be a scalar: f is assumed to be C^infty at a. It can be a two component vector [a[1],a[2]], where a[1] is the scalar, and a[2] is the singularity exponent (in ]-1,0]), logs being neglected. It can be a one component vector [1] or [-1] meaning +infty or -infty, slowly decreasing functions. It can be a two component vector [[1], z] or [[-1], z], where [1] or [-1] indicates +infty or -infty and z is coded as follows. If z is zero, slowly decreasing. If z is real positive, exponentially decreasing, of the type exp(-zX). If z<-1, very slowly decreasing like X^(-z). If z is complex nonreal, real part is ignored and if z = r+I*s then if s>0, cosine oscillation exactly cos(sX), while if s<0, sine oscillation exactly sin(sX). If f is exponentially decreasing times oscillating function, you have a choice, but it is in general better to choose the oscillating part. Finally tab is either 0 (let the program choose  the integration step), a positive integer m (choose integration step 1/2^m), or a table tab precomputed with intnuminit (depending on the type of interval: compact, semi-compact or R, very slow, slow, exponential, or cosine or sine-oscillating decrease).V=GGED0,L,pmatmultodiagonalmatsupplement(x): supplement the columns of the matrix x to an invertible matrixnfkermodprnfrootspolgaloispolrootspadicLLprinttexqfjacobiqflll(x,{flag=0}): LLL reduction of the vectors forming the matrix x (gives the unimodular transformation matrix). The columns of x must be linearly independent, unless specified otherwise below. flag is optional,  and can be 0: default, 1: assumes x is integral, columns may be dependent, 2: assumes x is integral, returns a partially reduced basis, 4: assumes x is  integral, returns [K,I] where K is the integer kernel of x and I the LLL reduced image, 5: same as 4 but x may have polynomial coefficients, 8: same as 0 but x may have polynomial coefficientsremoveprimes({x=[]}): remove primes in the vector x (with at most 100 components) from the prime table. x can also be a single integer. List the current extra primes if x is omittedrnfidealtwoeltrnfidealtwoelt(rnf,x): relative version of idealtwoelt, where rnf is a relative numberfieldrnflllgram(nf,pol,order): given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixwritetexfloating point exception: bug in PARI or calling programnot a definite form in minim0precision too low for generators, not given
#### Computing class group generators
cleanarchGaloisConj:val1=%ld val2=%ld
GaloisConj:Testing A4 first
testpermutation(%Z)negative index in polylogBeauzamy bound: %Z
incorrect permtuation to inversevaluation not equal to 1 in serreverseredimagsl2QuickNormL1
hnfall, final phase: Leaving hnffinal
new precision: %ld
(warn when using obsolete functions)(TeX output)psfileisanypower: now k=%ld, x=%Z
1/0 exponent in Fp_sqrtnincorrect bound type in bestapprhclassnopanic: more than %d tokens for optionsNo terminal specifiedCannot reset output routines to copy term list to a variable.pm
%s

couldn't open dynamic library '%s'plotrline(w,dx,dy): if the cursor is at (x1,y1), draw a line from (x1,y1) to (x1+dx,y1+dy) (and move the cursor) in the rectwindow wvLsD0,L,IFAC: Stop: Primary factor: %Z
Berlekamp matrix   %3ld factor of degree %3ld
too many iterations in rootsspec_FqXQ_pow(lifted) subgp of prime to %Z part:
subgroup: index bound must be positiveElliptic curves files not available for conductor %ld
[missing %s]detnegative size in fill_scalmat--> %Z : reduced absolute generator: %Z
  dedek: gcd has degree %ld
initial parameters p=%Z,
  f=%Z
bug in Decomp (not a factor), is p = %Z a prime?Rg_to_Fpbound 2^%ld. Goal 2^%ld	(extracted precomputed helix / baby step entries)
SQUFOF: blacklisting a = %ld on first cycle
OddPwrs: examining %Z
ifac_decompifac_omegaIFAC: found %Z =
	%Z ^%ld
factoringfactor has NULL exponent in ifac_findNot a group in group_identThis is not a valid colorD(6) = S(3)[x]2galois (bug2)not an integer type in dirzetaklog(n)%s/galdata/NAM%ldnth-root does not exist in gsqrtnreducebeta  Testing primes <= B (= %lu)

wrong type in too_big[apell1] sorting
MPQS: found factor = %ld whilst creating factor base
MPQS: bound for `large primes' = %ld
MPQS: Net yield: %4.3g full relations per 100 candidates
MPQS: found %ld factors =
MPQS: kN = %Z
MPQS: precomputing auxiliary primes up to %ld
MPQS: new bit pattern for primes for A: 0x%lX
ideal_two_eltnon-integral exponent in idealpownot a matrix in %snot a matrix of maximal rank in nfsmithnot a vector in idealredusing obsolete function %sray torsion unitsbernvec(x)=Vector of rational Bernoulli numbers B_0, B_2,... up to B_(2x)binbuchreal(D,...)=compute the structure of the class group and the regulator of the real quadratic field of discriminant D>0 in the wide sense. See manual for the other parameters (which can be omitted)coeffcoredisc(n)=discriminant of the quadratic field Q(sqrt(n))discrayrel(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdivreseigen(x)=eigenvectors of the matrix x given as columns of a matrixextract(x,y)=extraction of the components of the vector x according to the vector or mask y, from left to right (1, 2, 4, 8, ...for the first, second, third, fourth,...component)fact(x)=factorial of x (x C-integer), the result being given as a real numbergaloisconjforcehvector(n,X,expr)=row vector with n components of expression expr, the variable X ranging from 1 to nincgam1indexrankisincl(x,y)=tests whether the number field defined by the polynomial x is isomorphic to a subfield of the one defined by y; 0 if not, otherwise all the isomorphismsorderellpascal(n)=pascal triangle of order n (n C-integer)perf(a)=rank of matrix of xx~ for x minimal vectors of a gram matrix apolylogp(m,x)=P_m-modified m-th polylog of xpsi(x)=psi-function at xredrealnod(x,sq)=reduction of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]rootsold(x)=roots of the polynomial x using a modified Newton's methodsigmak(k,x)=sum of the k-th powers of the divisors of x (k C-integer)smallbasissubst(x,y,z)=in expression x, replace the variable y by the expression zincorrect archimedean component in Idealstarnot an element in zidealloggsubUnsupported interface %d for "direct-link" Pari function %s_flexible_longCannot load a Pari macro `%s': macros are unsupported; VALENCE=%#04x, code=<%s>, isFunction=%d, EpVAR=%dheap had %ld bytes (%ld items)
digits=0Math::Pari::EXISTSMath::Pari::pari2pvMath::Pari::interface_flexible_intMath::Pari::interface13Math::Pari::dumpHeapMath::Pari::PARI_DEBUG_setToo many args for a flexible-interface functionCannot process default argument %.*s of type %.1s for prototype '%s't_INTt_QUADmalformed binary file (no name)gen_0
chars:TMPDIR%s: Time ListStrarg(x): argument of x,such that -pi<arg(x)<=piatanh(x): inverse hyperbolic tangent of xLpbigomega(x): number of prime divisors of x, counted with multiplicitybnfunitcharpoly(A,{v=x},{flag=0}): det(v*Id-A)=characteristic polynomial of the matrix or polmod A. flag is optional and may be set to 1 (use Lagrange interpolation) or 2 (use Hessenberg form), 0 being the defaultconjvecellchangecurve(x,y): change data on elliptic curve according to y=[u,r,s,t]GD0,L,pellj(x): elliptic j invariant of xellsub(e,z1,z2): difference of the points z1 and z2 on elliptic curve eevalfloor(x): floor of x = largest integer<=xforellintfouriercosintfourierexpintlaplaceinvnfalgtobasis(nf,x): transforms the algebraic number x into a column vector on the integral basis nf.zknfbasis(x,{flag=0},{p}): integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithm. Second and third args are optional. Binary digits of flag mean 1: assume that no square of a prime>primelimit divides the discriminant of x, 2: use round 2 algorithm instead. If present, p provides the matrix of a partial factorization of the discriminant of x, useful if one wants only an order maximal at certain primes onlynfrootsof1(nf): number of roots of unity and primitive root of unity in the number field nfpermtonumLDnpolisirreduciblepollead(x,{v}): leading coefficient of polynomial or series x, or x itself if x is a scalar. Error otherwise. With respect to the main variable of x if v is omitted, with respect to the variable v otherwisepollegendre(n,{v=x}): legendre polynomial of degree n (n C-integer), in variable vqfbsolvequadgen(x): standard generator of quadratic order of discriminant xrnfidealdownrnfidealnormrel(rnf,x): relative norm of the ideal xrnfisnormsizebytesqrsuminftanvectorsmall(n,{X},{expr=0}): VECSMALL with n components of expression expr (X ranges from 1 to n) which must be small integers. By default, fill with 0svectorv(n,{X},{expr=0}): column vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0sznorder(x,{o}): order of the integermod x in (Z/nZ)*. Optional o is assumed to be a multiple of the order.minim0SPLIT: increasing factor base [%ld]
makematalclassgroup generators(%ld)for this relation
#### Computing regulator multiple
Be honest for %ld primes from %ld to %ld
GaloisConj:%d hop sur %Z iterations
negative or zero constant in veceint1sumnuminit0non integral index in suminfMignotte bound: %Z
iscomplexvariable must have higher priority in gtopolymatrix of non-maximal rank in matrixqzmit = %Z
B = %Z
C = %Z
unknown identifier valence, please reportdebug   [logfile was "%s"]
(bits 0x2/0x4 control output of \left/\PARIbreak)factor_add_primesnot an element of (Z/nZ)* in order%ld^%ld cglob = %ld. Bach constant <= 0 in buchquadget_lambda
Time %s rel [#rel/#test = %ld/%ld]: %ld

* Look for subfields of degree %ld

Y = %Z
prime %Z is not in primetabletoo many iterations in hqrSubCyclo: new conductor:%ld
extractempty matrix in deplinallbasenon-invertible polynomial in FpXQX_gcd for up to %ld roundsSQUFOF: second cycle exhausted after %ld iterations,
	dropping it
 5th%sfactoring 0 in ifac_decompgaloisindex for groups of order >127gtodblListchecknftschirnhausi0 = %ld
ODD	# rational integer roots = %ld:kummer for composite relative degreepolrel(beta) = %Z
powell for non integral, non CM, exponentsMPQS: striving for %ld relations
MPQS: smallest prime used for sieving FB[%ld] = %ld
MPQS: factoring this number will take %s hours:
N = %Zcannot rename file %s to %s  c10 = %Z
a6indexr1not an integerbreak not allowed in test expressionglobal variable: ray unitsboundcfbuchcertify(bnf)=certify the correctness (i.e. remove the GRH) of the bnf data output by buchinit or buchinitfucore2(n)=(long)gen_2-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecos(x)=cosine of xdeplin(x)=finds a linear dependence between the columns of the matrix xfibo(x)=fibonacci number of index x (x C-integer)forprime(X=a,b,seq)=the sequence is evaluated, X running over the primes between a and bgaloisapply(nf,aut,x)=Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfgaloisconjgetheap()=2-component vector giving the current number of objects in the heap and the space they occupyhermitehilb(x,y,p)=Hilbert symbol at p of x,y (integers or fractions)idealapprfact(nf,x)=x being a prime ideal factorization with possibly zero or negative exponents, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealmulred(nf,x,y)=reduced product of the two ideals x and y in the number field nfidealpow(nf,x,n)=n-th power of the ideal x in HNF in the number field nfintgenjellnfdiveuc(nf,a,b)=gives algebraic integer q such that a-bq is smallnupow(x,n)=n-th power of primitive positive definite quadratic form x using nucomp and nuduplpolredabs(x)=a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 normprime(n)=returns the n-th prime (n C-integer)principalidele(nf,x)=returns the principal idele generated by the algebraic number x in the number field nfrank(x)=rank of the matrix xrayclassnoresultant(x,y)=resultant of the polynomials x and y with exact entriesresultant2(x,y)=resultant of the polynomials x and yrootmodsmithpolsolve(X=a,b,expr)=real root of expression expr (X between a and b), where expr(a)*expr(b)<=0until(a,seq)=evaluate the expression sequence seq until a is nonzeroveclexsort(x): sort the elements of the vector x in ascending lexicographic orderelement_sqr_gcmprvx, nMath::Pari::FETCHMath::Pari::loadPariMath::Pari::setprecisionMath::Pari::longwordAssignment of a columns into a matrix of incompatible heightGDVDIV=GEpV=GGEDgp_readvec_stream: found %ld entries
t_INTMODinvalid range in print_functions_hash.:~:~/gpYou never gave me anything to read!outputnd%ld%s component = qfr( ( +changevaruncaught error: %ldasinh(x): inverse hyperbolic sine of xbnfdecodemodule(nf,fa): given a coded module fa as in bnrdisclist, gives the true moduleellgeneratorsellheightmatrixellordinateellzeta(om,z): om=[om1,om2], value of the Weierstrass zeta function of the lattice generated by om at zfactorial(x): factorial of x (x C-integer), the result being given as a real numberfactorint(x,{flag=0}): factor the integer x. flag is optional, whose binary digits mean 1: avoid MPQS, 2: avoid first-stage ECM (may fall back on it later), 4: avoid Pollard-Brent Rho and Shanks SQUFOF, 8: skip final ECM (huge composites will be declared prime)frac(x): fractional part of x = x-floor(x)lGGDGvSlistkill(list): kills listmatdetint(x): some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with mathnfmodmatsolvematsolvemod(M,D,B,{flag=0}): one solution of system of congruences MX=B mod D (M matrix, B and D column vectors). If (optional) flag is non-null return all solutionsnfeltmul(nf,a,b): element a. b in nfpolredordprimeprintp(a): outputs a (in beautified format) ending with newlineprod(X=a,b,expr,{x=1}): x times the product (X runs from a to b) of expressionsolvethue(tnf,a,{sol}): solve the equation P(x,y)=a, where tnf was created with thueinit(P), and sol, if present, contains the solutions of Norm(x)=a modulo units in the number field defined by P. If tnf was computed without assuming GRH (flag 1 in thueinit), the result is unconditionaltype(x): return the type of the GEN x.lllintpartialvector in new basis = %Z
not an algebraic number in isunit# ideals tried = %ld
weighted G matrices  small norms gave %ld relations.
field not Galois or Galois group not weakly super solvableGaloisAnalysis:p=%ld l=%ld group=%ld deg=%ld ord=%ld
GaloisConj:Sortie Init Test
A4GaloisConj:O=%Z 
step equal to zero in forstepneed positive degree in gdeflatelifting to prec %ld
### K = %d, %Z combinations
invalid data in qf_base_changenon existent component in truecoeffhnflll, k = %ld / %ldhnfall
i = %ld: conductor too large in ArtinNumbernot a polynomialinfinite precision (%ld digits displayed)   format = %c%ld.%ld
nodarkbg0(off)PARIpromptENDbroken prettyprinter: '%s'not a prime in Fp_sqrt
*** Bach constant: %f
class number = %ldnf_factor_boundname "%s" for terminal too longTerminal does not define resetperl -MConfig -wle %cuse Term::Gnuplot;print $INC{qq(Term/Gnuplot.pm)};print $Config{dlext}%cgaloissubgroup: not a WSS groupabs is not meromorphic at 0plotcursor(w): current position of cursor in rectwindow wploth(X=a,b,expr,{flags=0},{n=0}): plot of expression expr, X goes from a to b in high resolution. Both flags and n are optional. Binary digits of flags mean: 1=Parametric, 2=Recursive, 4=no_Rescale, 8=no_X_axis, 16=no_Y_axis, 32=no_Frame, 64=no_Lines (do not join points), 128=Points_too (plot both lines and points), 256=Splines (use cubic splines), 512=no_X_ticks, 1024= no_Y_ticks, 2048=Same_ticks (plot all ticks with the same length). n specifies number of reference points on the graph (0=use default value). Returns a vector for the bounding boxplothrawplotlinetypevLGGGGlserror in rootsold(): using roots2()lx<ly in Flx_addmul_inplacefactmod: %lu is not prime* Finding eigenvalues
SubCyclo: %ld not found
Please do not try to break PARI with ridiculous counterfeit data. Thanks!    lambda'= nf_to_ffnot the same variable in compositum new order:
%Z
%Z
Rg_to_FpXQpows [P,Q]non positive degree in ffinit	(giant step at p = %lu)
Rho: using X^2%+1ld for up to %ld rounds of 32 iterations
IFAC: cofactor = %Z
prime equals composite in ifac_sort_onecheckbnfplease apply bnrinit(,,1) and not bnrinit(,)incompatible modulus in %s:
  mod = %Z,
  nf  = %ZC(5) = 5nf_ADDZK flag when nf_ALL set (polredabs)need %Z coefficients in initzeta: computation impossiblenot a zeta number field in zetakallget_polargument must be positive in polcycloprecision<=0 in gprecrfix (conversion to t_REAL)odd exponent in p-adic sqrtsecond arg must be integer in gsqrtnBuchrayMinkowski bound is too large[ %ld, %ld, %ld ]: %Z
  *** testing p = %lu
    {%ld @ %s : %s}
  * {%ld @ %s : %s}
MPQS: X^2 - Y^2 != 0 mod N
MPQS: wrong relation found after Gaussvariable number too bigfutumodseqsymbol already in usebasis2buchimagcenterlift(x)=centered lift of x. Same as lift except for integermodscharGVIGLLforvec(x=v,seq)=v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1hell(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using theta-functionshermite(x)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, using a naive algorithmidealnorm(nf,x)=norm of the ideal x in the number field nfidealtwoelt2image2(x)=basis of the image of the matrix xincgam3minim(x,bound,maxnum)=number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0polylogpprod(x,X=a,b,expr)=x times the product (X runs from a to b) of expressionquadpoly(x)=quadratic polynomial corresponding to the discriminant xreverse(x)=reversion of the power series xsumalt2sumpos2trans(x)=x~=transpose of xveczideallog(nf,x,bid)=if bid is a big ideal as given by zidealstarinit or zidealstarinitgen , gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)not an element of (Z/pZ)* in znlogIdealstar needs an integral non-zero ideal: %Zratlift failed to catch d1 == 0
Import of Perl function with too many argumentssvarg1, arg2=0, arg3=gen_0Math::Pari::interface34Math::Pari::interface73Math::Pari::pari2boolCalling Perl via PARI with an unknown interface: avoiding loop Used :                         %ld  long words  (%ld K)
 Available :                    %ld  long words  (%ld K)
]
w}
}%s is a GP binary file. Please use writebin

%.8s%spopping %s (bloc no %ld)
 in %s; new prec = %ld
siginterModStrexpandabsbesselh1GDGpbnfisprincipaldenominator(x): denominator of x (or lowest common denominator in case of an array)ellisoncurve(e,x): true(1) if x is on elliptic curve e, false(0) if notellrootno(e,{p=1}): root number for the L-function of the elliptic curve e. p can be 1 (default), global root number, or a prime p (including 0) for the local root number at pellsigmaellzetaGDGD0,L,Dnidealaddidealfactor(nf,x): factorization of the ideal x given in HNF into prime ideals in the number field nfmatisdiagonalnfdiscnorm(x): norm of xnumdiv(x): number of divisors of xnumerator(x): numerator of xrnfbasisrnfdisc(nf,pol): given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfpseudobasissetunionshiftmul(x,n): multiply x by 2^n (n>=0 or n<0)GD1,L,substpolsubstpol(x,y,z): in expression x, replace the polynomial y by the expression z, using remainder decomposition of x.sumpostruncatevariable(x): main variable of object x. Gives p for p-adic x, error for scalarsk =dimension 0 in fincke_pohstadding vector = %Z
New bound: %ZBach constant <= 0 in buchplease apply galoisinit firstGaloisConj:denominator:%Z
GaloisConj: G[%d]=%Z of relative order %d
A4GaloisConj:orb=%Z 
Galoisconj:Subgroups list:%Z
qrom2: iteration %ld: %Z
negative number of columns in matrixnon integral index in sumaltKnapsackmissing nf in factorbackeltsrgcdnumerevaluation of a power seriesShanks distance must be a t_REAL in qfrreducible form in qfr_initsmithallaccuracy lost in matfrobeniusmatgen = %Z
can't deflate[%ld,%ld,%ld]readline   prettyprinter = "%s"
formatKC = %ld, need %ld relations
qrf5_rho_pow@2 * M = %Z
>not an integer modulus in cvtop[secure mode]: about to install '%s'. OK ? (^C if not)
vLD0,G,D0,G,D0,L,no such component in vecextractEntering gauss with inexact=%ld
gauss_pivot. k=%ld, n=%ldrnf functionfflgenPolynomials not irreducible in FpX_ffintersect	Pollard-Brent failed.
SQUFOF: giving up, time = %ld ms
 7th	stored (largest) factor no. %ld...
Classification of transitive groups of order > 30 is not known%ld %ld l
(%sincorrect bigray fieldC(4) = 4D_6(6) = [3]2gchnon-positive integer in glngammanot a p-adic argument in teichmullergcosStep 2
Step 13
not a rational curve in ellintegralmodelellinit data not accurate enough. Increase precision == {%s}
	index i = %ld
MPQS: got two factors, looking for more...
unknownAll solutions are <= %Z
idealhermitezero ideal in idealfactorquotient not integral in idealdivexactnot a module in %sa stringified flag does not start with an ida4unknown member functionroots (conjugates)bestappr(x,k)=gives the best approximation to the real x with denominator less or equal to kboundfact(x,lim)=partial factorization of the integer x (using primes up to lim)buchinitfubuchrayinitgen(bnf,ideal)=given a big number field as output by buchinitfu (only) and  an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data for computing in the ray class group  corresponding to this module. In particular, the fifth component is the ray class group structureceil(x)=ceiling of x=smallest integer>=xchangevar(x,y)=change variables of x according to the vector yconvoldiscrayrelfactoredpolredidealapprfactjacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xminideal(nf,ix,vdir)=minimum of the ideal ix in the direction vdir in the number field nfpf(x,p)=returns the prime form whose first coefficient is p, of discriminant xprimedec(nf,p)=prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a.Z_K, e,f as usual, a as vector of components on the  integral basis, b Lenstra's constantresultant2rhorealnod(x,sq)=single reduction step of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]simplefactmodsizesmallfact(x)=partial factorization of the integer x (using only the stored primes)not the same number field in algtobasisffshankstreating pr^%ld, pr = %Z
PARI: %s_flexible_void# Math::Pari::_to_intGGGD0,L,pV=GGEGGDVDVD[+++][;]matrix(0,%ld,j,k,0)\*lbot>ltop in gerepileDGMat({x=[]}): transforms any GEN x into a matrix. Empty matrix if x is omittedbezoutresbitnegbittestcontfracGDGDGvV=GDGIidealdivGGGD0,L,intcirc(X=a,R,s,{tab}): numerical integration of s on the circle  |z-a|=R, divided by 2*I*Pi. tab is as in intnum.lcm(x,{y}): least common multiple of x and y, i.e. x*y / gcd(x,y)GGLlistsort(list,{flag=0}): sort list in place. If flag is non-zero, suppress all but one occurence of each element in listlogmatkerintnextprime(x): smallest pseudoprime >= xnfeltdivrem(nf,a,b): gives [q,r] such that r=a-bq is smallnfroots({nf},pol): roots of polynomial pol belonging to nf (Q if omitted) without multiplicitypadicappr(x,a): p-adic roots of the polynomial x congruent to a mod pprimes(n): returns the vector of the first n primes (n C-integer)prodeulerrnfbasistoalg(rnf,x): relative version of nfbasistoalg, where rnf is a relative numberfieldsetissetweberGGGDGnpass = %ld, red. last time = %ld, log_2(det) ~ %ld

delta = %d, t = %d, cond = %lf
PHASE 1: check primes to Zimmert bound = %lu

*** p = %lu
fundamental units too largegaloisanalysis()jbesselhFourier transform of oscillating functionsincorrect beginning value in sumnumfactor for general polynomialsspecial_pivot output:
%Z
inversenonexistent componentqfr_unit_by_discEntering hnfspec
non integral matrix in smithallsmithcleancpl = 2^%ld
what's going on ?significant terms(bits 0x2/0x4 control matched-insert/arg-complete)lineshistsize   help = "%s"
form_to_idealUnsupported function map3d_xy calledTerminal does not define pointTR_POL(-1), i = %ld/%ldgaffect (gen_2)gaffect (gi)normalizepolplotpointtype(w,type): change the type of following points in rectwindow w. type -1 corresponds to a dot, larger values may correspond to something else. w=-1 changes highlevel plottingplotrboxIFAC: Stop: remaining %Z
 %ld incorrect length in sumhnfdividemissing eigenspace. Compute the matrix to higher accuracy, then restart eigen at the current precisiongauss    pass no %ld
resultant mod %ld (bound 2^%ld, stable = %d)QXQ_inv	(got [2]Q...[10]Q)
, oravoiding nonexistent factors in ifac_whoiswhorectdrawL(6):2 = PGL(2,5) = S_5(6)Found %ld minimal polynomials.
too many terms in dirzetaks = 1 is a pole (gzetakall)
    Output of isin_%ld_G_H(%ld,%ld): %ldindex too large in vecsortgpow: underflow or overflowp-adic argument out of range in gexpisvirtualunitcan't evaluate log(ellsigma) at lattice pointMPQS: %4.1f%% of the polynomials yielded no candidates
several FB[%ld]=%ld%s%s @ %s :%s
[1]: mpqs_solve_linear_systemnot a prime ideal factorization in idealchineseid too long in a stringified flaga1a2unknown parser codeuser function %s: variable %Z declared twicebuchgen(P,...)=compute the structure of the class group and the regulator for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)V=GGIDGdiscf(x)=discriminant of the number field defined by the polynomial x using round 4euler=euler()=euler's constant with current precisiongamh(x)=gamma of x+1/2 (x integer)gaussmodulo(M,D,Y)=(long)gen_1 solution of system of congruences MX=Y mod Didealappr(nf,x)=x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealpowredinitalginitzetainverseimage(x,y)=an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwiseisdiagonal(x)=true(1) if x is a diagonal matrix, false(0) otherwiseisqrt(x)=integer square root of x (x integer)issqfree(x)=true(1) if x is squarefree, false(0) if notlllrat(x)=lll reduction of the vectors forming the matrix x, computations done with rational numbers (gives the unimodular transformation matrix)matnfmul(nf,a,b)=element a.b in nfLGpprimes(n)=returns the vector of the first n primes (n C-integer)reorder(x)=reorder the variables for output according to the vector xresultantrnfsteinitz(nf,order)=given an order as output by rnfpseudobasis, gives [A,I,..] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialsetunion(x,y)=union of the sets x and ysort(x)=sort in ascending order of the vector xtaniyama(e)=modular parametrization of elliptic curve evecindexsort(x): indirect sorting of the vector xcv, name, numargs = 1, help = NULLExpected VOID return type, got code '%s'arg1arg1, arg2arg1, arg2=0, arg3=0_sinMath::Pari::FETCHSIZEMath::Pari::set_gnutermMath::Pari::setprimelimitMath::Pari::added_sectionsUnexpected syntax of default argument '%s' in prototype '%s'GGD0,L %ld objects on heap occupy %ld long words

II/O: new pariFILE %s (code %d) 
  I : 0.GPTMPDIRentering recover(), loc = %ld
acoshaliasbesseljh(n,x): J-bessel function of index n+1/2 and argument x, where n is a non-negative integerlGbnfclgp(P,{tech=[]}): compute the class group of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnfisunitdivisors(x): gives a vector formed by the divisors of x in increasing orderdivremellconvertnameellinitGGDGpellorder(e,p): order of the point p on the elliptic curve e over Q, 0 if non-torsionellsearchfactorcantorfactorcantor(x,p): factorization mod p of the polynomial x using Cantor-Zassenhausgaloisinitgaloissubcyclogammah(x): gamma of x+1/2 (x integer)idealprincipal(nf,x): returns the principal ideal generated by the algebraic number x in the number field nfintfuncinitintmellininvshortlistinsert(list,x,n): insert x at index n in list, shifting the remaining elements to the rightmathnfmodid(x,d): (upper triangular) Hermite normal form of x concatenated with d times the identity matrixmatid(n): identity matrix of order n (n C-integer)minmoebius(x): Moebius function of xnfhilbertlGGGDGnfhnfmod(nf,x,detx): if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jqfminimrnfalgtobasisrnfdedekindrnfeltreltoabsrnfeltupround(x,{&e}): take the nearest integer to all the coefficients of x. If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitssetintersectsetsearchsumnumtanhvecminvectorv K%ldalgdep0maximal number of vectors must be providedfinal sort & check...
insufficient precision for fundamental unitsprecision too low for generators, e = %ldcompleting bnf (building matal)
 ***** check = %f
GaloisConj:Back to Earth:%Z
S_2   bound: %Z^%ld
RgX_gcd_simplenfgcd: p=%d
MultiLift: bad argsgvarnot the same prime in padicprecnonexistent component in truecoeffinvalid vector in hqfevalnot a rational matrix in matrixqznon invertible matrix in gausssecure? primitive root mod 2^%ld does not existquadray needs a fundamental discriminantinitialf = %Z
pol. found = %Z
Group(plothsizes({flag=0}): returns array of 6 elements: terminal width and height, sizes for ticks in horizontal and vertical directions, width and height of characters. If flag=0, sizes of ticks and characters are in pixels, otherwise are fractions of the screen sizeplotkill(w): erase the rectwindow wplotmovepsplothnegative exponent in bitwise negationbitwise andbitwise xorfactorpadic2 for non-monic polynomialznstar_cosetsnot an integer matrix in detintgauss_pivotoriginal absolute generator: %Z
 treating %Z
powers is only [] or [1] in FpX_FpXQV_compoRho: restarting for remaining rounds...
 orifac_totientifac_sumdivkSQUFOF: found factor %ld from ambiguous form
	after %ld steps on the ambiguous cycle, time = %ld ms
rectpoints%.5g2A_4(6) = [2^3]3 = 2 wr 3S_4(6c) = 1/2[2^3]S(3)F_36(6):2 = [S(3)^2]2 = S(3) wr 2galois (bug1)polredabs0galois files %s not compatible
RES %dnot a series in convolno data in polinterpolategathproduct from 0 to N-1pol = %Z
Mahler bound for regulator: %Z
bnrdiscrayellpointtoz: %s square root
prime too large in apell2, use apellnot an integral model in akellpowell for nonintegral CM exponenttorsell (bug3)MPQS: sizing out of tune, FB too small or
	way too few primes in A :generic conversion to finite fieldJacobi sums and tables computed
b4elocal(); FFTanell(e,n)=computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<32768)buchgenbuchgenforcefu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units for the number field defined by the polynomial P, and insist until the units are obtained. See manual for the other parameters (which can be omitted)cf2compositum2(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2, with roots of pol1 and pol2 expressed on the compositum polynomialscyclofibogamhideallllred(nf,x,vdir)=LLL reduction of the ideal x in the number field nf along direction vdir, in HNFimag(x)=imaginary part of xker(x)=basis of the kernel of the matrix xkro(x,y)=kronecker symbol (x/y)lllgramintnfdiv(nf,a,b)=element a/b in nfnfpow(nf,a,k)=element a^k in nfpolredabs2GV=GGIrnfdiscfrnfequation2rnfhermitebasissmallpolred2(x)=partial reduction of the polynomial x (gives elements and minimal polynomials)torsell(e)=torsion subgroup of elliptic curve e: order, structure, generatorsvvectorwhile(a,seq)=while a is nonzero evaluate the expression sequence seq. Otherwise 0zidealstarinitgen(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*Pohlig-Hellman: DL mod %Z^%ld
Expected int return type, got code '%s'_sqrt_flexible_gen%s %2ld: %s
Math::Pari::interface21Math::Pari::interface2199Math::Pari::interface24Math::Pari::interface29_oldMath::Pari::interface84;$Math::Pari::lgefintvV=GID0,L,LV=GGIpD0,L,D0,L,%c[%ld;%ldmt_POL+ binary outputI can't remember before the big bang(precp=%ld,valp=%ld):Vecsmall([  ***   Error in the PARI system. End of program.
Set({x=[]}): convert x into a set, i.e. a row vector with strictly increasing coefficients. Empty set if x is omittedaddprimesatan(x): inverse tangent of xbnrconductor(a1,{a2},{a3},{flag=0}): conductor f of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc). flag is optional and can be 0: default, 1: returns [f, Cl_f, H], H subgroup of the ray class group modulo f defining the extension, 2: returns [f, bnr(f), H]bnrisconductorbnrstarkbreak({n=1}): interrupt execution of current instruction sequence, and exit from the n innermost enclosing loopscenterlift(x,{v}): centered lift of x. Same as lift except for integermodsellglobalred(e): e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_petagaloisinit(pol,{den}): pol being a polynomial or a number field as output by nfinit defining a Galois extension of Q, compute the Galois group and all neccessary informations for computing fixed fields. den is optional and has the same meaning as in nfgaloisconj(,4)(see manual)galoissubgroupsidealintersect(nf,x,y): intersection of two ideals x and y in the number field defined by nfidealstarmatmuldiagonal(x,d): product of matrix x by diagonal matrix whose diagonal coefficients are those of the vector d, equivalent but faster than x*matdiagonal(d)numtopermLGpoldiscreducedrnfhnfbasis(bnf,order): given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfisnorminit(pol,polrel,{flag=2}): let K be defined by a root of pol, L/K the extension defined by polrel. Compute technical data needed by rnfisnorm to solve norm equations Nx = a, for x in L, and a in K. If flag=0, do not care whether L/K is Galois or not; if flag = 1, assume L/K is Galois; if flag = 2, determine whether L/K is Galoisserconvol(x,y): convolution (or Hadamard product) of two power seriessqrtGGD&puntil(a,seq): evaluate the expression sequence seq until a is nonzerobus error: bug in PARI or calling programRoots: %Z
LIMC = %ld, LIMC2 = %ld
buchall (%s)*** Changing sub factor base
p too small in fixedfieldsympolFixedField: Found: %Z
FixedField: Sym: %Z
GaloisAnalysis:non Galois for p=%ld
MGaloisConj: Fixed field %Z
not an integer index in jbesselhhyperu's third argument must be positivenstop = %ld
zeta of power seriespolylog around a!=0forparifactpolsubresext, dr = %ldsubst: unexpected variable precedencedifferent number of variables and values in substvecincorrect argument in O()not a rational polynomial in conjvechnf_special[1]. i=%ldstarting SNF loopgsmithallFindModulusclass field not totally real in bnrstark(external prettyprint)primelimit[secure mode]: can't modify '%s' default (to %s)p = 1 in hilbert()discriminant too big in classnobe honest for primes from %ld to %ld
Hensel lift... mod p^k (avma - bot = %lu)
vecsmall_copyforbidden divisor %Z in ggvalgaffect (gen_m1)plotinit(w,{x=0},{y=0},{flag=0}): initialize rectwindow w to size x,y. If flag!=0, x and y express fractions of the size of the current output device. x=0 or y=0 means use the full size of the deviceplotscaledenominators not allowed in divisorspolfnf: choosing k = %ld
No such elliptic curve in databasenon-monic relative polynomialsZY_ZXY_rnfequationRho: searching small factor of %ld-bit integer
SQUFOF: first cycle exhausted after %ld iterations,
	dropping it
OddPwrs: passed modular checks
IFAC: prime %Z
	appears at least to the power %ld
incorrect rectwindow number in graphic function (%ld not in [0, %ld])stroke
A6gzetakallEVENtoo large precision in preci()non zero exponent in gsincosStep 6
r1>15 in discrayabslistarchStarting zidealstarunits computations
       column #%ld of the matrix log(b_j/Q): %Z
factordivexact is not exact!MPQS: next sort point at %3.1f%%
 %lu %luMPQS: combining
  c11 = %Z
  Entering CF...
LLL failed. Increasing kappa
a3t2incorrect type in %sthis code has to come firstFFTinitapell(e,p)=computes a_p for the elliptic curve e using Shanks-Mestre's methodarg(x)=argument of x,such that -pi<arg(x)<=pibytesizeclassno2(x)=class number of discriminant xdecodemoduledenom(x)=denominator of x (or lowest common denominator in case of an array)discrayabsdiscrayrelcond(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if module is not the conductorfactorpadic2(x,p,r)=p-adic factorization of the polynomial x to precision r, using Buchmann-Lenstrahermite2hilbp(x,y)=Hilbert symbol of x,y (where x or y is integermod or p-adic)hyperu(a,b,x)=U-confluent hypergeometric functionideallistunitgen(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, with generatorsidealpowred(nf,x,n)=reduced n-th power of the ideal x in HNF in the number field nfif(a,seq1,seq2)=if a is nonzero, seq1 is evaluated, otherwise seq2isfundlegendre(n)=legendre polynomial of degree n (n C-integer)lllint(x)=lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)matrixqz(x,p)=transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwisenfhermite(nf,x)=if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnorml2(x)=square of the L2-norm of the vector xnucomp(x,y,l)=composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedpfpolint(xa,ya,x)=polynomial interpolation at x according to data vectors xa, yapolredabsfast(x)=a smallest generating polynomial of the number field for the T2 norm on the rootspolsym(x,n)=vector of symmetric powers of the roots of x up to nprimroot(n)=returns a primitive root of n when it existsrayclassno(bnf,x)=ray class number of the module x for the big number field bnf. Faster than buchray if only the ray class number is wantedrnfpolred(nf,pol)=given a pol with coefficients in nf, finds a list of polynomials defining some subfields, hopefully simplersmallbuchinittheta(q,z)=Jacobi sine theta-functionelement_divelement_powNot a vector where column of a matrix expectedExpected long return type, got code '%s'gor_gandMath::Pari::lgt_QFIout of range in integer -> character conversion (%ld)                  
additionmultiplicationStr({str}*): concatenates its (string) argument into a single stringaddprimes({x=[]}): add primes in the vector x to the prime table to be used in trial division. x may also be a single integer. Composite "primes" are allowed, and in that case you may later get a message "impossible inverse", which will give you some factors. List the current extra primes if x is omitted. If some primes are added which intersect non trivially the existing table entries, suitable updating is doneGGpagm(x,y): arithmetic-geometric mean of x and yalias("new","old"): new is now an alias for oldasinhbnfissunit(bnf,sfu,x): bnf being output by bnfinit (with flag<=2), sfu by bnfsunit, gives the column vector of exponents of x on the fundamental S-units and the roots of unity if x is a unit, the empty vector otherwiseD1,L,ceilD"",r,D"",s,D0,L,ellchangecurveelljeulerphi(x): Euler's totient function of xGGLD0,L,idealcoprimeidealpow(nf,x,n,{flag=0}): n-th power of the ideal x in HNF in the number field nf If (optional) flag is non-null, reduce the resultimagincgamintfouriersin(X=a,b,x,s,{tab}): numerical integration from a to b of sin(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the sine-Fourier transform if a=-infty and b=+infty.kronecker(x,y): kronecker symbol (x/y)listcreatenfeltreducenfkermodpr(nf,x,pr): kernel of the matrix x in Z_K/pr, where pr is in modpr format (see nfmodprinit)nfmodprinitnfsubfieldsGGDnD0,L,poltchebi(n,{v=x}): Tchebitcheff polynomial of degree n (n C-integer), in variable vqfbcompraw(x,y): Gaussian composition without reduction of the binary quadratic forms x and yrnfeltabstorel(rnf,x): transforms the element x from absolute to relative representationrnfequation(nf,pol,{flag=0}): given a pol with coefficients in nf, gives the absolute equation apol of the number field defined by pol. flag is optional, and can be 0: default, or non-zero, gives [apol,th], where th expresses the root of nf.pol in terms of the root of apolrnfidealreltoabsrnfidealupserlaplacesetrand(n): reset the seed of the random number generator to nteichmullerwrite(filename,a): write the string expression a (same output as print) to filenamewritetex(filename,a): write the string expression a (same format as print) to filename, in TeX formatzeta(s): Riemann zeta function at s with s a complex or a p-adic numbernot enough memory, new stack %luuser interruptpslq     is %Z
non-monic polynomial. Change of variables discardedcodeprime
++++ cglob = %ld: new relation (need %ld)truncation error in bestappr
+Not a Galois field in a Galois related functionMonomorphismLift: trying early solution %Z
GaloisConj:Frobenius:%Z
S4GaloisConj:Computing isomorphisms %d:%Z
incorrect permutation in permtopoltwistpartialzeta (2), j = %ld/%ldqpsolublepolratliftrelatively prime polynomials expectedLLL_cmbf: rank decrease
Time setup: %ld
invalid data in hqfevalnot a t_QFI in powimag    2nd phase done
[3]: smithalleasycharpolrelnum = %Z
not enough precomputed primes, need primelimit ~ [%ld,,%ld]   %s = "%s"
prompt_cont\ifx\%s\undefined
  \def\%s{%s}\fi
not a polynomial of degree 2 in quadrayp = %lu, q = %lu, e = %ld
  3) Final bound: %Z
Terminal does not define linetypeTR_POL(1), i = %ld/%ld2 * Hadamard bound * ind = %Z
lg(Z) = %ld, lg(Y) = %ld
gaffect (gen_1)inverse mod %ld (stable=%ld)empty matrix in supplcompositum: %Z inseparablereducible polynomial in allbase content in fastnu is %Z
Rho: hang on a second, we got something here...
	found %sfactor = %Z
ifac_moebiusIFAC: factoring %Z
	factor no. %ld is a duplicate%s
incorrect number of components in rectdrawF_21(7) = 7:3L(7) = L(3,2)reverse polmod does not existgacosgpow: need integer exponent if series valuation != 0[rnfkummer] compositumnaive reduction mod U^l: unit exp. = %Z
conductorincorrect hash code in decodemoduleStarting discrayabs computations
Default bound for regulator: 0.2
     p divides w(K)
FNEW
MPQS: starting Gauss over F_2 on %ld relations
MPQS: wrapping, more primes for A now chosen near FB[%ld] = %ld
get_embCF failed. Increasing kappa
incorrect idele in idealaddtooneChoosing t = %ld
Individual Fermat powerings:
b2zknot enough flags in string function signatureadjbuchgenfucoredisc2(n)=(long)gen_2-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integerdivsum(n,X,expr)=sum of expression expr, X running over the divisors of neta(x)=eta function without the q^(1/24)factoreddiscf(x,p)=discriminant of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactornf(x,t)=factorization of the polynomial x over the number field defined by the polynomial thermitehavas(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Havas's algorithmidealcoprime(nf,x,y)=gives an element b in nf such that b.x is an integral ideal coprime to the integral ideal yideallistunitisprincipalforce(bnf,x)=same as isprincipal, except that the precision is doubled until the result is obtainedissquare(x)=true(1) if x is a square, false(0) if notjell(x)=elliptic j invariant of xkerintlog(x)=ln(x)=natural logarithm of xnfhermitemod(nf,x,detx)=if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfpowoordellpermutation2num(vect)=ordinal (between 1 and n!) of permutation vectpolredabs2(x)=gives [pol,a] where pol is as in polredabs, and alpha is the element whose characteristic polynomial is polreal(x)=real part of xrnfhermitebasis(bnf,order)=given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisesmalldiscf(x)=discriminant of the number field defined by the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallpolredsum(x,X=a,b,expr)=x plus the sum (X goes from a to b) of expression exprtchebizetak(nfz,s)=Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)element_mulelement_invmodideal  treating a = %ld, b = %ld
gpowaMath::Pari::pari2numMath::Pari::pari_printMath::Pari::typMath::Pari::lgefMath::Pari::initmemToo few args %d for PARI function %sGDGDGD0,L,pvLGr  p : pari.psarcherpaddhelp(symbol,"message"): add/change help message for a symbolbigomegabnrrootnumber(bnr,chi,{flag=0}); returns the so-called Artin Root Number, i.e. the constant W appearing in the functional equation of the Hecke L-function associated to chi. Set flag = 1 if the character is known to be primitiveellconvertname(name): convert an elliptic curve name (as found in the elldata database) from a string to a triplet [conductor, isogeny class, index]. It will also convert a triplet back to a curve name.ellpointtozfactorff(x,p,a): factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]fibonaccifordiv(n,X,seq): the sequence is evaluated, X running over the divisors of nintnumstep(): gives the default value of m used by all intnum and sumnum  routines, such that the integration step is 1/2^m.matker(x,{flag=0}): basis of the kernel of the matrix x. flag is optional, and may be set to 0: default; non-zero: x is known to have integral entriesnfeltreducemodpr(nf,a,pr): element a modulo pr in nf, where pr is in modpr format (see nfmodprinit)omegapolcoeffpolcoeff(x,s,{v}): coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[]). With respect to the main variable if v is omitted, with respect to the variable v otherwisepoldiscpolrootsmod(x,p,{flag=0}): roots mod p of the polynomial x. flag is optional, and can be 0: default, or 1: use a naive search, useful for small ppoltschirnhaus(x): random Tschirnhausen transformation of the polynomial xpolylogV=GGEpqfbpowrawquaddiscrnfeltdown(rnf,x): expresses x on the base field if possible; returns an error otherwisesetminus(x,y): set of elements of x not belonging to yiGsimplify(x): simplify the object x as much as possiblesizebyte(x): number of bytes occupied by the complete tree of the object xsolve(X=a,b,expr): real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sumnum(X=a,sig,expr,{tab},{flag=0}): numerical summation of expr from  X = ceiling(a) to +infinity. sig is either a scalar or a two-component vector coding the function's decrease rate at infinity. It is assumed that the scalar part of sig is to the right of all poles of expr. If present, tab must be initialized by sumnuminit. If flag is nonzero, assumes that conj(expr(z)) = expr(conj(z)).thetanullk(q,k): k'th derivative at z=0 of theta(q,z)variableznprimroot(n): returns a primitive root of n when it existsResetting all trapsK%ld list = %Z
monomorphismlift()Q_denomnot a series in serreversereducible form in qfr_rhogtracehnfspec[1]mit = %Z
collecting garbage inimpossible assignment I-->Iimpossibleforbiddendefault: inexistent format\vskip\medskipamount\bgroup\bfunknown default: %stest if polynomial is square-free
  2) Conversion from T_2 --> | |^2 bound : %Z
nf_LLL_cmbf: checking factor %ld (avma - bot = %lu)
Expect a number, got a stringLow-level terminals of Gnuplot.  Query available terminals via
	plotterm("??")
Specify size (if it changable via scaling) as in "termname=300,200".
Add terminal options (if applicable) after the name and SPACE char.
See documentation of options in gnuplot, or, if via Term::Gnuplot, via
    perldoc GnuplotTerminals
greffeplotrpoint(w,dx,dy): draw a point (and move cursor) at position dx,dy relative to present position of the cursor in rectwindow wpadicsqrtnlift.degree does not divide phi(n) in subcyclo  alpha_lambda(mu,p) = %Z
    column selection:deplinFpX_FpXQV_compo: %d FpXQ_mul [%d]
[caller of] snextprIFAC: trying MPQS
chk_gen_init: subfield %Z
_%ldtwo abcissas are equal in polintplease apply bnrinit(,,1) and not bnrinit(,,0)bad subgroup in conductor or discray%Z - (%Z)
expecting a simple variable in ellwptwo vector/matrix types in bilhellc14 = %Z
nfsmith for non square matricesnot a variable:break not allowed in print()can't allow allocatemem() in loopsconformal_polbezoutres(x,y)=gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbinary(x)=gives the vector formed by the binary digits of x (x C-integer)bittest(x,n)=gives bit number n (coefficient of 2^n) of the integer xboundcf(x,lmax)=continued fraction expansion of x with at most lmax termsbuchgenfu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units (if they are not too large) for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)buchinitforcefubuchrayinitbuchrayinit(bnf,ideal)=same as buchrayinitgen, except that the generators are not explicitly computeddirdiv(x,y)=division of the Dirichlet series x by the Dir. series ydireuler(p=a,b,expr)=Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s)idealdiv(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nfidealfactor(nf,x)=factorization of the ideal x given in HNF into prime ideals in the number field nfidealhermite2(nf,a,b)=hermite normal form of the ideal aZ_K+bZ_K in the number field K defined by nf, where a and b are elementsideallistarchgenideallistzstar(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, without generatorsintgen(X=a,b,s)=general numerical integration of s from a to b with respect to X, to be used after removing singularitiesintinf(X=a,b,s)=numerical integration of s from a to b with respect to X, where a or b can be plus or minus infinity (1.0e4000), but of same signkerint2(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLlexsort(x)=sort the elements of the vector x in ascending lexicographic orderlindep(x)=Z-linear dependencies between components of x (Hastad et al)lllgrammodpnfmod(nf,a,b)=gives r such that r=a-bq is small with q algebraic integernucomppolredabsfastrayclassnolistredrealrndtoi(x)=take the nearest integer to all the coefficients of x, without taking into account loss of integer part precisionrnfequation(nf,pol)=given a pol with coefficients in nf, gives the absolute equation of the number field defined by polroots(x)=roots of the polynomial x using Schonhage's method modified by Gourdonsetsigmaksignunitsmalldiscfsmallpolred2Math::Pari::Eparg1, arg2, arg3gcmp0_lexMath::Pari::pari2ivMath::Pari::interface2099Math::Pari::__dumpStackMath::Pari::setseriesprecisionMath::Pari::initprimes$Math::Pari::initprimes not defined!intiter%iGD0,L,D0,G,t_COL
 Top : %lx   Bottom : %lx   Current stack : %lx
 %ld variable names used out of %d

 /%s/%s%s(lg=%ld%s):   next = %s ])	-reorderPolrevQfbGpbnrclassnolist(bnf,list): if list is as output by ideallist or similar, gives list of corresponding ray class numbersGDGdilogelllocalredfactorback(f,{e},{nf}): given a factorisation f, gives the factored object back. If this is a prime ideal factorisation you must supply the corresponding number field as last argument. If e is present, f has to be a vector of the same length, and we return the product of the f[i]^e[i]forprimefracgaloispermtopolGD0,L,Dngamma(x): gamma function at xglobalhyperuhyperu(a,b,x): U-confluent hypergeometric functionidealchinese(nf,x,y): x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealinv(nf,x,{flag=0}): inverse of the ideal x in the number field nf. If flag is omitted or set to 0, use the different. If flag is 1 do not use itideallistarchintfouriercos(X=a,b,x,s,{tab}): numerical integration from a to b of cos(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the cosine-Fourier transform if a=-infty and b=+infty.intnuminitmathnfmodidmatindexrank(x): gives two extraction vectors (rows and columns) for the matrix x such that the extracted matrix is square of maximal rankmatinverseimagematsolvemodmodreversenfbasisnfeltdivremnfeltmod(nf,a,b): gives r such that r=a-bq is small with q algebraic integernffactor(nf,x): factor polynomial x in number field nfnfgaloisapply(nf,aut,x): Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfnfsolvemodprnumdivpadicapprpadicprec(x,p): absolute p-adic precision of object xpolhenselliftpolroots(x,{flag=0}): complex roots of the polynomial x. flag is optional, and can be 0: default, uses Schonhage's method modified by Gourdon, or 1: uses a modified Newton methodprintpsi(x): psi-function at xqfbclassno(x,{flag=0}): class number of discriminant x using Shanks's method by default. If (optional) flag is set to 1, use Euler productsrandom({N=2^31}): random integer between 0 and N-1rnfequationrnfpolredsetintersect(x,y): intersection of the sets x and ysuminf(X=a,expr): infinite sum (X goes from a to infinity) of real or complex expression exprGD0,L,D1,L,psignificant pointers lost in gerepile! (please report)Init: trying delta = %d, t = %d
overflow in real shiftbound = 0 in minim2########## FACTORBASE ##########

base change =
bestappr/regulatorrootpadicfast()GaloisConj:%Z
GaloisConj: not found, %d hops 
code error in intnumRgX_powersidentical index variables in matrixsubresall, dr = %lddiscsr* Time LLL: %ld
* Time Check Factor: %ld
non-positive argument in O()lift_internhnf_special[2]. i=%ld    after phase1:
variable must have higher priority in matfrobenius    first pass in hnffinal done
Compute Cl(k)S & Tnot a square matrixreducible polynomialtrying to overwrite a universal object\egroup\bgroup\ttZ_issquaremissing exponentFp_sqrtincorrect size in pnqnFp_sqrtlpolynomial variable must have highest priority in nfrootserror setting terminal "%s"lifting embedding mod p^k = %Z^%ld
coeff too big for embedding
coset not found in cosets_perm_searchvGD0,L,V=GGIpD0,M,D0,L,
Parametric|1; Recursive|2; no_Rescale|4; no_X_axis|8; no_Y_axis|16; no_Frame|32; no_Lines|64; Points_too|128; Splines|256; no_X_ticks|512; no_Y_ticks|1024; Same_ticks|2048plothraw(listx,listy,{flag=0}): plot in high resolution points whose x (resp. y) coordinates are in listx (resp. listy). If flag is 1, join points, other non-0 flags should be combinations of bits 8,16,32,64,128,256 meaning the same as for ploth()psplothraw(listx,listy,{flag=0}): same as plothraw, except that the output is a postscript program in psfile (pari.ps by default)removeprimeprime too big in rootmod2non-positive precision in rootpadicpolynomial variable must have higher priority in factorffQpX_to_ZX* Eigenvalues computed

group:Incorrect curve name in ellconvertname%s/elldata/ell%ldmatextractimage2det2  new order: %Z
  fx = %Z, gx = %Zno root in nilord. Is p = %Z a prime?  entering DecompZZ_%Z[%Z]/(%Z) is not a field in FpX_ffintersectp = %lu, setting up for B2
composite squfof [caller of] (5n is a square)	- ruled out
IFAC: found %ld large prime (power) factor%s.
found factor
	%Z
currently lost to the factoring machineryRho: time = %6ld ms,	%3ld round%s
...partial impossibly short in ifac_sort_onenot a row vector in plothF_18(6):2 = [1/2.S(3)^2]2nfiso or nfinclgalois files not available
[missing %s]    ----> Group # %ld/%ld:
gen_sortsqrtrnon positive argument in mplogzero argument in palognon Galois extension in rnfnormgroupconductorofcharMQPS: short of space -- another buffer for sorting
error whilst writing to file %sMPQS: resplitting a factor after %ld kernel vectors
, looking for more...MPQS: wrapping up vector of %ld factors
expected an integer in bnfisintnorminithueunif_mod_fZelement not in ideal in ideal_two_elt2element does not belong to ideal in ideal_two_elt2Junk after an id in a stringified flagb6can't replace an existing symbol by an aliasbreak not allowed here (reading arguments)incorrect type or length in matrix assignmentbreak not allowed here (reading long)basistoalg(nf,x)=transforms the vertical vector x on the integral basis into an algebraic numberbuchinitforcefu(P,...)=compute the necessary data for future use in ideal and unit group computations, and insist on having fundamental units. See manual for detailsbuchinitfu(P,...)=compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. See manual for detailsbuchrayGDGDGD1,G,diagonaldiscf2(x)=discriminant of the number field defined by the polynomial x using round 2factmod(x,p)=factorization mod p of the polynomial x using Berlekamphermitemod(x,d)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is the non-zero determinant of this latticeincgam1(s,x)=incomplete gamma function (for debugging only)initell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,q,area]label(n)=THIS FUNCTION HAS BEEN SUPPRESSEDlex(x,y)=compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)lllgramkerimgenlllgramkerimgen(x)=kernel and lll reduction of the lattice whose gram matrix is the matrix x with polynomial coefficientslnlogagm(x)=natural logarithm of x, computed using agm (faster than log for more than a few hundred decimal digits)modulargcd(x,y)=gcd of the polynomials x and y using the modular methodnfdivres(nf,a,b)=gives [q,r] such that r=a-bq is smallnfmulV=GGIpsimplefactmod(x,p)=same as factmod except that only the degrees of the irreducible factors are givensturmpart(x,a,b)=number of real roots of the polynomial x in the interval (a,b]sumpos(X=a,expr)=sum of positive series expr, the formal variable X starting at athreetotwotrace(x)=trace of xweberf(x)=Weber's f function of x (j=(f^24-16)^3/f^24)wf2module too large in ffshanksratliftg, n, eltin, dummy1, dummy2interface$;@Math::Pari::interface29Math::Pari::interface4Math::Pari::interface31Math::Pari::interface35Math::Pari::interface48Math::Pari::interface19Math::Pari::interface86Math::Pari::ifactMath::Pari::have_graphicsV=GIpt_COMPLEXno such function 1)\cr
inputunexpected closing bracenum = coef of degree %ld = can't trap memory errorsaccureruserColVec({x=[]}): transforms the object x into a vector. Empty vector if x is omittedaddhelpalgdep(x,n,{flag=0}): algebraic relations up to degree n of x, using lindep([1,x,...,x^(n-1)], flag).bitandbnfclassunitbnfnarrowbnfsunitconjvec(x): conjugate vector of the algebraic number xellidentify(E): look up the elliptic curve E in the elldata database and return [[N, M, ...], C] where N is the name of the curve in J. E. Cremona database, M the minimal model and C the coordinates change (see ellchangecurve).galoissubfields(G,{flags=0},{v}):Output all the subfields of G. flags have the same meaning as for galoisfixedfieldgcd(x,{y}): greatest common divisor of x and y.getstackglobal(x): declare x to be a global variableidealchineseintfourierexp(X=a,b,x,s,{tab}): numerical integration from a to b of exp(-2*I*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the ordinary Fourier transform if a=-infty and b=+infty. Note the minus sign.intnumromb(X=a,b,s,{flag=0}): numerical integration of s (smooth in  ]a,b[) from a to b with respect to X. flag is optional and mean 0: default.  s can be evaluated exactly on [a,b]; 1: general function; 2: a or b can be  plus or minus infinity (chosen suitably), but of same sign; 3: s has only  limits at a or blistinsertmatfrobeniusmoebiusnewtonpolynfeltmulmodprpolleadpolresultant(x,y,{v},{flag=0}): resultant of the polynomials x and y, with respect to the main variables of x and y if v is omitted, with respect to the variable v otherwise. flag is optional, and can be 0: default, assumes that the polynomials have exact entries (uses the subresultant algorithm), 1 for arbitrary polynomials, using Sylvester's matrix, or 2: using a Ducos's modified subresultant algorithmpolrootsmodquadunitrnfidealnormabssigmasizedigitsumnuminittanh(x): hyperbolic tangent of xthetanullkvecextractincorrect data in eltreltoabsdifferent pointers in ff_poltypegmulsgGaloisFixedField:den=%Z mod=%Z 
GaloisConj: Solution too large, discard it.
S4GaloisConj:Testing %d/8 %d:%d:%d
s4test()sum from 1 to N-1p-adic jbessel functionexponential increase in integral transform0 argument in nfhilbertnfhilbert not soluble at real place %ld
not a factorization in polhenselliftnot an integral factorization in polhenselliftleftright_powinexact computation in RgX_extgcdLLL_cmbfDDF: wrong numbers of factorsisintpositive discriminant in %sqfr_unithnfspec[2]main variable in bnrstark must not be xquickpolRecCoeffimpossible inverse modulo: not enough memory   colors = "%s"
1TeXstyle"/usr/local/lib/pari/gphelp"FB = %Z
rootexponent: %ld
gtofpplotrecthraw(w,data,{flags=0}): plot graph(s) for data in rectwindow w, where data is a vector of vectors. If plot is parametric, length of data should be even, and pairs of entries give curves to plot. If not, first entry gives x-coordinate, and the other ones y-coordinates. Admits the same optional flags as plotrecth, save that recursive plot is meaninglessbitwise orzero argument in an arithmetic functionnon-cyclic case in polsubcyclo: use galoissubcyclo insteadIncorrect vector in ellconvertnameIndexPartial: discriminantdisc. factorisation	B2 = %6lu,	gss = %4ld*420
	(baby step table complete)
Rho: fast forward phase (%ld rounds of 64)...
SQUFOF: found factor %ld^2
compositeIFAC: a factor was a power of another prime factor
misaligned partial detected in ifac_sort_oneGaloisIndex: Using hash value s=%ld
you must initialize the rectwindow firstincorrect prime idealTschirnhaus transform. New pol: %ZF_18(6) = [3^2]2 = 3 wr 2S_4(6d) = [2^2]S(3)matricesCOSopening %sdifferent lengths in polinterpolaten too small (%ld) in numtopermagm of two vector/matriceszero argument in mplogsorry, too many primes to checkdiscrayabslist     Beta list = %Z
[apell1] baby steps, s = %ldMPQS: done sorting%s, time = %ld ms
MPQS: decomposed a %s
Partial = %Z
Number of Fermat powerings = %lu
[install] identifier '%s' already in usetatebreak not allowed in array contextexpected character: '%c' instead of.fualgdep2(x,n,dec)=algebraic relations up to degree n of x where dec is as in  lindep2apell2divres(x,y)=euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainderexp(x)=exponential of xfordiv(n,X,seq)=the sequence is evaluated, X running over the divisors of ngcd(x,y)=greatest common divisor of x and yhermitemodhvectoridealaddmultoneidealval(nf,x,p)=valuation at p given in primedec format of the ideal x in the number field nfinitalgredintnum(X=a,b,s)=numerical integration of s from a to b with respect to Xisirreduciblelll1(x)=old version of lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lllgram1(x)=old version of lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)order(x)=order of the integermod x in (Z/nZ)*polred(x)=reduction of the polynomial x (gives minimal polynomials only)polzag(n,m)=Zagier's polynomials of index n,mqfi(a,b,c)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c<0rootpadic(x,p,r)=p-adic roots of the polynomial x to precision rsylvestermatrix(x,y)=forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowstan(x)=tangent of xteich(x)=teichmuller character of p-adic number xweipell(e)=formal expansion in x=z of Weierstrass P functionzidealstarthis function has been suppressedtruncr (precision loss in truncation)arg1, arg2, arg3, arg4=0arg1, arg2, arg3, arg4, arg5, arg6=0, arg7=0gpui_gaddUnsupported interface %d and no code for a Pari function %s<null>newvalue = -1Bad PARI variable name "%s" specifiedI/O: leaked file descriptor (%d): %sLINES,CLONEint = pol = mod = 
 - 0.E%ldE%ldTeX variable name too long  ***   this trap keywordCol({x=[]}): transforms the object x into a column vector. Empty vector if x is omittedSer(x,{v=x}): convert x (usually a vector) into a power series with variable v, starting with the constant coefficientatanbesselh2(nu,x): H^2-bessel function of index nu and argument xbnrconductorofchar(bnr,chi): conductor of the character chi on the ray class group bnrbnrdisc(a1,{a2},{a3},{flag=0}): absolute or relative [N,R1,discf] of the field defined by a1,a2,a3. [a1,{a2},{a3}] is of type [bnr], [bnr,subgroup], [bnf, module] or [bnf,module,subgroup], where bnf is as output by bnfclassunit (with flag<=2), bnr by bnrclass (with flag>0), and subgroup is the HNF matrix of a subgroup of the corresponding ray class group (if omitted, the trivial subgroup). flag is optional whose binary digits mean 1: give relative data; 2: return 0 if module is not the conductorbnrinit(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data linked to the ray class group structure corresponding to this module. flag is optional, and can be 0: default (same as bnrclass with flag = 1), 1: compute also the generators (same as bnrclass with flag = 2). The fifth component is the ray class group structurebnrstark(bnr,{subgroup}): bnr being as output by bnrinit(,,1), finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the trivial subgroup if omitted) using Stark's units. The ground field and the class field must be totally real.centerliftdirdiv(x,y): division of the Dirichlet series x by the Dirichlet series yellorderfactorialfactormodffinit(p,n,{v=x}): monic irreducible polynomial of degree n over F_p[v]idealnormidealval(nf,x,p): valuation at p given in idealprimedec format of the ideal x in the number field nfisfundamental(x): true(1) if x is a fundamental discriminant (including 1), false(0) if notispseudoprime(x,{n}): true(1) if x is a strong pseudoprime, false(0) if not. If n is 0 or omitted, use BPSW test, otherwise use strong Rabin-Miller test for n randomly chosen basesissquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is given puts the exact square root there if it was computedmatdetintmatfrobenius(M,{flag},{v=x}): Return the Frobenius form of the square matrix M. If flag is 1, return only the elementary divisors as a vector of polynomials in the variable v. If flag is 2, return a two-components vector [F,B] where F is the Frobenius form and B is the basis change so that M=B^-1*F*B.mathnf(A,{flag=0}): (upper triangular) Hermite normal form of A, basis for the lattice formed by the columns of A. flag is optional whose value range from 0 to 4 (0 if omitted), meaning : 0: naive algorithm. 1: Use Batut's algorithm. Output 2-component vector [H,U] such that H is the HNF of A, and U is a unimodular matrix such that AU=H. 3: Use Batut's algorithm. Output [H,U,P] where P is a permutation matrix such that P A U = H. 4: as 1, using a heuristic variant of LLL reduction along the waymathnfmod(x,d): (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is a multiple of the non-zero determinant of this latticematrixqzmatsize(x): number of rows and columns of the vector/matrix x as a 2-vectorminpoly(A,{v=x}): minimal polynomial of the matrix or polmod A.nfeltdivGGGGnfeltval(nf,a,pr): valuation of element a at the prime pr as output by idealprimedecnfisincl(x,y): tests whether the number field x is isomorphic to a subfield of y (where x and y are either polynomials or number fields as output by nfinit). Return 0 if not, and otherwise all the isomorphisms. If y is a number field, a faster algorithm is usedpolredabs(x,{flag=0}): a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 norm. flag is optional, whose binary digit mean 1: give the element whose characteristic polynomial is the given polynomial. 4: give all polynomials of minimal T2 norm (give only one of P(x) and P(-x)). 16: partial reductionquadclassunit(D,{flag=0},{tech=[]}): compute the structure of the class group and the regulator of the quadratic field of discriminant D. If flag is non-null (and D>0), compute the narrow class group. See manual for the optional technical parametersreadvec({filename}): create a vector whose components are the evaluation of all the expressions found in the input file filenamernfconductorrnfidealhnfrnfinitrnfisnorm(T,x,{flag=0}): T is as output by rnfisnorminit applied to L/K. Tries to tell whether x is a norm from L/K. Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-integer, with S a list of places in K containing the ramified primes, generators of the class group of ext, as well as those primes dividing x. If L/K is Galois, omit flag, otherwise it is used to add more places to S: all the places above the primes p <= flag (resp. p | flag) if flag > 0 (resp. flag < 0). The answer is guaranteed (i.e x norm iff b=1) if L/K is Galois or, under GRH, if S contains all primes less than 12.log(disc(M))^2, where M is the normal closure of L/Ksinsqrtint(x): integer square root of x (x integer)sqrtn(x,n,{&z}): nth-root of x, n must be integer. If present, z is set to a suitable root of unity to recover all solutions. If it was not possible, z is set to zerosumdivthueinitbroken pipeCandidate: %Z
negative accuracy in lindep2
#### Computing fundamental units
Computing powers for subFB: %Z

#### Tentative regulator : %Z
NumberOfConjugates:Nbtest=%ld,card=%ld,p=%ld
GaloisFixedField:cosets=%Z 
Bernoulli sumjbesselp-adic kbessel functionnot a correct factorization in polhenselliftforbidden substitution by a vectorpolevalpoleval: i = %ldnot an integer exponent in nupowdiscriminant not congruent to 0,1 mod 4 in primeformd must be 0 or 3 mod 4RecpolnumWarning: increasing precnot an integer argument in an arithmetic functiondivision by zerolightbg[1,,1], [5,,1], [3,,1], [7,,1], [6,,1], , [2,,1]compatiblestrictmatchincorrect variables in rnfcharpolybound computationGNUPLOT_DRAW_DLLGNUPLOT_DRAW_DLL_NO_PERLauto/Term/Gnuplot/forbidden or conflicting type in gvalempty vector in vecmininstall(name,code,{gpname},{lib}): load from dynamic library 'lib' the function 'name'. Assign to it the name 'gpname' in this GP session, with argument code 'code'. If 'lib' is omitted use 'libpari.so'. If 'gpname' is omitted, use 'name'plotcursorplotpointtypeeuclidean division (poldivrem)FqX_split_Trager failed!Subcyclo: prime l=%Z
impossible inverse: %Z  fastnu: HNF(G) is computed
different variables in modulargcdgcd mod %lu (bound 2^%ld)IFAC: checking for pure square
IFAC: factor %Z
	is %s
IFAC: a factor was divisible by another prime factor,
	leaving a cofactor = %Z
i = %ld: %Z
	%2ld: %Z
legendreincorrect lextype in vecsortsqrtnrlog is not meromorphic at 0gcotan     p divides h(K)
MPQS: found multiplier %ld for N
ftell error on full relations filecannot seek FREL file  - norm sol. no %ld/%ld
Not enough precision in thuecannot invert zero idealgengroupbreak not allowed after ^local() bloc must appear before any other expressiona 0x0 matrix has no elementsrefine_Hbuchfu(bnf)=compute the fundamental units of the number field bnf output by buchinitbuchnarrowchar(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using the comatrixchell(x,y)=change data on elliptic curve according to y=[u,r,s,t]compimag(x,y)=Gaussian composition of the binary quadratic forms x and y of negative discriminantcompositum(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2core2cosh(x)=hyperbolic cosine of xfactoredpolred(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives minimal polynomials only)frac(x)=fractional part of x=x-floor(x)hermitehavasjbesselh(n,x)=J-bessel function of index n+1/2 and argument x, where n is a non-negative integerlllkerimmatextract(x,y,z)=extraction of the components of the matrix x according to the vector or masks y (for the rows) and z (for the columns) from left to right (1,2,4,8,...for the first, second, third, fourth, ...rows or columns)minim2(x,bound)=looks for vectors of square norm <= bound, return the first one and its normperfpermutation2numrnflllgram(nf,pol,order)=given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixsignatsmallpolred(x)=partial reduction of the polynomial x (gives minimal polynomials only)tchebi(n)=Tchebitcheff polynomial of degree n (n C-integer)threetotwo(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]zero element in zsigneColumns of input matrix are of different heightgdivxMath::Pari::interface109$$$$Math::Pari::type_namevLLGDVDEMat(%s is not a GP binary filekilling bloc (no %ld): %08lx
pari.logOagmbitor(x,y): bitwise "or" of two integers x and y. Negative numbers behave as if modulo big power of 2bnfcertify(bnf): certify the correctness (i.e. remove the GRH) of the bnf data output by bnfclassunit or bnfinitbnfisintnorm(bnf,x): compute a complete system of solutions (modulo units of positive norm) of the absolute norm equation N(a)=x, where a belongs to the maximal order of big number field bnf (if bnf is not certified, this depends on GRH)bnrclassnolistbnrdisccos(x): cosine of xdivrem(x,y,{v}): euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainder, with respect to v (to main variable if v is omitted)ellchangepoint(x,y): change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]ellpoweval(x): evaluation of x, replacing variables by their valueforprime(X=a,b,seq): the sequence is evaluated, X running over the primes between a and bgaloispermtopol(gal,perm): gal being a galois field as output by galoisinit and perm a element of gal.group, return the polynomial defining the corresponding Galois automorphismgetstack(): current value of stack pointer avmaintnuminitgenispseudoprimelift(x,{v}): lifts every element of Z/nZ to Z or T[x]/PT[x] to T[x] for a type T if v is omitted, otherwise lift only polymods with main variable v. If v does not occur in x, lift only intmodslog(x): natural logarithm of x.matimagecomplGGDVDVDImatsnfnfhnfpolsym(x,n): vector of symmetric powers of the roots of x up to nprecisionprodinfquadray(D,f,{lambda}): relative equation for the ray class field of conductor f for the quadratic field of discriminant D (which can also be a bnf). For D < 0, lambda (if supplied) is the technical element of bnf  necessary for Schertz's method. In that case, return 0 if lambda is not suitable.rnfidealnormabs(rnf,x): absolute norm of the ideal xrnfidealup(rnf,x): lifts the ideal x (of the base field) to the relative fieldrnfpolred(nf,pol): given a pol with coefficients in nf, finds a list of relative polynomials defining some subfields, hopefully simplersqr(x): square of x. NOT identical to x*xsumnuminit(sig, {m=0}, {sgn=1}): initialize tables for numerical summation. sgn is 1 (in fact >= 0), the default, for sumnum (ordinary sums)  or -1 (in fact < 0) for sumnumalt (alternating sums). sig is as in sumnum and m is as in intnuminit.trace(x): trace of xtruncate(x,{&e}): truncation of x; when x is a power series,take away the O(X^). If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitswrite1segmentation fault: bug in PARI or calling programmid = %ZFincke-Pohst, final LLL: prec = %ld
getfusub factorbase (%ld elements)different modulus in ff_poltypevandermondeinversemod()%d.%Zjbessel around a!=0intnuminit0incorrect abscissa in sumnumincorrect table length in intnum initializationnon monic relative equationpolhensellift: factors %Z and %Z are not coprimepseudodiv dx = %ld >= %ldfactorsthis object is a leaf. It has no componentsinvalid data in qfevalgtopolycompositiongnormintersectallhnfmod[1]. i=%ldbase field not totally real in bnrstarkpolrel = %Z
error opening overflow in R->dbl conversionPARIpromptSTARTtimerdatadir[quadhilbert] incorrect values in pq: %lu... lifted (avma - bot = %lu)
negative length in listcreateplotlines(w,listx,listy,{flag=0}): draws an open polygon in rectwindow w where listx and listy contain the x (resp. y) coordinates of the vertices. If listx and listy are both single values (i.e not vectors), draw the corresponding line (and move cursor). If (optional) flag is non-zero, close the polygonplotrmoveV=GGIpD0,L,D0,L,panic: set_optimizeaddprimecan't accept 0 in addprimesZp_apprSubcyclo: orbits=%Z
generators must be prime to conductor in galoissubcyclosubgroup    mu = negative size in matid_FlmIdeals to consider:
resultant mod %ld (bound 2^%ld, stable=%ld)Miller-Rabin: testing base %ld
snextpr: %lu != prc210_rp[%ld] mod 210
IFAC: unfactored composite declared primeToo few points (%ld) for spline plotincorrect idealgpolcomp (different degrees)D(4)F(5) = 5:4LLL basis    Output of isin_%ld_G_H(%ld,%ld): not included.
doubling stack in dirmul
non-positive integer argument in cxpsi
PHASE 2: are all primes good ?

M0 = %Z
Yvaluation of j must be negative in p-adic ellinitMPQS: primes for A to be chosen near FB[%ld] = %ld
	%Z%s
comp.1, 0, x1 -> %Z
Checking (\pm %Z, \pm %Z)
Cannot negate id=value in a stringified flagcan't pop gp variablefujpositive integer expectedcauchy_boundbuchfubuchray(bnf,ideal)=given a big number field as output by buchinitfu (only) and  an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this modulebytesize(x)=number of bytes occupied by the complete tree of the object xconjvec(x)=conjugate vector of the algebraic number xcore(n)=unique (positive of negative) squarefree integer d dividing n such that n/d is a squarederiv(x,y)=derivative of x with respect to the main variable of yfactfqfactoredbasis(x,p)=integral basis of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoreddiscffpn(p,n)=monic irreducible polynomial of degree n over F_p[x]isfund(x)=true(1) if x is a fundamental discriminant (including 1), false(0) if notisisomfastlaplace(x)=replaces the power series sum of a_n*x^n/n! by sum of a_n*x^npermutationprecision(x)=real precision of object xprimrootprincipalideleV=GID0,L,pranksturmpartteichvecmax(x)=maximum of the elements of the vector/matrix xelement_mulizidealij done
invmodrtodblVariable in sv2pari is not of known typegdiventggt_gbitorMath::Pari::interface10Math::Pari::interface15Math::Pari::interface12$;$$$;$$$Math::Pari::interface49Math::Pari::interface44Math::Pari::pari_texprintvV=GGGt_REALt_RFRACList((lgeflist=%ld):imag = /var/tmp%lu.: %ld
Euler=Euler(): Euler's constant with current precisionStrchr(x): converts x to a string, translating each integer into a characterbernfracbinarybinomial(x,y): binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbitnegimply(x,y): bitwise "negated imply" of two integers x and y, in other words, x BITAND BITNEG(y). Negative numbers behave as if modulo big power of 2bnfcertifybreakceil(x): ceiling of x=smallest integer>=xGDnD0,L,coredireuler(p=a,b,expr,{c}): Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s). If c is present, output only the first c termsellztopointeta(x,{flag=0}): if flag=0, eta function without the q^(1/24), otherwise eta of the complex number x in the upper half plane intelligently computed using SL(2,Z) transformationsfactorpadic(x,p,r,{flag=0}): p-adic factorization of the polynomial x to precision r. flag is optional and may be set to 0 (use round 4) or 1 (use Buchmann-Lenstra)for(X=a,b,seq): the sequence is evaluated, X going from a up to bvV=GGGIforsubgroup(H=G,{bound},seq): execute seq for each subgroup H of the abelian group G (in SNF form), whose index is bounded by bound. H is given as a left divisor of G in HNF formgaloisisabeliangetrandhilbert(x,y,{p}): Hilbert symbol at p of x,y.ideallogissquarefreelcmmatidmatsnf(x,{flag=0}): Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector d. Binary digits of flag mean 1: returns [u,v,d] where d=u*x*v, otherwise only the diagonal d is returned, 2: allow polynomial entries, otherwise assume x is integral, 4: removes all information corresponding to entries equal to 1 in dnfgaloisconj(nf,{flag=0},{den}): list of conjugates of a root of the polynomial x=nf.pol in the same number field. flag is optional (set to 0 by default), meaning 0: use combination of flag 4 and 1, always complete; 1: use nfroots; 2 : use complex numbers, LLL on integral basis (not always complete); 4: use Allombert's algorithm, complete if the field is Galois of degree <= 35 (see manual for detail). nf can be simply a polynomial with flag 0,2 and 4, meaning: 0: use combination of flag 4 and 2, not always complete (but a warning is issued when the list is not proven complete); 2 & 4: same meaning and restrictions. Note that only flag 4 can be applied to fields of large degrees (approx. >= 20)norml2poldegree(x,{v}): degree of the polynomial or rational function x with respect to main variable if v is omitted, with respect to v otherwise. For scalar x, return 0 is x is non-zero and a negative number otherwiseprime(n): returns the n-th prime (n C-integer)print1D"",s,reorder({x=[]}): reorder the variables for output according to the vector x. If x is void or omitted, print the current list of variablesrnfbasistoalgrnfeltdownrnfinit(nf,pol): pol being a non constant irreducible polynomial defined over the number field nf, initializes a vector of data necessary for working in relative number fields (rnf functions). See manual for technical detailsrnfsteinitzserreverse(x): reversion of the power series xsign(x): sign of x, of type integer, real or fractionsubstvecthueinit(P,{flag=0}): initialize the tnf corresponding to P, that will be used to solve Thue equations P(x,y) = some-integer. If flag is non-zero, certify the result unconditionnaly. Otherwise, assume GRH (much faster of course)whilewritebin(filename,{x}): write x as a binary object to file filename. If x is omitted, write all session variablesbad component %ld in object %Znot a factorization matrix in isunitbnfnewpreccompute_Rrootspriority of optional variable too high in galoisfixedfieldGaloisConj: exp %d: s=%ld [%ld] a=%ld w=%ld wg=%ld sr=%ld
%d non invertible polynomial in Flxq_invm too large in intnuminitnot an integral polynomial in polhenselliftNaive recombinationinexact computation in subresextpolsturm, dr = %ldgfloorgconjmatrixqz3    1st phase done
allhnfmod[2]. i=%ldCompute polrelnumunexpected charactersorry, not yet available on this system\ifx\%s\undefined
  \def\%s#1#2{%s}\fi
classno with too small orderproduct, error bits = %ldquadhilbert (pq)incorrect data in findquadsquarefree testsplitting mod %ZTerminal does not define vectorChosen prime: p = %ld
perm_to_GAPempty group in group_domaingtolongcan't find symbol '%s' in library '%s'plotcopyplotstringpsploth(X=a,b,expr,{flags=0},{n=0}): same as ploth, except that the output is a postscript program in psfile (pari.ps by default)bitwise negated implyIFAC: (Partial fact.) Initial stop requested.
kernelfrobeniusspec_FpXQ_powincorrect vector in matmuldiagonaldetintnot a pseudo-basis in nfsimplifybasis  (Fa, Ea) = (%ld,%ld)
  ** switching to normal mode
factor_irred_matECM: %lu should have been prime but isn't
SQUFOF: square form (%ld^2, %ld, %ld) on first cycle
	after %ld iterations, time = %ld ms
ifac_numdivinconsistent data in rectplothinincorrect nf in nfnewprec%s group
(argument too large in ggamdggammalngamma around a!=1gpow: modulus %Z is not primegsinStep 8
beta reduced = %Z
   BOUND = %ld
MPQS: found factors = %Z
	and %Z
MQPS: line wrap -- another buffer for sorting
 %ld %ldMPQS: combined %ld full relation%s
%Z eliminated because of sign
bug2 in nfsmith0 in get_arch_realregray regulatorapprpadic(x,a)=p-adic roots of the polynomial x congruent to a mod pchelldirzetak(nf,b)=Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1disc(x)=discriminant of the polynomial xfactgauss(a,b)=gaussian solution of ax=b (a matrix,b vector)hilbidealinv(nf,x)=inverse of the ideal x in the number field nf not using the differentideallistarchgen(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, with generatorsidealmul(nf,x,y)=product of the two ideals x and y in the number field nfincgam(s,x)=incomplete gamma functionintersect(x,y)=intersection of the vector spaces whose bases are the columns of x and yV=GGID1,L,pisprincipalray(bnf,x)=bnf being output by buchrayinit, gives the vector of exponents on the ray class group generators of x. In particular x is principal if and only if the result is the zero vectorlllintpartial(x)=partial (hence faster) lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllratminidealmodp(x,y)=creates the integer x modulo y as a permanent object (on the heap)principalideal(nf,x)=returns the principal ideal generated by the algebraic number x in the number field nfqfrredrealnodrndtoisign(x)=sign of x, of type integer, real or fractionsmallbuchinit(pol)=small buchinit, which can be converted to a big one using makebigbnfsumalt(X=a,expr)=Villegas-Zagier's acceleration of alternating series expr, X starting at atwototwo(nf,a,b)=returns a 3-component vector [d,e,U] such that U is a unimodular 2x2 matrix with algebraic integer coefficients such that [a,b]*U=[d,e] and d,e are hopefully smallernot an integer exponent in nfpowarch_to_permnf_Pohlig-Hellman: DL mod %Z^%ld
Expected GEN return type, got code '%s'_gbitshiftl_gorMath::Pari::interface_flexible_gen%d unused args for PARI function %svGVtempfile %s already existsthqfi(close pipecan't expand ~/tmp in %s.GGLD0,L,pGGD1,L,pbnfmakeconj(x): the algebraic conjugate of xdefaultdirzetak(nf,b): Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1ellglobalredellisoncurveerfceulerphifactorpadicfloornfalgtobasisnfeltpownfmodprinit(nf,pr): transform the 5 element row vector pr representing a prime ideal into modpr format necessary for all operations mod pr in the number field nf (see manual for details about the format)nfrootsof1polhensellift(x, y, p, e): lift the factorization y of x modulo p to a factorization modulo p^e using Hensel lift. The factors in y must be pairwise relatively prime modulo pGDGDGD&polredabspolsympolzagierqfbcomprawqfbnupow(x,n): n-th power of primitive positive definite quadratic form x using nucomp and nuduplGD0,L,DGDGDGqfgaussredqflllquadhilbertquadhilbert(D,{pq}): relative equation for the Hilbert class field of the quadratic field of discriminant D (which can also be a bnf). If D<0, pq (if supplied) is a 2-component vector [p,q], where p,q are the prime numbers needed for Schertz's method. In that case, return 0 if [p,q] not suitable.rnfalgtobasis(rnf,x): relative version of nfalgtobasis, where rnf is a relative numberfieldrnfbasis(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setsubgrouplist(bnr,{bound},{flag=0}): bnr being as output by bnrinit or a list of cyclic components of a finite Abelian group G, outputs the list of subgroups of G (of index bounded by bound, if not omitted), given as HNF left divisors of the SNF matrix corresponding to G. If flag=0 (default) and bnr is as output by bnrinit, gives only the subgroups for which the modulus is the conductorV=GGEDGD0,L,pznprimrootlllfp (exact)powFBgenbe_honest() failure on prime %Z
Best lift: %d
non-real argument in eint1|gpolvarintegnot a real quadratic form in redrealgnorml1sqred1H = %Z
C = %Z
[1]: smithall i = %ldPermutation: %Z
incorrect subgrp in bnrstarkAllStarkprecision too lowincorrect typehelpnarrow class groupnumber of factor(s) found: %ld
PermutationGroup<1|>gaffect (gnil)negative index (%ld) in listputno more room in this L (size %ld)plotcolorvLLGGD0,L,D0,L,plotpoints(w,listx,listy): draws in rectwindow w the points whose x (resp y) coordinates are in listx (resp listy). If listx and listy are both single values (i.e not vectors), draw the corresponding point (and move cursor)FpXQYQ_powroots2not a prime in polrootsmod[FqX_split] splitting time: %ld (%ld trials)
to_Fqnot a HNF matrix in galoissubcycloN must be a bnrinit or a znstar if H is a matrix in galoissubcyclogtransrnfdet2ordmaxFpXQ_sqrtnmatrix cycloOddPwrs: testing for exponent %ld
ifac_issquarefreeIFAC: main loop: %ld factor%s left
IFAC: repeated factor %Z
	detected in ifac_sort_one
postscriptCikGalois names files not available, please upgrade galdata
[missing %s]not a set in setsearchgashsingular argument in atanh[2]: discrayabslistarchlocalred (p | c6)apell (f^(i*s) = 1)MPQS: creating factor base and allocating arrays...
MPQS: sieve threshold = %u
LPTMP
MPQS: time in Gauss and gcds = %ld ms
,%lu7th powergcd f_P  does not divide n_p
non-integral exponent in idealpowredpanicc6unused charactersaddprimes(x)=add primes in the vector x (with at most 20 components) to the prime tableclassno(x)=class number of discriminant xdirmul(x,y)=multiplication of the Dirichlet series x by the Dir. series ygetstack()=current value of stack pointer avmahess(x)=Hessenberg form of xmakebigbnfmatrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by the columns of xqfismithtransvector(n,X,expr)=row vector with n components of expression expr (X ranges from 1 to n)garg1, arg2=0, arg3=0, arg4=0gegalDo not know how to work with Pari control structure `%s'%sstack size is %ld bytes (%d x %ld longs)
Pari.cMath::Pari::interface2Math::Pari::interface26$$$$$Math::Pari::PARIvarpanic: PARI narg value not attachedDid not get a variablet_SERt_VECt_STRunknown type %ld, \cdotI can't see into the futureNULL
%ld%s column =  \left(
For full compatibility with GP 1.39.15, type "default(compatible,3)", or set "compatible = 3" in your GPRC fileSerasinbinary(x): gives the vector formed by the binary digits of x (x integer)bitneg(x,{n=-1}): bitwise negation of an integers x truncated to n bits. n=-1 means represent infinite sequences of bit 1 as negative numbers. Negative numbers behave as if modulo big power of 2bitxorbnfisnormbnrL1(bnr, {subgroup}, {flag=0}): bnr being output by bnrinit(,,1) and subgroup being a square matrix defining a congruence subgroup of bnr (the trivial subgroup if omitted), for each character of bnr trivial on this subgroup, compute L(1, chi) (or equivalently the first non-zero term c(chi) of the expansion at s = 0). The binary digits of flag mean 1: if 0 then compute the term c(chi) and return [r(chi), c(chi)] where r(chi) is the order of L(s, chi) at s = 0, or if 1 then compute the value at s = 1 (and in this case, only for non-trivial characters), 2: if 0 then compute the value of the primitive L-function associated to chi, if 1 then compute the value of the L-function L_S(s, chi) where S is the set of places dividing the modulus of bnr (and the infinite places), 3: return also the charactersbnrconductorcontent(x): gcd of all the components of x, when this makes sensecosderiv(x,{y}): derivative of x with respect to the main variable of y, or to the main variable of x if y is omittedellakellheightmatrix(e,x): gives the height matrix for vector of points x on elliptic curve e using theta functionslincgamclistkillmatadjointmatcompanion(x): companion matrix to polynomial xmatindexrankmatkerint(x,{flag=0}): LLL-reduced Z-basis of the kernel of the matrix x with integral entries. flag is optional, and may be set to 0: default, uses a modified LLL, 1: uses matrixqznfbasistoalg(nf,x): transforms the column vector x on the integral basis into an algebraic numbernfdetint(nf,x): multiple of the ideal determinant of the pseudo generating set xnfeltpow(nf,a,k): element a^k in nfnfnewprec(nf): transform the number field data nf into new data using the current (usually larger) precisionpolcycloLLDnqfbhclassnoreadvecremoveprimessetunion(x,y): union of the sets x and ysimplifysumaltvectorbitsize bound: %ld
qzer[%ld]=%ld
bound too largeisprincipal (incompatible bnf generators)
#### Looking for %ld relations (small norms)
relation cancelled: D = %Z
den = %Z
be honestrnfinitalgNumberOfConjugates:card=%ld,p=%ld
MonomorphismLift: lift to prec %dincgam2kbesselDDF_roots, m = %ldQ_div_to_intnon-invertible polynomial in RgXQ_invdegreegnorml2not enough precomputed primesrealprecisiondebugmem(prettymatrix)(prettyprint)colors   %s = %lu
composite modulus in Fp_sqrt: %Zcontfrac0sorry, couldn't deal with this field. PLEASE REPORTquadhilbertimag%3ld %s at prime
  %Z
Time: %ld
Terminal does not define optionsDISPLAYBuffer overflow finding gnuplot DLLget_term_ftablep = %ld,	lcm = %ld,	orbits: %Z

***** Entering subfields

pol = %Z
PermutationGroup<gaffect (geuler)vLToo large primelimit     real coefficients polynomial, using zrhqr()
* [mod p]absolute basisincorrect coeff in rnf functionDegree list for ERS (trials: %ld) = %Z
pol[Frobenius]ECM: working on %ld curves at a time; initializing	(got initial helix)
IFAC: factor %Z
	is prime (no larger composite)
rectlines) %ld %ld m 90 rotate show -90 rotate
incorrect bigidealD(5) = 5:2D(7) = 7:2rootsof1a(i,j)gtanmpsc1can't compute tan(Pi/2 + kPi)[rnfkummer] conductorStep 16
isprincipalray       prime ideal Q: %Z
not a prime in apellw1 and w2 R-linearly dependent in elliptic functionMPQSMPQS: found factor = %Z

MPQS: restarting sieving ...
	(estimated memory needed: %4.1fMBy)
MPQS: renamed file %s to %s
[3]: mpqs_solve_linear_system  c13 = %Z
SmallSolssol = %Z
Maximal number of nondeterministic steps = %lu
poltufuaddellassmat(x)=associated matrix to polynomial xGfcompositum2decodefactor(fa)=given a factorisation fa, gives the factored object backdiagonal(x)=creates the diagonal matrix whose diagonal entries are the entries of the vector xdiscf2GD0,G,D0,G,D2,L,divisors(x)=gives a vector formed by the divisors of x in increasing orderGGffactoredpolred2goto(n)=THIS FUNCTION HAS BEEN SUPPRESSEDhermite2(x)=2-component vector [H,U] such that H is an (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, and U is a unimodular matrix such that xU=H, using Batut's algorithmidealdivexactidealhermite(nf,x)=hermite normal form of the ideal x in the number field nf, whatever form x may haveideallistunitarchgen(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitgenideallllredidealmulredidealtwoelt2(nf,x,a)=(long)gen_2-element representation of an ideal x in the number field nf, with the first element equal to ainitalg(x)=x being a nonconstant irreducible polynomial, gives the vector: [x,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]isprime(x)=true(1) if x is a strong pseudoprime for 10 random bases, false(0) if notissqfreelambdaklllgramgenpolylogdold(m,x)=D_m-modified m-th polylog of xrnfpseudobasis(nf,pol)=given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rootmod2(x,p)=roots mod p of the polynomial x, when p is smallsin(x)=sine of xtanh(x)=hyperbolic tangent of xtschirnhaus(x)=random Tschirnhausen transformation of the polynomial xincompatible variables in algtobasisnot the same number field in rnfalgtobasisinPari.xs %2ld: %s
Math::Pari::is_gnilMath::Pari::installPerlFunctionCVMath::Pari::interface9900GD0,L,DGpvV=GED0,L,GD0,G,D0,G,D0,L,pt_LISTrun-away comment. Closing itcouldn't find a suitable name for a tempfile (%s)rd(%c,varn=%ld):[secure mode]: about to write to '%s'. OK ? (^C if not)
/usr/local/lib/pari/Cannot initialize kernelGDnPol(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientSetLbesselkbnfinitbnfmake(sbnf): transforms small sbnf as output by bnfinit with flag=3 into a true big bnfconjdirzetakellminimalmodelellsubelltors(e,{flag=0}): torsion subgroup of elliptic curve e: order, structure, generators. If flag = 0, use Doud's algorithm; if flag = 1, use Lutz-NagellGDGD0,L,pPerrorfactorgaloisidentifygaloissubcyclo(N,H,{fl=0},{v}):Compute a polynomial (in variable v) defining the subfield of Q(zeta_n) fixed by the subgroup H of (Z/nZ)*. N can be an integer n, znstar(n) or bnrinit(bnfinit(y),[n,[1]],1). H can be given by a generator, a set of generator given by a vector or a HNF matrix (see manual). If flag is 1, output only the conductor of the abelian extension. If flag is 2 output [pol,f] where pol is the polynomial and f the conductor.gammahidealadd(nf,x,y): sum of two ideals x and y in the number field defined by nfidealprincipalV=GGGEDGpV=GGED0,L,D0,L,pisprimevGmatdet(x,{flag=0}): determinant of the matrix x using Gauss-Bareiss. If (optional) flag is set to 1, use classical gaussian elimination (slightly better for integer entries)mathilbertmathilbert(n): Hilbert matrix of order n (n C-integer)mathnfmodmatimage(x,{flag=0}): basis of the image of the matrix x. flag is optional and can be set to 0 or 1, corresponding to two different algorithmsmax(x,y): maximum of x and ynfbasistoalgnfeltdivmodprnfeltdivmodpr(nf,a,b,pr): element a/b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltmulmodpr(nf,a,b,pr): element a. b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltreduce(nf,a,id): gives r such that a-r is in the ideal id and r is smallnfeltreducemodprnfinit(pol,{flag=0}): pol being a nonconstant irreducible polynomial, gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]. flag is optional and can be set to 0: default; 1: do not compute different; 2: first use polred to find a simpler polynomial; 3: outputs a two-element vector [nf,Mod(a,P)], where nf is as in 2 and Mod(a,P) is a polymod equal to Mod(x,pol) and P=nf.pol; 4: as 2 but use a partial polred; 5: is to 3 what 4 is to 2nfsolvemodpr(nf,a,b,pr): solution of a*x=b in Z_K/pr, where a is a matrix and b a column vector, and where pr is in modpr format (see nfmodprinit)polgalois(x): Galois group of the polynomial x (see manual for group coding). Return [n, s, k, name] where n is the order, s the signature, k the index and name is the GAP4 name of the transitive group.polrecip(x): reciprocal polynomial of xpolsylvestermatrixprodinf(X=a,expr,{flag=0}): infinite product (X goes from a to infinity) of real or complex expression. flag can be 0 (default) or 1, in which case compute the product of the 1+expr insteadquaddisc(x): discriminant of the quadratic field Q(sqrt(x))quadpoly(D,{v=x}): quadratic polynomial corresponding to the discriminant D, in variable vquadregulatorrnfconductor(bnf,polrel,{flag=0}): conductor of the Abelian extension of bnf defined by polrel. The result is [conductor,rayclassgroup,subgroup], where conductor is the conductor itself, rayclassgroup the structure of the corresponding full ray class group, and subgroup the HNF defining the norm group (Artin or Takagi group) on the given generators rayclassgroup[3]. If flag is non-zero, check that polrel indeed defines an Abelian extensionserreverseshiftsinh(x): hyperbolic sine of xvecmax(x): maximum of the elements of the vector/matrix xvecmin(x): minimum of the elements of the vector/matrix xvector(n,{X},{expr=0}): row vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0snegative number of vectors in minim0smallvectorsisunitincorrect parameters in classgroupFixedField: Size: %ldx%ld
Polynomial not squarefree in galoisinitA4GaloisConj:sigma=%Z 
lim, nn: [%ld, %ld]
too many iterations in solvecoeff bound: %Z^%ld
RecombinationQ_muli_to_intresultantducos, degpol Q = %ldnon-monic polynomial in poldiscreducedsturmTotal Time: %ld
===========
inv_sergroundnot a t_QFI in nuduplmatrixqz0incorrect character in bnrrootnumberthe ground field must be distinct from QArtinNumberLooking for a modulus of norm: Compute Wzetavalues = %Z
* conductor no %ld/%ld (N = %ld)
	Init: sorry,[secure mode]: Do you want to modify the 'secure' flag? (^C if not)
tex2mail -TeX -noindent -ragged -by_parzero modulus in znprimrootquadhilbertimag (can't find p,q)dumblistconcatplotstring(w,x,{flags=0}): draw in rectwindow w the string corresponding to x. Bits 1 and 2 of flag regulate horizontal alignment: left if 0, right if 2, center if 1. Bits 4 and 8 regulate vertical alignment: bottom if 0, top if 8, v-center if 4. Can insert additional gap between point and string: horizontal if bit 16 is set, vertical if bit 32 is setpolynomial has probably multiple roots in zrhqrbnr must be over Q in bnr_to_znstarSubcyclo: elements:wrong modulus in galoissubcycloinverseimage%Z^%ld
  Increasing Fa
FpX_resultant (da = %ld)False prime number %Z in plisprime	found factors = %Z, %Z,
	and %Z
ifac_bigomega	yielded `factor' %Z
	which isn't!
A4S5F_36(6) = 1/2[S(3)^2]22S_4(6) = [2^3]S(3) = 2 wr S(3)galois (bug4)round4i = %ld
precision too low in chk_gen_initdiscriminant = %Z
$$$$$ New prec = %ld
not a set in setunionzero argument in gargp-adic lngamma functiongpow: nth-root does not exist1/0 exponent in gsqrtnp-adic argument out of range in gcospadic_sqrtnot an n-th power in idealsqrtn  Testing primes | h(K)


Starting bnrclassno computations
zellMPQS: number to factor N = %Z
MPQS: largest prime in FB = %ld

MPQS: passing the %3.1f%% sort point, time = %ld ms
MPQS: no factors found.
error whilst flushing file %slongernot a matrix of maximal rank in nfhermitenfsmith0th power in idealpowprime_specnfshift operand too biginteger too bigtruc(): n = %ldlocalformal derivationasinh(x)=inverse hyperbolic sine of xboundfactbuchinit(P,...)=compute the necessary data for future use in ideal and unit group computations. See manual for detailscf2(b,x)=continued fraction expansion of x (x rational,real or rational function), where b is the vector of numerators of the continued fractionchar1coeff(x,s)=coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[])factpol2gamma(x)=gamma function at xgaussmodulo2(M,D,Y)=all solutions of system of congruences MX=Y mod Dideallistzstaridealtwoelt(nf,x)=(long)gen_2-element representation of an ideal x in the number field nfisoncurveisprincipalgen(bnf,x)=bnf being output by buchinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorkaramullift(x)=lifts every element of Z/nZ to Z or Z[x]/PZ[x] to Z[x]ln(x)=log(x)=natural logarithm of xlngamma(x)=logarithm of the gamma function of xnfdetint(nf,x)=multiple of the ideal determinant of the pseudo generating set xnfsmith(nf,x)=if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xorderorderell(e,p)=order of the point p on the elliptic curve e over Q, 0 if non-torsionpipolvarrecip(x)=reciprocal polynomial of xsmallinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j]smith2sqredelement_invnot a matrix in matalgtobasis%Z not a nfeltPARI: %.*s%*s%.*s%*s%sPerl function exported into PARI did not return a valueg, narg1, arg2, invgmod_gmul_gpuiUnsupported Pari function %s, interface 0 code NULL$Math::Pari::interface20Math::Pari::interface28_oldMath::Pari::interface47Math::Pari::interface59Math::Pari::memUsageMath::Pari::PARI_DEBUGt_PADIC + setting %s
p^l : :
   hash = %3ld, valence = %3ld, menu = %2ld, code = %s
undefined environment variable: %s%s%sassignmentvSsGGD1,L,bnfisunit(bnf,x): bnf being output by bnfinit (with flag<=2), gives the column vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisechinesecontfracpnqndirmuleint1ellanelltorsfactorffforell(E,a,b,seq): execute seq for each elliptic curves E of conductor between a and b in the elldata database.galoissubgroups(G):Output all the subgroups of Gif(a,seq1,seq2): if a is nonzero, seq1 is evaluated, otherwise seq2. seq1 and seq2 are optional, and if seq2 is omitted, the preceding comma can be omitted alsointfouriersinissquarefree(x): true(1) if x is squarefree, false(0) if notkill(x): kills the present value of the variable or function x. Returns new value or 0matalgtobasis(nf,x): nfalgtobasis applied to every element of the matrix xmatbasistoalg(nf,x): nfbasistoalg applied to every element of the matrix xmatinverseimage(x,y): an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwisenfeltdiveucnffactormodnfgaloisconjnfisisompolred(x,{flag=0},{p}): reduction of the polynomial x (gives minimal polynomials only). Second and third args are optional. The following binary digits of flag are significant 1: partial reduction, 2: gives also elements. p, if present, contains the complete factorization matrix of the discriminantqfbsolve(Q,p): Return [x,y] so that Q(x,y)=p where Q is a binary quadratic form and p a prime number, or 0 if there is no solution.rnfkummer(bnr,{subgroup},{deg=0}): bnr being as output by bnrinit, finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the ray class field if subgroup is omitted). deg can be zero (default), or positive, and in this case the output is the list of all relative equations of degree deg for the given bnrrnfsteinitz(nf,order): given an order as output by rnfpseudobasis, gives [A,I,D,d] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialGLptypewhile(a,seq): while a is nonzero evaluate the expression sequence seq. Otherwise 0writebin
Recomputing Gram-Schmidt, kmax = %ld
  Testing P = %Z
isprincipalnot a vector/matrix in cleanarchnormalizing a series with 0 leading termCalcul polynomesS4GaloisConj:Testing %d/3:%d/4:%d/4:%d/4:%Z
inv_szeta_euler, p = %lu/%lutrueetazpsolubleqpsolublenfnon integral index in sumprodinf1negative number of rows in matrixnon integral index in sumpos2...tried prime %3ld (%-3ld %s). Time = %ld
recipgceilvectosmalldenomt_SER with negative valuation in gtopolyredimagnot a positive definite matrix in sqred1incompatible matrices in hnf_specialhnfadd (%ld + %ld)extendedgcdhnfpermhnfall[1], li = %ldhnffinal, i = %ldNot enough precomputed primes (need all p <= %ld)seriesprecision1, 5, 3, 7, 6, 2, 3user functions re-initializedPARIoutpathdefault: incorrect value for %s [%lu-%lu]pnqngeneratorscoeff too big for pol g(x)
)plotclipplotcopy(sourcew,destw,dx,dy,{flag=0}): copy the contents of rectwindow sourcew to rectwindow destw with offset (dx,dy). If flag's bit 1 is set, dx and dy express fractions of the size of the current output device, otherwise dx and dy are in pixels. dx and dy are relative positions of northwest corners if other bits of flag vanish, otherwise of: 2: southwest, 4: southeast, 6: northeast cornersplotrlineplotrpointn-th prime meaningless if n = %ldBerlekamp_matrixnot a prime in factmodforsubgroup (alpha != countsub)Elliptic files %s not compatible
incorrect range in extractFpM_gauss_pivot  Increasing Ea
primedec: %Z is not primebad degrees in FpX_ffintersect: %d,%d,%dFpV_polintFinal lambda = %ld
	(got [p]Q, p = %lu = prc210_rp[%ld] mod 210)
SQUFOF: square form (%ld^2, %ld, %ld) on second cycle
	after %ld iterations, time = %ld ms
IFAC: found %Z =
	%Z ^2
IFAC: forcing ECM, may take some time
(\)please apply nfinit firstincorrect matrix for idealordredpolred for non-monic polynomial  generator: %Z
chk_gen_init: new prec = %ld (initially %ld)
%s/galdata/%s%ld_%ld_%ldmore than %ld rational integer roots
invalid bound in randomgarggthgachBernoulliStep 12
Step 14, 15 and 17
%ld(%ld) badMPQS: coefficients A will be built from %ld primes each
%s :%s
FREL file truncated?!prec = %d
  Entering LLL...
[2]: nfhermitemod, i = %ld  %-3ld: %3ld
renaming a GP variable is forbidden) = break not allowed here (defining global var)[none]unfinished stringacosh(x)=inverse hyperbolic cosine of xbigomega(x)=number of repeated prime divisors of xdecodemodule(nf,fa)=given a coded module fa as in discrayabslist, gives the true moduleGD0,G,D0,G,D1,L,eval(x)=evaluation of x, replacing variables by their valueglobalred(e)=e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pi=i()=square root of -1idealintersect(nf,x,y)=intersection of two ideals x and y in HNF in the number field defined by nfideallistzstargen(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, with generatorsV=GGID2,L,pisisomlcm(x,y)=least common multiple of x and y=x*y/gcd(x,y)nfdivrespermutation(n,k)=permutation number k (mod n!) of n letters (n C-integer)phipolypolyrev(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termquaddisc(x)=discriminant of the quadratic field Q(sqrt(x))setminus(x,y)=set of elements of x not belonging to ysturm(x)=number of real roots of the polynomial xwith a = %Z
The longword %ld ordinal out of boundMath::Pari::convertedXSUB call through interface did not provide *functiongneg$$;$$$%c[0mt_POLMODt_VECSMALLthis object uses debugging variablescould not open requested file %sI/O: checking output pipe...
.
 %s is not yet implemented.divisionQfb(a,b,c,{D=0.}): binary quadratic form a*x^2+b*x*y+c*y^2. D is optional (0.0 by default) and initializes Shanks's distance if b^2-4*a*c>0s*Strchrbernreal(x): Bernoulli number B_x, as a real number with the current precisionbesselh2binomialbnfunit(bnf): compute the fundamental units of the number field bnf output by bnfinit when they have not yet been computed (i.e. with flag=2)bnrdisclist(bnf,bound,{arch}): gives list of discriminants of ray class fields of all conductors up to norm bound, in a long vector The ramified Archimedean places are given by arch; all possible values are taken if arch is omitted. Supports the alternative syntax bnrdisclist(bnf,list), where list is as output by ideallist or ideallistarch (with units)bnrisprincipalGDGD0,L,coredisc(n,{flag=0}): discriminant of the quadratic field Q(sqrt(n)). If (optional) flag is non-null, output a two-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integerellwp(e,{z=x},{flag=0}): Complex value of Weierstrass P function at z on the lattice generated over Z by e=[om1,om2] (e as given by ellinit is also accepted). Optional flag means 0 (default), compute only P(z), 1 compute [P(z),P'(z)], 2 consider om as an elliptic curve and compute P(z) for that curve (identical to ellztopoint in that case). If z is omitted or is a simple variable, return formal expansion in zfactorbackfactormod(x,p,{flag=0}): factorization mod p of the polynomial x using Berlekamp. flag is optional, and can be 0: default or 1: simple factormod, same except that only the degrees of the irreducible factors are givenffinitidealappr(nf,x,{flag=0}): x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other p. If (optional) flag is non-null x must be a prime ideal factorization with possibly zero exponentsintmellininv(X=sig,x,s,{tab}): numerical integration on the  line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin  transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). tab is as in intnum. Use  intmellininvshort if several values must be computed.liftmatadjoint(x): adjoint matrix of xmatbasistoalgmateigenmathess(x): Hessenberg form of xmatpascal(n,{q}): Pascal triangle of order n if q is omited. q-Pascal triangle otherwisematrix(m,n,{X},{Y},{expr=0}): mXn matrix of expression expr, the row variable X going from 1 to m and the column variable Y going from 1 to n. By default, fill with 0snewtonpoly(x,p): Newton polygon of polynomial x with respect to the prime ppolcompositum(pol1,pol2,{flag=0}): vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2. If (optional) flag is set (i.e non-null), output for each compositum, not only the compositum polynomial pol, but a vector [pol,al1,al2,k] where al1 (resp. al2) is a root of pol1 (resp. pol2) expressed as a polynomial modulo pol, and a small integer k such that al2+k*al1 is the chosen root of polpollegendrepolsturmpoltschirnhausV=GED0,L,ppsiquadrayrnfidealmulrnfpseudobasis(nf,pol): given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2setisset(x): true(1) if x is a set (row vector with strictly increasing entries), false(0) if notsumsumdiv(n,X,expr): sum of expression expr, X running over the divisors of nzetakzetakinit(x): compute number field information necessary to use zetak, where x is an irreducible polynomialzncoppersmith(P, N, X, {B=N}): finds all integers x0 with |x0| <= X such that  gcd(N, P(x0)) > B. X should be smaller than exp((log B)^2 / (deg(P) log N)).bad object %Zlllfpnot a p-adic vector in plindephigher degree than expected in algdepclassgroupallv[%ld]=%.4g galoisconj2polp-adic jbesselh functionkbessel around a!=0
found factor %Z
remaining modular factor(s): %ld
forbidden substitution by a non square matrixinvalid quadratic form in qfbevalinvalid quadratic form in hqfevalsqred3hnfspec[3], (i,j) = %ld,%ldincorrect subgroup in bnrL1S&T\egroup   %s = %lu %s
discriminant too large in classno...need %ld more relations
computeP2
Entering nffactor:
Using Trager's method
nf_LLL_cmbf/blib/lib
***** Leaving subfields

wrong argument in galoisisabelianl <= 2 in greffecvtopplothplotlinesnot a prime in rootmodto_Fq_poldegree <= 0 in galoissubcycloIncorrect curve name in ellsearchkeriprimedecfactor_irredECM: time = %6ld ms
ECM: dsn = %2ld,	B1 = %4lu,ECM: time = %6ld ms,	p <= %6lu,
	found factor = %Z
Rho: time = %6ld ms,	Pollard-Brent giving up.
SQUFOF: ...but the root form seems to be on the principal cycle
 3rd%s	   %3ld:  %3ld   (3rd %ld, 5th %ld, 7th %ld)
miller(rabin)IFAC: main loop: this was the last factor
`washere' out of bounds in ifac_sort_one%10s%-9.7g%*.7g
single vector in gtodblList%ld %ld m %ld %ld l %ld %ld l %ld %ld l closepath
A5C(7) = 7S7non-monic polynomial. Result of the form [nf,c]chk_gen_init: difficult field, trying random elements
incorrect value in bin()not a vector in permtonumnot a polmod in modreversegatannon-positive integer argument in ggammaStep 9, 10 and 11
       generator of (Zk/Q)^*: %Z
arg to partition function must be < 10^15torselllocalred (nu_D - nu_j != 0,6)MPQS: number too big to be factored with MPQS,
	giving upLPNEWCOMBMPQS: found %lu candidate%s
MPQS: done sorting one file.
	packaging %ld: %Z ^%ld (%s)
cube  c6  = %Z
nored + denominator in idealapprfact%s already exists with incompatible valenceunknown functionnot a proper member definitionincorrect vector or matrixbreak not allowed in assignmentcan't modify a pre-defined member: akell(e,n)=computes the n-th Fourier coefficient of the L-function of the elliptic curve eapellcompimagconj(x)=the algebraic conjugate of xcvtoidiscrayabslistarchalldiscrayabslistlong(bnf,bound)=gives list of discrayabscond of all modules up to norm bound without archimedean places, in a longvector formatgalois(x)=Galois group of the polynomial x (see manual for group coding)gaussmodulo2hellhermitemodid(x,d)=(upper triangular) Hermite normal form of x concatenated with d times the identity matrixidealaddone(nf,x,y)=when the sum of two ideals x and y in the number field K defined by nf is equal to Z_K, gives a two-component vector [a,b] such that a is in x, b is in y and a+b=1idealinv2(nf,x)=inverse of the ideal x in the number field nf using the differentindexrank(x)=gives two extraction vectors (rows and columns) for the matrix x such that the exracted matrix is square of maximal rankinteg(x,y)=formal integration of x with respect to the main variable of yintopenisideal(nf,x)=true(1) if x is an ideal in the number field nf, false(0) if notisprincipalgenforceisprincipalgenforce(bnf,x)=same as isprincipalgen, except that the precision is doubled until the result is obtainedissetlength(x)=number of non code words in xlllgramgen(x)=lll reduction of the lattice whose gram matrix is x with polynomial coefficients (gives the unimodular transformation matrix)max(x,y)=maximum of x and yordell(e,x)=y-coordinates corresponding to x-ordinate x on elliptic curve epi=pi()=the constant pi, with current precisionpointellpolred2(x)=reduction of the polynomial x (gives elements and minimal polynomials)polredabsall(x)=complete list of the smallest generating polynomials of the number field for the T2 norm on the rootspolvar(x)=main variable of object x. Gives p for p-adic x, error for scalarsrootmod2subell(e,z1,z2)=difference of the points z1 and z2 on elliptic curve esupplementtaniyamaunitvaluation(x,p)=valuation of x with respect to pzidealstar(nf,I)=3-component vector v, giving the structure of (Z_K/I)^*. v[1] is  the order (i.e. phi(I)), v[2] is a vector of cyclic components, and v[3]  is a vector giving the corresponding generatorsleaving
geqMath::Pari::interface0Math::Pari::interface2091Math::Pari::interface22GD0,L,D mod \pmatrix{ run-away string. Closing itcouldn't find a suitable name for a tempdir (%s)COLUMNS%s fileabs(x): absolute value (or modulus) of xatanhbesselj(nu,x): J-bessel function of index nu and argument xbezoutGGbitnegimplydenominatorellan(e,n): computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<2^24 on a 32-bit machine)elleta(om): om=[om1,om2], returns the two-component row vector [eta1,eta2] of quasi-periods associated to [om1,om2]ellinit(x,{flag=0}): x being the vector [a1,a2,a3,a4,a6] defining the curve Y^2 + a1.XY + a3.Y = X^3 + a2.X^2 + a4.X + a6, gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,disc,j,[e1,e2,e3],w1,w2,eta1,eta2,area]. If the curve is defined over a p-adic field, the last six components are replaced by root,u^2,u,q,w,0. If optional flag is 1, omit them altogether. x can also be a string, in this case the coefficients of the curve with matching name are looked in the elldata database if available.ellpow(e,x,n): n times the point x on elliptic curve e (n in Z)lGDGgettime(): time (in milliseconds) since last call to gettimeGLD4,L,matimageGD0,L,DGpolsturm(x,{a},{b}): number of real roots of the polynomial x in the interval]a,b] (which are respectively taken to be -oo or +oo when omitted)polsubcyclo(n,d,{v=x}): finds an equation (in variable v) for the d-th degree subfields of Q(zeta_n). Output is a polynomial or a vector of polynomials is there are several such fields, or none.print(a): outputs a (in raw format) ending with newlineprint1(a): outputs a (in raw format) without ending with newlineqfbprimeformrnfidealnormrelrnflllgramrnfpolredabs(nf,pol,{flag=0}): given a pol with coefficients in nf, finds a relative simpler polynomial defining the same field. Binary digits of flag mean: 1: return also the element whose characteristic polynomial is the given polynomial, 2: return an absolute polynomial, 16: partial reductionsubstsum(X=a,b,expr,{x=0}): x plus the sum (X goes from a to b) of expression exprtheta(q,z): Jacobi sine theta-functionD"",r,DIDIvecsort(x,{k},{flag=0}): sorts the vector of vectors (or matrix) x in ascending order, according to the value of its k-th component if k is not omitted. Binary digits of flag (if present) mean: 1: indirect sorting, return the permutation instead of the permuted vector, 2: sort using lexicographic order, 4: use descending instead of ascending orderlllint_marked
Checking LLL basis...lllfp[2], kmax = %ldEnd of PHASE 1.

bnfinit: %s%s, not givenfor this ideal
m = %Z
polynomial not in Z[X] in galoisconj4galoisborne()conjugates list may be incomplete in nfgaloisconjGaloisConj: B=%Z
A4GaloisConj:I will test %ld permutations
GaloisConj:next p=%ld
GaloisConj:p=%ld deg=%ld fp=%ld
argument equal to one in zetaintegral transformnfhilbertpconstant term != 1 in direulerHensel lift (mod %Z^%ld)nfrootsQnon positive valuation in a series substitutiongtrunc; no non-trivial character in bnrL1CplxModulusWarning: failed tobug inthe PARI stack overflows !exponent (expo) overflow1, 6, 3, 4, 5, 2, 3(on)I was expecting an integer herecomposite modulus in Fl_sqrt: %lunegative nmax in sfcontsfcontlambda = %Z
choice of a prime idealTerminal does not define movesorry, too many block systems in nfsubfieldsgaffect (ghalf)plotbox(w,x2,y2): if the cursor is at position (x1,y1), draw a box with diagonal (x1,y1) and (x2,y2) in rectwindow w (cursor does not move)plotclip(w): clip the contents of the rectwindow to the bounding box (except strings)plotrecth(w,X=xmin,xmax,expr,{flags=0},{n=0}): plot graph(s) for expr in rectwindow w, where expr is scalar for a single non-parametric plot, and a vector otherwise. If plotting is parametric, its length should be even and pairs of entries give points coordinates. If not, all entries but the first are y-coordinates. Both flags and n are optional. Binary digits of flags mean: 1 parametric plot, 2 recursive plot, 4 do not rescale w, 8 omit x-axis, 16 omit y-axis, 32 omit frame, 64 do not join points, 128 plot both lines and points. n specifies the number of reference points on the graph (0=use default value). Returns a vector for the bounding boxpsdraw(list, {flag=0}): same as plotdraw, except that the output is a postscript program in psfile (pari.ps by default), and flag!=0 scales the plot from size of the current output device to the standard postscript plotting sizeznstar_conductor    lambda =   subgroup:ZM_inv donegauss_pivot_kerIndexPartial: factorizationpolcompositum0  beta = %Z
  ** switching to fast mode
nilordbound for resultant coeffs: 2^%ld
PL: proving primality of N = %Z
IFAC: (Partial fact.)Stop requested.
[caller of] elladd0	factor no. %ld was unique%s
IFAC: prime %Z
	appears with exponent = %ld
stroke showpage
%ld %ld m
S2A7
    Reordering of the roots: different variables in convolnot an invertible dirseries in dirdivggamdgamd of a power seriesBernoullisgsincosStep 1
reducing beta = %Z
not an Abelian extension in rnfnormgroupnon-positive bound in Discrayabslist       new rank: %ld
  z  = %Z
  z2 = %Z
orderell for nonrational elliptic curvestorsell (bug1)FRELSemirat. reduction: B0 -> %Z
thue (totally rational case)nfhermite, i = %ld
Step5: testing conditions lp
array index (%ld) out of allowed range curve not defined over a p-adic fieldalgtobasis(nf,x)=transforms the algebraic number x into a column vector on the integral basis nf[7]buchinitGD0,G,D0.1,G,D0.1,G,D5,G,pfactfq(x,p,a)=factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]gettime()=time (in milliseconds) since last call to gettimehell2(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using Tate's methodhermitemodididealadd(nf,x,y)=sum of two ideals x and y in the number field defined by nfidealaddoneidealhermite2idmat(n)=identity matrix of order n (n C-integer)intopen(X=a,b,s)=numerical integration of s from a to b with respect to X, where s has only limits at a or bisprincipal(bnf,x)=bnf being output by buchinit, gives the vector of exponents on the class group generators of x. In particular x is principal if and only if the result is the zero vectorkeri(x)=basis of the kernel of the matrix x with integer entrieslllnextprime(x)=smallest prime number>=xnfvalordred(x)=reduction of the polynomial x, staying in the same orderprodeuler(X=a,b,expr)=Euler product (X runs over the primes between a and b) of real or complex expressionprodinf1(X=a,expr)=infinite product (X goes from a to infinity) of real or complex 1+expressionnot the same number field in basistoalgratlift: must have 2*amax*bmax < m, found
	amax=%Z
	bmax=%Z
	m=%Z
arg1, arg2, arg3, arg4, arg0=0arg1, arg2, arg3, arg4, arg5stack size is %ld bytes (%d x %ld longs)
$$$Math::Pari::PARImatMath::Pari::interface27$$$;$Math::Pari::interface37Math::Pari::interface83Math::Pari::dumpStack(%c,varn=%ld,prec=%ld,valp=%ld):) ^{%ld}variable out of range in reorder  ###   user error: Matbesseli(nu,x): I-bessel function of index nu and argument xbesselk(nu,x,{flag=0}): K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type). flag is optional, and may be set to 0: default; 1: use hyperubitorGDGDGD0,L,changevar(x,y): change variables of x according to the vector yellbilellbil(e,z1,z2): canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsellminimalmodel(e,{&v}): return the standard minimal integral model of the rational elliptic curve e. Sets v to the corresponding change of variableserfc(x): complementary error functionerror("msg"): abort script with error message msgfactornf(x,t): factorization of the polynomial x over the number field defined by the polynomial tvV=GGIvVLLIforvecgettimeidealcoprime(nf,x,y): gives an element b in nf such that b. x is an integral ideal coprime to the integral ideal yideallistideleprincipal(nf,x): returns the principal idele generated by the algebraic number x in the number field nfifispowerlength(x): number of non code words in x, number of characters for a stringlindeplngamma(x): logarithm of the gamma function of xmatintersectnfinitnumbpartomega(x): number of distinct prime divisors of xpoldegreeqfgaussred(x): square reduction of the (symmetric) matrix x (returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)qfrepqfsign(x): signature of the symmetric matrix xrnfcharpolyGGGDnrnfdetrnfdiscrnfnormgroup(bnr,polrel): norm group (or Artin or Takagi group) corresponding to the Abelian extension of bnr.bnf defined by polrel, where the module corresponding to bnr is assumed to be a multiple of the conductor. The result is the HNF defining the norm group on the given generators in bnr[5][3]shift(x,n): shift x left n bits if n>=0, right -n bits if n<0.GVEthuetrap({err}, {rec}, {seq}): try to execute seq, trapping error err (all of them if err ommitted); sequence rec is executed if the error occurs and is the result of the command. When seq is omitted, define rec as a default handler for error err (a break loop will be started if rec omitted). If rec is the empty string "" pop out the last default handlerunknown signal (%ld)lllfp[1]**** Testing Different = %Z
    is %Z
initalg & rootsof1looking hard for %Z
vandermondeinverseA4GaloisConj:tau=%Z 
Trying degre %d.
GaloisConj:Testing %Ztwistpartialzeta (3), j = %ld/%ldincorrect a or b in intnumRgX_RgXQ_compopolsym of a negative nnot a prime number in polhenselliftbuilding treeLLL_cmbf: checking factor %ld
splitting mod p = %ldpseudorem dx = %ld >= %ld%simplify_iinvalid data in qfbevalqfi_unit_by_dischessmathnfspec with large entriesB = %Z
    hnflll done
No, we're done!
Modulus = %Z and subgroup = %Z
Checking the square-root of the Stark unit...
too many iterations for desired precision in integration routinenegative exponentdebugfilesPARIinputEND_contpromptisanypowerexponent overflow in regulaquadray: looking for [a,b] != unit mod 2f
[a,b] = nfissplit  1) T_2 bound for %s: %Z
X11Cannot reset output routines back...Gnuplot
ns = %ld
gtolistplot(X=a,b,expr,{ymin},{ymax}): crude plot of expression expr, X goes from a to b, with Y ranging from ymin to ymax. If ymin (resp. ymax) is not given, the minima (resp. the maxima) of the expression is used insteadplotdraw(list, {flag=0}): draw vector of rectwindows list at indicated x,y positions; list is a vector w1,x1,y1,w2,x2,y2,etc. . If flag!=0, x1, y1 etc. express fractions of the size of the current output deviceplotpointsize(w,size): change the "size" of following points in rectwindow w. w=-1 changes global valueplottermplotterm("termname"): set terminal to plot in high resolution to. Ignored by some drivers. In gnuplot driver possible terminals are the same as in gnuplot, terminal options can be put after the terminal name and space; terminal size can be put immediately after the name, as in "gif=300,200". If term is "?", lists possible values. Positive return value means successtoo many divisors (more than %ld)zero argument in factorinttoo many iterations in rootsold(): using roots2()
polished roots = %Z  alpha = %Z
mask too large in vecextractinseparable relative equation in rnfequationROUND2: epsilon = %ld	avma = %ld
FpXQ_sqrtlFpXQ_matrix_powsZX_resultantFF l-Gen:next %Z
ECM: number too small to justify this stage
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[2] ifac_decompnon-existent factor class in ifac_mainIFAC: factor %Z
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 (so far)...GaloisIndex: Using hash value w=%ld
%s %%!
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galois of reducible polynomialrootsof1 (bug1)beta reduced via ell-th root = %Z
not an integral modelMPQS: got %ld factors%s
MPQS: decomposed a square
epsilon_3 -> %Z
  B0  = %Z
incorrect ideal in idealtypideals don't sum to Z_K in idealaddmultooneincompatible variables in idealinvbnfbreak not allowed in O()algtobasisbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithmbernreal(x)=Bernoulli number B_x, as a real number with the current precisionbuchimag(D,...)=compute the structure of the class group of the complex quadratic field of discriminant D<0. See manual for the other parameters (which can be omitted)det2(x)=determinant of the matrix x (better for integer entries)discrayabslistlongdiscrayrelcondeulerfactcantorfor(X=a,b,seq)=the sequence is evaluated, X going from a up to bidealchinese(nf,x,y)=x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pideallist(nf,bound)=vector of vectors of all ideals of norm<=bound in nfindsortinitalgred2intinfisprincipalforceisunit(bnf,x)=bnf being output by buchinit, gives the vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisekerint1(x)=LLL-reduced Z-basis of the kernel of the matrix x with rational entries using matrixqz3 and the HNFlambdak(nfz,s)=Dedekind lambda function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)lllgram(x)=lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)lllgram1lllgramkerim(x)=kernel and lll reduction of the lattice whose gram matrix is the integral matrix xlllkerimgenmakebigbnf(sbnf)=transforms small sbnf as output by smallbuchinit into a true big bnfminim2munfreducepnqn(x)=[p_n,p_{n-1};q_n,q_{n-1}] corresponding to the continued fraction xpolred2powrealrawqfr(a,b,c,d)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c>0 and distance dredimag(x)=reduction of the binary quadratic form x with D<0redreal(x)=reduction of the binary quadratic form x with D>0round(x)=take the nearest integer to all the coefficients of xsmallbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmith2(x)=gives a three element vector [u,v,d] where u and v are square unimodular matrices such that d=u*x*v=diagonal(smith(x))sqr(x)=square of x. NOT identical to x*xsqrt(x)=square root of xznstar(n)=3-component vector v, giving the structure of (Z/nZ)^*. v[1] is  the order (i.e. phi(n)), v[2] is a vector of cyclic components, and v[3]  is a vector giving the corresponding generatorselement_mulidNaN or Infinity in dbltorլլ*6BNnzNNQ8l/8lllsOfWyWDyyEEvv]\5;N\\p\hB%%XBBBBA:Rf:<<PP)<<#b#b#$$"$$#s#8#$$##$$"V;V;VU;VU;VU;VU;V;VU]]]W^W^W^W^]%gd&b&b&bdfhEh&b&bd%gdd&b&b&b&b&b&b&b&b&bd&b&b&bh%gdddd&b&b&bFgFg&bd&bddhhhhhghhhhhwwwwwwwwwxwxWWXX2XXXXXpYYYYYZYYZ[S(%v,d%*{11{1111111111111J(H(s(z((((^L[5Sq3;M&^G>@@@/u/[/////////f)8888R5KVxwKKKKKKKKKnnn
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