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	bmax=%Z
ratilft: amax must be >= 0, found
	amax=%Z
ratlift: must have 2*amax*bmax < m, found
	amax=%Z
	bmax=%Z
	m=%Z
ratlift: must have 0 <= x < m, found
	x=%Z
	m=%Z
ratlift failed to catch d1 == 0


	Coverflow in real shiftFlx_to_FlvFlxqX_safegcdnon invertible polynomial in Flxq_invzero discriminant in %ssquare discriminant in %spositive discriminant in %snegative discriminant in %sreducible form in qfr_rhoreducible form in qfr_initcornacchiad must be positived must be 0 or 3 mod 4quadpolynegative definite t_QFIzero discriminant in Qfbqfbredcompositionredimagnot a t_QFI in nuduplnot a t_QFI in nucompqfi_unit_by_discqfi_unitnot a t_QFI in powimagqfr_unit_by_discqfr_unitnot a t_QFR in powrealrawredimagsl2qfbsolvediscriminant not congruent to 0,1 mod 4 in %sdifferent discriminants in qfb_compdiscriminant not congruent to 0,1 mod 4 in primeformShanks distance must be a t_REAL in qfrnot a real quadratic form in redrealnot an integer exponent in nupownot a t_REAL in 4th component of a t_QFRRgX_powersRgX_RgXQ_componormalizing a polynomial with 0 leading termnegative size in fill_scalcolincorrect length in sumgauss_pivot. k=%ld, n=%ldgauss_pivot_ker. k=%ld, n=%lddet. col = %lddet, col %ld / %ldmatdet* [mod p]FlxqM_kerFpM_kerFqM_kerFpM_invimagegauss_pivot_kerkeridetint. k=%ldnegative size in matid_Flmhnf_invimagehnfdividegaddmatmattodiagonalmatmuldiagonalnegative size in fill_scalmatmatmultodiagonalmask too large in vecextractincorrect range in extractincorrect mask in vecextractFpM_gauss_pivotgtransmatextractshallowtransempty matrix in supplFqM_gauss_pivotimage2matimageinverse mod %ld (stable=%ld)ZM_invZM_inv donegaussFpM_gauss. i=%ldincorrect object in diagonalempty matrix in deplingaussmoduloimpossible concatenation: %s %Z . %s %Zmissing eigenspace. Compute the matrix to higher accuracy, then restart eigen at the current precisionnot an integer matrix in detintincorrect vector in matmuldiagonalno such component in vecextractSolving the triangular system
Entering gauss with inexact=%ld
trying to concat elements of an empty vectormathnfQuickNormL1gnorml1sqred1assmatgconjgtracegnormgnorml2hess, m = %ldsqred2conjvecsmithcleangsmithallPermutation: %Z
matgen = %Z
hnfallhnfall[1], li = %ldhnfall[2], li = %ld
hnfall, final phase: hnfall[3], j = %ldhnfpermhnflllhnflll (reducing), i = %ldhnflll, k = %ld / %ldEntering hnffinal:
dep = %Z
mit = %Z
    hnflll done
hnffinal, i = %ldLeaving hnffinal
mit = %Z
B = %Z
C = %Z
    1st phase done
    2nd phase done
hnfadd (%ld + %ld)H = %Z
C = %Z
Entering hnfspec
    after phase1:
hnfspec[1]hnfspec[2]    after phase2:
hnfspec[3], (i,j) = %ld,%ldhnf_specialhnf_special[1]. i=%ldhnf_special[2]. i=%ldhnf[1]. i=%ldhnf[2]. i=%ldmathnfspec with large entriesmatrixqzsqred3easycharincorrect variable in caradjmatrixqz_auxmatrixqz3matrixqz2matrixqz0extendedgcdaccuracy lost in matfrobeniusallhnfmodallhnfmod[1]. i=%ldallhnfmod[2]. i=%ldstarting SNF loop
i = %ld: [1]: smithall i = %ld[2]: smithall, i = %ld[3]: smithallmatsnfminpolyqqq/0000/5000111/1///111|2"3"3|2"3|2H3|2S33H4H4|24|2|2|24425Z6{656565670707507555555G)IG)I)IGHGH<HGGGGGGGbGbGGnot a positive definite matrix in sqred1incompatible field degrees in conjvecnot a rational polynomial in conjvec    first pass in hnffinal done
    matb cleaned up (using Id block)
hnfspec [%ld x %ld] --> [%ld x %ld]incompatible matrices in hnf_specialmore rows than columns in matrixqzmatrix of non-maximal rank in matrixqznot a rational matrix in matrixqzmatrixqz when the first 2 dets are zeronon coprime ideals in hnfmergenb lines > nb columns in hnfmodnon integral matrix in smithallvariable must have higher priority in matfrobenius?333333?insufficient precision for p = 2 in hilbertcomposite modulus in Fl_sqrt: %luincorrect bound type in bestapprforbidden or incompatible types in hildiscriminant too large in classnodiscriminant too big in classnolist of numerators too short in sfcontf2negative argument in factorial functionintegral part not significant in sfcontlarge exponent in Mod(a,N)^n: reduce n mod phi(N)composite modulus in Fp_sqrt: %Zispower for non-rational argumentsprimitive root mod 2^%ld does not existprimitive root mod %Z does not existnot an element of (Z/nZ)* in ordernegative integer in sqrtintassociationbestapprp = 1 in hilbert()isanypowerisanypower: now k=%ld, x=%Z
exponent overflow in regulaclassno2classno with too small ordersfcont2missing exponentbestappr0Z_issquaregisprimeqfbclassnofundunitincorrect size in pnqnnegative nmax in sfcontcontfrac01/0 exponent in Fp_sqrtnFp_sqrtlnot a prime in Fp_sqrtispowerzero modulus in znprimroot]mo3p]mIn]mm]m]m]mmm]mm]m]m]mmmm1l=11"ygC^p77777!777777pRh\~t  !t  !t t !t t !t t W!Gz?E,W@@Too large primelimitremoveprimeprime %Z is not in primetableIFAC: Stop: remaining %Z
panic: set_optimizebitwise xorbitwise negated implybitwise andbitwise orprimepibinaryaddprimecan't accept 0 in addprimesbitwise negationzero argument in factorintdivisorsIFAC: Stop: Primary factor: %Z
zero argument in an arithmetic functionn-th prime meaningless if n = %ldnegative exponent in bitwise negationIFAC: (Partial fact.) Initial stop requested.
denominators not allowed in divisorstoo many divisors (more than %ld)q=
ףp???p=
ף@?@@j\P|ZX

tTmTmbmsmmCm	get_bnfpolpolynomial not in Z[X] in %sincorrect prime idealincorrect bigidealincorrect idealincorrect matrix for idealplease apply nfinit firstchecknfplease apply bnfinit firstcheckbnfmissing units in %sincorrect bigray fieldcheckrnffalse nf in nf_get_r2i0 = %ld
get_pol  generator: %Z
gpolcomp (different degrees)nsiso0nfiso or nfinclgzetakalls = 1 is a pole (gzetakall)s = 0 is a pole (gzetakall)
initzeta: N0 = %Z
false nf in nf_get_r1round4false nf in nf_get_signgalois (bug1)galois (bug3)galois (bug2)galois (bug4)a(i,j)coefa(n)log(n)Ciktoo many terms in dirzetakmatrix Mchk_gen_init: generator %Z
chk_gen_init: subfield %Z
polredabs (precision problem)incorrect nf in nfnewprecget_red_GLLL basismult. tablexbest = %Z
ordredrootsof1 (bug1)polredabs0S1S2A3S3C(4) = 4E(4) = 2[x]2D(4)A4S4C(5) = 5D(5) = 5:2F(5) = 5:4A5S5C(6) = 6 = 3[x]2D_6(6) = [3]2D(6) = S(3)[x]2A_4(6) = [2^2]3F_18(6) = [3^2]2 = 3 wr 22A_4(6) = [2^3]3 = 2 wr 3S_4(6d) = [2^2]S(3)S_4(6c) = 1/2[2^3]S(3)F_18(6):2 = [1/2.S(3)^2]2F_36(6) = 1/2[S(3)^2]2L(6) = PSL(2,5) = A_5(6)L(6):2 = PGL(2,5) = S_5(6)A6S6C(7) = 7D(7) = 7:2F_21(7) = 7:3F_42(7) = 7:6L(7) = L(3,2)A7S7polynomial not in Z[X,Y] in %splease apply bnrinit(,,1) and not bnrinit(,)incompatible modulus in %s:
  mod = %Z,
  nf  = %Znot a zeta number field in zetakallneed %Z coefficients in initzeta: computation impossibleincorrect galois automorphism in galoisapplyTschirnhaus transform. New pol: %Zgalois of degree higher than 11galois of reducible polynomialnot an integer type in dirzetakchk_gen_init: difficult field, trying random elements
precision too low in chk_gen_initchk_gen_init: skipfirst = %ld
chk_gen_init: new prec = %ld (initially %ld)
get_red_G: starting LLL, prec = %ld (%ld + %ld)
polred for non-monic polynomialnon-monic polynomial. Result of the form [nf,c]you found a counter-example to a conjecture, please report!Found %ld minimal polynomials.
nf_ADDZK flag when nf_ALL set (polredabs)2S_4(6) = [2^3]S(3) = 2 wr S(3)F_36(6):2 = [S(3)^2]2 = S(3) wr 2^Q?ɴK?ɴK?x2?9B.???primedec: %Z is not primernflllgramrnfdet2not a pseudo-matrix in rnfdetincorrect modpr formatget_normrnf functionallbasedisc. factorisationnf_to_ffrowred j=%ldnewtonsums  f = %Z,
  a = %Z
  new order: %Z
IndexPartial: discriminantIndexPartial: factorizationIndexPartial: factor %Z^%ld --> %Z : Treating p^k = %Z^%ld
polcompositum0compositum: %Z inseparablek =   beta = %Z
  ** switching to fast mode
 content in fastnu is %Z
  fastnu: G is computed
  fastnu: HNF(G) is computed
  (eq,er) = (%ld,%ld)
  Increasing Fa
  Increasing Ea
nilord  entering Nilord with parameters: %Z^%ld
  fx = %Z, gx = %Z  (Fa, Ea) = (%ld,%ld)
  dedek: gcd has degree %ld
  entering Decomp, parameters: %Z^%ld
  f = %Zimpossible inverse: %ZResult for prime %Z is:
%Z
rnfdedekindIdeals to consider:
 treating %Z
    pass no %ld
 new order:
%Z
%Z
rnfordmaxrnfpolredabsrelative basis computed
absolute basisrnfpolrednot a pseudo-matrix in %srnfisfreernfbasisrnfsteinitzincorrect polynomial in rnf functionincorrect coeff in rnf functionnon-monic relative polynomialsinseparable relative equation in rnfequationreducible polynomial in allbasemodpr initialized for integers only!not a pseudo-basis in nfsimplifybasisnewtonsums: result doesn't fit in cache
  entering Dedekind Basis with parameters p=%Z
ROUND2: epsilon = %ld	avma = %ld
not the same variable in compositum  ** switching to normal mode
no root in nilord. Is p = %Z a prime?initial parameters p=%Z,
  f=%Z
bug in Decomp (not a factor), is p = %Z a prime?  leaving Decomp: f1 = %Z
f2 = %Z
e = %Z
de= %Z
not a factorisation in nfbasisthis combination of flags in rnfpolredabsoriginal absolute generator: %Z
reduced absolute generator: %Z
%Z not a nfeltelement_muliarch_to_permideallistarchnot a matrix in matbasistoalgelement_mulidmodule too large in Fp_shanksFp_shanks, k = %ldmodule too large in ffshankszero element in zsignenot a matrix in matalgtobasiselement_sqrnfmulelement_mulalgtobasis_ielement_invelement_powelement_divnfdiventering zlog, with a = %Z
leaving
element_invmodidealnot an element in zideallogtreating pr^%ld, pr = %Z
  treating a = %ld, b = %ld
zidealij done
Ideallistideallistnot the same number field in basistoalgPohlig-Hellman: DL mod %Z^%ld
not an element of (Z/pZ)* in znloga not invertible in ff_PHlog_Fpnf_Pohlig-Hellman: DL mod %Z^%ld
not the same number field in rnfalgtobasisnot the same number field in algtobasisincompatible variables in algtobasisnot an integer exponent in nfpownegative power in element_powid_mod_pincorrect archimedean component in IdealstarIdealstar needs an integral non-zero ideal: %Z?
?


+
JrMJJrMJJJJ\M\MJJJJJJJKLnot a module in %snot a matrix in %sincorrect ideal in idealtypnfsolvemodprnfkermodprget_archunif_mod_fZnot a vector in idealred0 in get_arch_realprincipalidealidealnormidealvalzero ideal in idealfactorideal_two_eltcannot invert zero idealidealinvidealdivnfhermitemod[1]: nfhermitemod, i = %ld[2]: nfhermitemod, i = %ldnot a module in nfsmithnot a matrix in nfsmithbug2 in nfsmithnfhermitenfhermite, i = %ldnfdetintnot a correct ideal list in %sgeneric conversion to finite field0th power in idealpowprime_specincorrect vector length in idealrednot a vector of ideals in idealaddmultooneideals don't sum to Z_K in idealaddmultoonenon-integral exponent in idealpownot a prime ideal factorization in idealchinesenot a suitable vector of elements in idealchinesenored + denominator in idealapprfactelement not in ideal in ideal_two_elt2element does not belong to ideal in ideal_two_elt2not a factorization in idealapprfactincompatible variables in idealinvquotient not integral in idealdivexactincorrect idele in idealaddtoonenon-integral exponent in idealpowrednot a correct ideal list in nfsmithnot a matrix of maximal rank in nfsmithnfsmith for non square matricesnot a matrix of maximal rank in nfhermite::::::::::::::EEEEEEEEEEEEEE_please apply rnfequation(,,1)incorrect data in eltreltoabsrnfinitalgrnfidealabstorelrnfidealhermiteelement is not in the base field in rnfelementdownmain variable must be of higher priority in rnfinitalgsmallvectors looking for norm < %Z
time [max,t12,loop,reds,fin] = [%ld, %ld, %ld, %ld, %ld]
npass = %ld, red. last time = %ld, log_2(det) ~ %ld

maximal number of vectors must be providednegative number of vectors in minim0
...LLL reducing precision to %ld

Recomputing Gram-Schmidt, kmax = %ld
inconsistent primes in plindepnot a p-adic vector in plindepnegative polynomial degree in algdephigher degree than expected in algdepnegative bound in zncoppersmithdelta = %d, t = %d, cond = %lf
Init: trying delta = %d, t = %d
Entering LLL
bitsize bound: %ld
expected shvector bitsize: %ld
Increasing dim, delta = %d t = %d
Fincke-Pohst, final LLL: prec = %ld
smallvectorsNew bound: %Zsorting...
final sort & check...
time for ct = %ld : %ld
lllintpartialtm1 = %Zmid = %Zlllintpartialalllllintlllint_markedK%ld  (%ld)lllint[1], kmax = %ldlllint[2], kmax = %ldbound = 0 in minim2not a definite form in minim0adding vector = %Z
vector in new basis = %Z
base change matrix =
minim0, rank>=%ldlllgramallgenincrementalGSgenlindepqzer[%ld]=%ld
pslqInitialization time = %ld
pslqL2 K%ldlllfp[1]dependent vectors in lllfplllfp (exact)count_max = %ld
lllfp giving upk =lllfp[1], kmax = %ld
Checking LLL basis...in precision %ld

Checking LLL basis
lllfp[2], kmax = %ldqflllqflllgrammatkerintnegative accuracy in lindep2algdep0zncoppersmithzero polynomial forbiddenModified P: %Z
bound too largeMatrix to be reduced:
%Z
Candidate: %Z
bitsize Norm: %ld
bitsize bound: %ld
Roots: %Z
dimension 0 in fincke_pohstfirst LLL: prec = %ld
qfminimd]]ХܥǥLXajCffffff?-C6?RQ??@I@yPD?not a set in setsearchnegative degree in legendreinvalid bound in randomdirmulreverse polmod does not existgen_sortincorrect lextype in vecsortnegative index in vecsortindex too large in vecsortnot a polmod in modreversenot a vector in permtonumnot a set in setminusnot a set in setintersectnot a set in setunionnot vectors in polinterpolateno data in polinterpolatebinomialpolrecipprecision<=0 in gprecnot a series in convoldifferent variables in convolnot a series in laplacenegative valuation in laplacedoubling stack in dirmul
argument must be positive in polcyclonot an invertible dirseries in dirdivn too small (%ld) in numtopermtwo abcissas are equal in polintdifferent lengths in polinterpolategqy	O?
rel = %ld^%ld 
*** Bach constant: %f
qrf5_rho_powcglob = %ld. regulator is zero.
#### Tentative regulator: %Z
initialreal_relations %ldPbe honestpowsubFBquadbuchquadfactor baseFB = %Z
subFBquad (%ld elt.)*** Changing sub factor base
KC = %ld, need %ld relations
...need %ld more relations
smith/class groupnarrow class groupform_to_idealclass number = %ldquadhilbert (pq)p = %lu, q = %lu, e = %ld
product, error bits = %ldquadhilbertimagincorrect data in findquad[%ld,%ld] lambda = %Z
get_lambdaquadraycomputeP2
Time %s rel [#rel/#test = %ld/%ld]: %ld
sorry, couldn't deal with this field. PLEASE REPORT[quadhilbert] incorrect values in pq: %lube honest for primes from %ld to %ld
Bach constant <= 0 in buchquad
#### Tentative class number: %Z
incorrect parameters in quadclassunitquadhilbertimag (can't find p,q)quadray: looking for [a,b] != unit mod 2f
[a,b] = not a polynomial of degree 2 in quadrayquadray needs a fundamental discriminantnot a polynomial of degree 2 in quadhilbertquadhilbert needs a fundamental discriminant??&DT!	@LXz??@?
m = %Z
base change =
codeprime
#### Computing check
truncation error in bestappr
D = %Z
den = %Z
bestappr/regulator
 ***** check = %f
v[%ld]=%.4g BOUND = %.4g
for this idealsmall norm relationssub factorbase (%ld elements)looking hard for %Z
relation cancelled: (jid=%ld,jdir=%ld)for this relationred_mod_unitsfundamental units too largebnfinit: %s%s, not givengetfuincorrect big number fieldclassgroup generatorsbnfnewprec# ideals tried = %ld
bnfmakepowFBgeninitalg & rootsof1weighted G matricesBach constant <= 0 in buchR1 = %ld, R2 = %ld
D = %Z
LIMC = %ld, LIMC2 = %ld
++ LV[%ld] = %ZKCZ = %ld, KC = %ld, n = %ld
compute_Rbuchall (%s)cleanarchclassgroupallbnfinitbnfclassunitzero ideal in isprincipal*%ld makematal**** Testing Different = %Z
     is %Z
*** p = %lu
  Testing P = %Z
    Norm(P) > Zimmert bound
    #%ld in factor base
End of PHASE 1.


#### Tentative regulator : %Z

#### Looking for %ld relations (small norms)

*** Ideal no %ld: [%Z, %Z, %Z, %Z]
small_norm (precision too low)  small norms gave %ld relations.
  nb. fact./nb. small norm = %ld/%ld = %.3f
Be honest for %ld primes from %ld to %ld
be_honest() failure on prime %Z

#### Computing regulator multiple

(more relations needed: %ld)

++++ cglob = %ld: new relation (need %ld)insufficient precision for fundamental unitsunknown problem with fundamental units
#### Computing fundamental units
not a vector/matrix in cleanarchnot a factorization matrix in isunitnot an algebraic number in isunit
#### Computing class group generators
SPLIT: increasing factor base [%ld]
Computing powers for subFB: %Z
non-monic polynomial. Change of variables discarded########## FACTORBASE ##########

KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld

#### Looking for random relations
incorrect parameters in classgroupinsufficient precision for generators, not givencompleting bnf (building matal)completing bnf (building cycgen)isprincipal (incompatible bnf generators)precision too low for generators, e = %ldprecision too low for generators, not givenPHASE 1: check primes to Zimmert bound = %lu

*m mmllll??UUUUUU? @P@zo?ư>&DT!@+m0_?(@4@@333333?Lbnrclassnoincorrect subgroup in %swrong type in too_bigbnrclassnolist  *** testing p = %lu
     p divides h(K)
     p divides w(K)
     Beta list = %Z
       prime ideal Q: %Z
       new rank: %ld
bnfcertify   BOUND = %ld
M* = %Z
pol = %Z
old method: y = %Z, M0 = %Z
[ %ld, %ld, %ld ]: %Z
bnrdiscrayfactordivexact is not exact!discrayabslistMinkowski bound is too large  Testing primes | h(K)

isprincipalrayrnfnormgroupr1>15 in discrayabslistarch[1]: discrayabslistarch[2]: discrayabslistarchBuchrayrnfconductorbnrinitbnrclassDefault bound for regulator: 0.2
not a factorisation in decodemoduleincorrect hash code in decodemodule       generator of (Zk/Q)^*: %Z
       column #%ld of the matrix log(b_j/Q): %Z
Searching minimum of T2-form on units:
(lower bound for regulator) M = %Z
large Minkowski bound: certification will be VERY longMahler bound for regulator: %Z
sorry, too many primes to check
PHASE 2: are all primes good ?

  Testing primes <= B (= %lu)

please apply bnrinit(,,1) and not bnrinit(,,0)not an Abelian extension in rnfnormgroup?non Galois extension in rnfnormgroupnot an Abelian extension in rnfnormgroupnon-positive bound in DiscrayabslistStarting zidealstarunits computations

Starting bnrclassno computations
Starting discrayabs computations
incorrect character length in KerCharneither bnf nor bnr in conductor or discraybad subgroup in conductor or discrayףۤD;vq
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ó9@ЗE8@bG8@RK)7@xT6@/%06@tHb5@8:4@x *4@;_ <@qVqf;@<\iD:@c9@H*@9@֫而8@RI97@HPW(7@0(x6@uIh&5@Vl5@rZ|
?{Gz?qpsolublezpsolublebnfissunitnon monic relative equationrnfisnorminitzpsolublenfqpsolublenf0 argument in nfhilbertp0 argument in nfhilbertbnfsunitrnfisnormmain variable must be of higher priority in rnfisnorminitnfhilbert not soluble at real place %ld
nfhilbert not soluble at finite place: %Z
please apply rnfisnorminit firstuseless flag in rnfisnorm: the extension is Galoisplease apply galoisinit firstgaloisconj2polconjugate %ld: %Z
galoisanalysis()FixedField: Size: %ldx%ld
FixedField: Weight: %Z
FixedField: Sym: %Z
FixedField: Found: %Z
%d.GaloisConj:Entree Init Test
GaloisConj:Sortie Init Test
frobenius powerf=%Z
 borne=%Z
 l-borne=%Z
s4test()monomorphismlift()vandermondeinverseGaloisConj:val1=%ld val2=%ld
GaloisConj: Bound %Z
GaloisFixedField:cosets=%Z 
galoisfixedfieldgaloispermtopolTrying degre %d.
Galoisconj:Subgroups list:%Z
GaloisConj:Testing %ZBest lift: %d
GaloisConj:next p=%ld
S4GaloisConj:sigma=%Z
S4GaloisConj:pj=%Z
A4GaloisConj:sigma=%Z 
A4GaloisConj:tau=%Z 
A4GaloisConj:orb=%Z 
A4GaloisConj:O=%Z 
galoisisabelianGaloisConj:denominator:%Z
GaloisConj:Testing A4 first
GaloisConj:Testing S4 first
GaloisConj:Orbite:%Z
GaloisConj:Frobenius:%Z
GaloisConj:Back to Earth:%Z
GaloisConj: B=%Z
%d%% testpermutation(%Z)GaloisConj:Fini!
galoisconj4galoisborne()rootpadicfast()vandermondeinversemod()GaloisConj:%Z
%d Calcul polynomesnfgaloisconjGaloisConj: Fixed field %Z
Not a Galois field in a Galois related functionincorrect denominator in initgaloisborne: %ZPolynomial not squarefree in galoisinitGaloisAnalysis:non Galois for p=%ld
GaloisAnalysis:Nbtest=%ld,p=%ld,o=%ld,n_o=%d,best p=%ld,ord=%ld,k=%ld
Galois group almost certainly not weakly super solvableGaloisAnalysis:p=%ld l=%ld group=%ld deg=%ld ord=%ld
incorrect permutation in permtopolp too small in fixedfieldsympolGaloisConj: Solution too large, discard it.
MonomorphismLift: trying early solution %Z
MonomorphismLift: true early solution.
MonomorphismLift: false early solution.
MonomorphismLift: lift to prec %dGaloisFixedField:den=%Z mod=%Z 
GaloisConj:increase prec of p-adic roots of %ld.
priority of optional variable too high in galoisfixedfieldNumberOfConjugates:Nbtest=%ld,card=%ld,p=%ld
NumberOfConjugates:card=%ld,p=%ld
GaloisConj:p=%ld deg=%ld fp=%ld
GaloisConj:I will try %Z permutations
Combinatorics too hard : would need %Z tests!
I will skip it, but it may induce an infinite loopGaloisConj: %d hops on %Z tests
GaloisConj: not found, %d hops 
galoisconj _may_ hang up for this polynomialS4GaloisConj:Computing isomorphisms %d:%Z
S4GaloisConj:Testing %d/3:%d/4:%d/4:%d/4:%Z
S4GaloisConj:Testing %d/3:%d/2:%d/2:%d/4:%Z:%Z
S4GaloisConj:Testing %d/8 %d:%d:%d
A4GaloisConj:I will test %ld permutations
A4GaloisConj: %ld hop sur %ld iterations
A4GaloisConj:%ld hop sur %d iterations max
GaloisConj: G[%d]=%Z of relative order %d
GaloisConj: exp %d: s=%ld [%ld] a=%ld w=%ld wg=%ld sr=%ld
Combinatorics too hard : would need %Z tests!
 I'll skip it but you will get a partial result...GaloisConj:%d hop sur %Z iterations
polynomial not in Z[X] in galoisconj4non-monic polynomial in galoisconj4Second arg. must be integer in galoisconj4field not Galois or Galois group not weakly super solvableconjugates list may be incomplete in nfgaloisconj$08<HKP`lx6,@Y@different modulus in ff_poltypedifferent pointers in ff_poltypenormalizing a series with 0 leading termincompatible variables in gredgmulsggmul2nʹ#_*߱-]ѵ;{ȲӺ[
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|	|		gsignelistkillbad index in listinsertno more room in this listlistputgexpogtolongnormalizepolcomparisonp = 1 in Z_lvalreml <= 2 in greffegtofpempty vector in vecminempty vector in vecmaxabs is not meromorphic at 0gabsnegative length in listcreatelistconcatgtolistlistsortforbidden divisor %Z in ggvalnormalizegaffect (gen_0)gaffect (gen_1)gaffect (gen_m1)gaffect (gen_2)gaffect (gnil)gaffect (gpi)gaffect (geuler)gaffect (ghalf)gaffect (gi)gaffect (pol_1/pol_x)matsizegcvtopcvtop2negative index (%ld) in listputno more room in this L (size %ld)forbidden or conflicting type in gvalnot an integer modulus in cvtop




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ۭͮ..4ìì!б4бббббббvʷʷʷVʷ2ʷʷƷ+Y+,L~~~~ooolift_internnonexistent componentgvargpolvarinvalid data in qfbevalinvalid vector in qfbevalpolevalpoleval: i = %ldiscomplexgdivmodpadicprecisintinvalid data in hqfevalinvalid vector in hqfevalinvalid data in qfevalinvalid vector in qfeval%simplify_iinv_sergreal/gimagcenterliftpolcoeffvectosmallgtruncgfloor2ngrndtoigroundgceilgfloor\evaluation of a power seriesgevalnumergtopolynon-positive argument in O()zero argument in O()incorrect object in O()incorrect argument in O()deriva log appears in intformalinteggtosergsubstnot a series in serreversenot a variable in substvecpolleadthis object is a leaf. It has no componentsinvalid quadratic form in qfbevalnot the same prime in padicprecinvalid quadratic form in hqfevalinvalid quadratic form in qfevalinvalid data in qf_base_changeinvalid base change matrix in qf_base_changeincorrect permtuation to inversenon existent component in truecoeffnonexistent component in truecoeffvariable must have higher priority in gtopolyt_SER with negative valuation in gtopolya log/atan appears in intformalmain variable must have higher priority in gtoserforbidden substitution by a non square matrixforbidden substitution by a vectorforbidden substitution in a scalar typenon positive valuation in a series substitutionnon polynomial or series type substituted in a seriesvaluation not equal to 1 in serreversedifferent number of variables and values in substvecsubst: unexpected variable precedenceGyPD3@factor has NULL exponent in ifac_findRho: time = %6ld ms,	%3ld round%s
snextpr: %lu != prc210_rp[%ld] mod 210
snextpr: %lu should have been prime but isn't
snextpr: integer wraparound after prime %lu
SQUFOF: found factor %ld from ambiguous form
	after %ld steps on the ambiguous cycle, time = %ld ms
SQUFOF: ...found nothing on the ambiguous cycle
	after %ld steps there, time = %ld ms
SQUFOF: squfof_ambig returned %ld
OddPwrs: passed modular checks
IFAC: new partial factorization structure (%ld slots)
 power?
	modulo: resid. (remaining possibilities)
	   %3ld:  %3ld   (3rd %ld, 5th %ld, 7th %ld)
	But it nevertheless wasn't a %ld%s power.
found factor
	%Z
currently lost to the factoring machineryavoiding nonexistent factors in ifac_whoiswhoIFAC: factor %Z
	is prime (no larger composite)
IFAC: prime %Z
	appears with exponent = %ld
Rho: searching small factor of %ld-bit integer
Rho: restarting for remaining rounds...
Rho: using X^2%+1ld for up to %ld rounds of 32 iterations
Rho: time = %6ld ms,	Pollard-Brent giving up.
Rho: fast forward phase (%ld rounds of 64)...
Rho: time = %6ld ms,	%3ld rounds, back to normal mode
Rho: hang on a second, we got something here...
	found factors = %Z, %Z,
	and %Z
ECM: number too small to justify this stage
ECM: working on %ld curves at a time; initializingECM: stack tight, using heap space
ECM: time = %6ld ms
ECM: dsn = %2ld,	B1 = %4lu,ECM: time = %6ld ms, B1 phase done, ECM: %lu should have been prime but isn't
	(got [p]Q, p = %lu = prc210_rp[%ld] mod 210)
ECM: time = %6ld ms, entering B2 phase, p = %lu
ECM: finishing curves %ld...%ld
	(extracted precomputed helix / baby step entries)
ECM: time = %6ld ms,	ellfacteur giving up.
ECM: time = %6ld ms,	p <= %6lu,
	found factor = %Z
Miller-Rabin: testing base %ld
PL: proving primality of N = %Z
False prime number %Z in plisprimesqufof [caller of] (n or 3n is a square)squfof [caller of] (5n is a square)SQUFOF: entering main loop with forms
	(1, %ld, %ld) and (1, %ld, %ld)
	of discriminants
	%Z and %Z, respectively
SQUFOF: blacklisting a = %ld on first cycle
SQUFOF: blacklisting a = %ld on second cycle
SQUFOF: first cycle exhausted after %ld iterations,
	dropping it
SQUFOF: square form (%ld^2, %ld, %ld) on first cycle
	after %ld iterations, time = %ld ms
SQUFOF: ...but the root form seems to be on the principal cycle
SQUFOF: second cycle exhausted after %ld iterations,
	dropping it
SQUFOF: square form (%ld^2, %ld, %ld) on second cycle
	after %ld iterations, time = %ld ms
SQUFOF: giving up, time = %ld ms
OddPwrs: testing for exponent %ld
IFAC: checking for pure square
IFAC: trying Pollard-Brent rho method
IFAC: trying Shanks' SQUFOF, will fail silently if input
      is too large for it.
IFAC: trying Lenstra-Montgomery ECM
IFAC: forcing ECM, may take some time
IFAC: unfactored composite declared primeIFAC: untested integer declared primeIFAC: incorporating set of %ld factor(s)
	stored (largest) factor no. %ld...
	factor no. %ld is a duplicate%s
	yielded `factor' %Z
	which isn't!
ifac_crack [Z_issquarerem miss]IFAC: main loop: repeated old factor
	%Z
IFAC: main loop: repeated new factor
	%Z
IFAC: a factor was a power of another prime factor
IFAC: a factor was divisible by another prime factor,
	leaving a cofactor = %Z
IFAC: prime %Z
	appears at least to the power %ld
IFAC: main loop: another factor was divisible by
	%Z
partial impossibly short in ifac_sort_one`*where' out of bounds in ifac_sort_one`washere' out of bounds in ifac_sort_onemisaligned partial detected in ifac_sort_oneIFAC: repeated factor %Z
	detected in ifac_sort_one
composite equals prime in ifac_sort_oneprime equals composite in ifac_sort_onenon-existent factor class in ifac_mainIFAC: after main loop: repeated old factor
	%Z
IFAC: main loop: %ld factor%s left
IFAC: main loop: this was the last factor
IFAC: (Partial fact.)Stop requested.
IFAC: found %ld large prime (power) factor%s.
[caller of] snextpr	checking modulo %ld
	- ruled out
	But it wasn't a pure power.
OddPwrs: is %Z
	...a 3rd%s, or 5th%s 7thmiller(rabin)LucasModcompositeIFAC: factor %Z
	is %s
	found factor = %Z
	Pollard-Brent failed.
composite 	found %sfactor = %Z
[caller of] elladd0 for one round for up to %ld rounds	B2 = %6lu,	gss = %4ld*420
p = %lu, setting up for B2
	(got [2]Q...[10]Q)
ellfacteur	(got initial helix)
	(baby step table complete)
	(giant step at p = %lu)
factoring 0 in ifac_startPL: N-1 factored!
SQUFOF: found factor %ld^2
OddPwrs: examining %Z
IFAC: cracking composite
	%Z
IFAC: found %Z =
	%Z ^2
IFAC: factor %Z
	is prime
IFAC: checking for odd power
IFAC: found %Z =
	%Z ^%ld
IFAC: trying MPQS
 (so far)...	factor no. %ld was unique%s
IFAC: factoring %Z
factoringIFAC: cofactor = %Z
ifac_sort_oneifac_sumdivkifac_sumdivifac_numdivifac_totientifac_bigomegaifac_omegaifac_issquarefreeifac_moebiusfactoring 0 in ifac_decomp[2] ifac_decompXXXXXXXXXXXXXXXXXXXXXXXXXX      ? .`* .!e ;(g:i.`e*4e*e.;
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Z&/ma~h/7?empty group in group_domainvecsmall_copyPermutationGroup<1|>PermutationGroup<|perm_to_GAPGroup(())Group(galoisexportcoset not found in cosets_perm_searchwrong argument in galoisisabeliangaloissubgroup: not a WSS groupnot a prime in factmodspec_FqXQ_powQpX_to_ZXapprpadicspec_FpXQ_pow   %3ld fact. of degree %3ld
Berlekamp matrixnot a prime in rootmodnot a prime in polrootsmodprime too big in rootmod2lx<ly in Flx_addmul_inplace* Finding eigenvalues
too many iterations in hqr* Eigenvalues computed

polished roots = %ZFpXQYQ_powBerlekamp_matrixBerlekamp_kerrootpadicZp_apprpadicapprQpXQ_to_ZXYpolfnfpolfnf: choosing k = %ld
reducible modulus in factornfFpX_factor_2factmod: %lu is not primeFqX_factorFqX_split_Trager failed!factorffto_Fq_polto_Fqfactorpadicroots2too many iterations in rootsrootsoldzero polynomial in FpXQ_pow. %Z not prime   %3ld factor of degree %3ld
[FqX_split] splitting time: %ld (%ld trials)
polynomial has probably multiple roots in zrhqreuclidean division (poldivrem)non-positive precision in rootpadicFqX_split_Trager: choosing k = %ld
polynomial variable must have higher priority in factorffnon-positive precision in factorpadicfactorpadic2 for non-monic polynomialtoo many iterations in roots2() ( laguer() ):
     real coefficients polynomial, using zrhqr()
too many iterations in rootsold(): using roots2()error in rootsold(): using roots2()>Q>Q>>|ܿp=
ף?RQ?ףp=
?)\(?subresallleftright_powprod: remaining objects %ld
Q_denomRgX_gcd_simplemissing case in gdivexactLLL_cmbf [no factor]S_2   bound: %Z^%ld
coeff bound: %Z^%ld
Mignotte bound: %Z
Beauzamy bound: %Z
polratliftmatratliftnewtonpolypolsym of a negative npolsympolsym_genMultiLift: bad argsbuilding treelifting to prec %ldreduceddiscsmithpoldiscreducedpseudorem dx = %ld >= %ldQ_divmuli_to_intQ_muli_to_intQ_div_to_intQ_contentpolsturm, dr = %ldnextSousResultant j = %ld/%ldsubresall, dr = %lddiscsrpolresultantsrgcdsrgcd: dr = %ld
missing nf in factorbackeltRoot boundHensel lift (mod %Z^%ld)DDF_roots, m = %ldRecombinationfor this block of tracesLLL_cmbf: rank decrease
special_pivot output:
%Z
LLL_cmbf: chk_factors failed
### K = %d, %Z combinations
Naive recombinationKnapsackDDF: wrong numbers of factorssplitting mod p = %ldTime setup: %ld
Total Time: %ld
===========
nfrootsQfactpolfactor of general polynomialcan't factor %Zgisirreduciblegdeflatenfgcd: p=%d
nfgcdpseudodiv dx = %ld >= %ldsubresextRgX_extgcd, dr = %ldsubresext, dr = %ldginvmod,888 'X' '' ' ' ' ' ''' ' ' ' ' ' '(((`Z[`Z`Z`Z`Z`Z`Z`Z`Z\`Z`Z`Z`Z`Z`Z`Z[4\`Z`ZZkBkkkkkkkkkkkkՃcVVVRLR&RB2P̏(8ɗ#ɗɗɗɗɗ!+ɗɗɗɗɗɗ


͝Pv7XbXuuͽ??'PP;;;&\&'Qakkk%'%%'%r'%%%F&%%&'())@&@&:((relatively prime polynomials expectednot a polynomial in polhenselliftnot a factorization in polhenselliftnot a prime number in polhenselliftnot a positive exponent in polhenselliftnot an integral polynomial in polhenselliftnot an integral factorization in polhenselliftnot a correct factorization in polhenselliftpolhensellift: factors %Z and %Z are not coprimenon-monic polynomial in poldiscreducednot the same variables in sylvestermatrixnot a squarefree polynomial in sturmresultantducos, degpol Q = %ldnot a factorisation in factorback
LLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)
LLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld
LLL_cmbf: checking factor %ld
* Time LLL: %ld
* Time Check Factor: %ld

found factor %Z
remaining modular factor(s): %ld
last factor still to be checked
...tried prime %3ld (%-3ld %s). Time = %ld
factor for general polynomialspartial factorization is not meaningful hereneed positive degree in gdeflatecan't deflate this power series (d = %ld): %Zinexact computation in RgX_extgcdinexact computation in subresextnon-invertible polynomial in RgXQ_inv h|5@h㈵>@@Rg_to_FlRg_to_Fpbound for resultant: 2^%ld
ZX_resultantQXQ_invQXQ_inv: char 0 check failedmodulargcdgcd mod %lu (bound 2^%ld)bound 2^%ld. Goal 2^%ldTrying lambda = %ld
Final lambda = %ld
ZY_ZXY_rnfequationFpV_polintFpX_resultant (da = %ld)non positive degree in ffinitpol[Frobenius]matrix cycloFpXQ_sqrtl1/0 exponent in FpXQ_sqrtnFF l-Gen:next %Z
fflgenFpXQX_gcdRg_to_FpXQpows [P,Q]FpXQ_matrix_powsfactor_irred_matfactor_irredpolint_triv2 (i = %ld)resultant mod %ld (bound 2^%ld, stable = %d)QXQ_inv: mod %ld (bound 2^%ld)different variables in modulargcdmodulargcd: trial division failedFFInit: using subcyclo(%ld, %ld)
ZY_ZXY_resultant_all: LERS needs lambdabound for resultant coeffs: 2^%ld
Degree list for ERS (trials: %ld) = %Z
resultant mod %ld (bound 2^%ld, stable=%ld)non invertible polynomial in FpXQ_invZZ_%Z[%Z]/(%Z) is not a field in FpX_ffintersectnon-invertible polynomial in FpXQX_gcdpowers is only [] or [1] in FpX_FpXQV_compoFpX_FpXQV_compo: %d FpXQ_mul [%d]
bad degrees in FpX_ffintersect: %d,%d,%dPolynomials not irreducible in FpX_ffintersectX@\(\?isrealapprcauchy_boundconformal_polrefine_Frefine_HFFTinitparametersFFTinvalid coefficients in rootsroots (conjugates)polrootsall_roots: restarting, i = %ld, e = %ld
j$@j@?i@Q	@@?9B.?ht@{Gz??]9?&?@@r0?	?+eG@SubCyclo: testing %ld^%ld
SubCyclo: %ld not found
SubCyclo: new conductor:%ld
SubCyclo: conductor:%ld
wrong type in galoissubcycloSubcyclo: prime l=%Z
Subcyclo: borne=%Z
Subcyclo: val=%ld
padicsqrtnlift.subcyclo_rootssubcyclo_cyclicroots_to_poldegree <= 0 in galoissubcycloSubcyclo: elements:Subcyclo: complex=%ld
znstar_conductorznstar_cosetsSubcyclo: orbits=%Z
wrong modulus in galoissubcyclogenerators must be prime to conductor in galoissubcyclobnr must be over Q in bnr_to_znstardegree does not divide phi(n) in subcyclonon-cyclic case in polsubcyclo: use galoissubcyclo insteadPlease do not try to break PARI with ridiculous counterfeit data. Thanks!not a HNF matrix in galoissubcycloN must be a bnrinit or a znstar if H is a matrix in galoissubcycloMatrix of wrong dimensions in galoissubcycloSubcyclo: %ld orbits with %ld elements each
gmgmggggggggggggglllljggiexact type in subgrouplist %ld     column selection:not a group in forsubgroupinfinite group in forsubgroup
group:    lambda =     lambda'=     mu =     mu'=   alpha_lambda(mu,p) = %Z
  subgroup:  countsub = %ld
for this type  alpha = %Z
nb subgroup = %ld
subgroup: index bound must be positive(lifted) subgp of prime to %Z part:
forsubgroup (alpha != countsub)not a p-adic argument in teichmullernon positive argument in mplogp-adic argument out of range in gexpp-adic argument out of range in gsinp-adic argument out of range in gcos%Z should divide valuation (= %ld) in sqrtngpow: need integer exponent if series valuation != 0gpow: 0 to a non positive exponentsecond arg must be integer in gsqrtnnth-root does not exist in gsqrtnvaluation overflow in sqrtnodd exponent in p-adic sqrtpadic_sqrtgpowdegree overflow in pow_monomeincorrect precision in transca transcendental functionzero argument in palogsqrtrmpsc1can't compute tan(Pi/2 + kPi)gexpnon zero exponent in gsincosgsingcosgcotan0 argument in cotangtanser_powzero argument in mploggloglog is not meromorphic at 0agm of two vector/matricesgpow: 0 to a forbidden powergpow: underflow or overflowgpow: modulus %Z is not primegpow: nth-root does not existsqrtnr1/0 exponent in gsqrtnaaɵDDaDa\aaaabjqs}}MMMdr!6	ziC,322,32,*.,-I:;l<:;I:y<:9?@LA?@9?YA?DDkEyEDDEDXFGGGGXFGGq4t4tr4tq4tcrOr9B.F@9B.?8,6V?+eG?O@UUUUUU%@9B.&@|?5^@rfix (conversion to t_REAL)Bernoulligashlim, nn: [%ld, %ld]
sum from 0 to N-1Bernoulli sumgpsipsi of power seriesproduct from 0 to N-1Bernoullisglngammalngamma around a!=1p-adic lngamma functionargument too large in ggammaargument too large in ggamdgamd of a power seriesgachgchgacoszero argument in garggasinsingular argument in atanhgathgatancaching Bernoulli numbers 2*%ld to 2*%ld, prec = %ld
non-positive integer argument in cxpsilim, nn: [%ld, %ld], la = %lf
non-positive integer argument in cxgammanon-positive integer in glngammanon-positive integer argument in ggammaGamma not defined for non-integral p-adic numberЩueuuRqx`qx}W}}}jq<-<Oz&DT!?&DT!x?i@b@,4l?k < 0 in thetanullkq >= 1 in thetanon-positive valuation in etajbesseljbessel around a!=0p-adic jbessel functionkbesselkbessel around a!=0incgam2non-real argument in eint1veceint1Entering veceint1:
nstop = %ld
inv_szeta_euler, p = %lu/%luszetanegative index in polylogpadic polylogarithmpolylog around a!=0gpolyloggpolylogztab[q^-s] from 1 to N-1sum from 1 to N-1czetaargument equal to one in zetagzetazeta of power seriesincgamcerfctrueetaweberp-adic jbesselh functionjbesselhp-adic kbessel functionbesselkargument must belong to upper half-planebad argument for modular functiontwistpartialzeta (1), j = %ld/%ldtwistpartialzeta (2), j = %ld/%ldtwistpartialzeta (3), j = %ld/%ldnegative or zero constant in veceint1too large negative arg %ld in zetanot an integer index in jbesselhhyperu's third argument must be positivezero argument in a k/n bessel functioncannot give a power series result in k/n bessel functioni88i8i8 8iiiiiii$$$S'W'WW'WS'WW'W5VSSSSSSSUUUz{^|[zzz^|nzzzz{~~~~~~~~~~$B
?Ǝ%N@9B.6@p=
ף@?8dg?(\?-"l?UUUUUU?Gz?Q?e2Jd?G?ۦx\T?NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNu,,Lu{-_++ttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt4w##X"" G!,-)-)-)-)+-)-)-)-)+-)-)-)-)-)-)-)-)-)-)-)-),-)*-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)-)*CDB@C@;BwF@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@8EmPKKKKK<PKKKKKKKKKKKKKKKKPKKKKKKOOKxNKOKKONMKKNKKNKKCOKKKKKKKKKKKKKKKMKKKKKKKlMKLMKLGLKKKKL8W+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T+T/VWnU+T+T+T+T+T?T
d'@Bʚ;.%s = local(); positive integer expectedincorrect type in %sinteger too bigusing obsolete function %scan't kill thatglobal variable not allowedskipidentifier (unknown code); or ] expectedthis code has to come firstunknown parser codenot a valid identifiererror in parse_option_stringunknown functiononly functions can be aliasedcan't pop gp variablepanicno more variables availablevariable number too bigbreak not allowedunfinished stringglobal variable: a 0x0 matrix has no elementsbreak not allowed after ^this should be an integernot an integershift operand too bigexprbreak not allowed in print()%s[1-%ld]sequnknown member functionformal derivationdeep recursionnot a variable:can't derive thisObreak not allowed in O()ifwhileuntilglobalsymbol already in uselocalbreak not allowed after !break not allowed after #truc(): n = %ldincorrect vector or matrixunused charactersunused characters: %sa1a2a3a4a6areab2b4b6b8bidc4c6codiffcycfutuindexorderstatetufuzkzkstcan't allow allocatemem() in loopsthis function uses a killed variableexpected character: '%c' instead oftoo many parameters in user-defined function call[install] identifier '%s' already in use[install] updating '%s' prototype; module not reloadedid too long in a stringified flaga stringified flag does not start with an idnumeric id in a stringified flagUnrecognized id '%s' in a stringified flagCannot negate id=value in a stringified flagUnrecognized action in a templateNon-numeric argument of an action in a templateJunk after an id in a stringified flagcan't replace an existing symbol by an aliasrenaming a GP variable is forbidden%s already exists with incompatible valencenot a suitable VECSMALL componentincorrect type or length in matrix assignmentnot a proper member definitionbreak not allowed in assignmentbreak not allowed in array contextbreak not allowed here (expanding string)break not allowed here (reading long)array index (%ld) out of allowed range can't modify a pre-defined member: break not allowed here (reading arguments)not enough flags in string function signaturebreak not allowed in test expressionbreak not allowed here (defining global var)break not allowed here (reading function args)local() bloc must appear before any other expressionunknown function '%s', expected '=' instead ofuser function %s: variable %Z declared twiceI can't remember before the big bangthis function has been suppressedO(a^b)=o(a^b)=p-adic or power series zero with precision given by babs(x)=absolute value (or modulus) of xacosh(x)=inverse hyperbolic cosine of xaddell(e,z1,z2)=sum of the points z1 and z2 on elliptic curve eaddprimes(x)=add primes in the vector x (with at most 20 components) to the prime tableagm(x,y)=arithmetic-geometric mean of x and yakell(e,n)=computes the n-th Fourier coefficient of the L-function of the elliptic curve ealgdep(x,n)=algebraic relations up to degree n of xalgdep2(x,n,dec)=algebraic relations up to degree n of x where dec is as in  lindep2algtobasis(nf,x)=transforms the algebraic number x into a column vector on the integral basis nf[7]anell(e,n)=computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<32768)apell(e,p)=computes a_p for the elliptic curve e using Shanks-Mestre's methodapell2(e,p)=computes a_p for the elliptic curve e using Jacobi symbolsapprpadic(x,a)=p-adic roots of the polynomial x congruent to a mod parg(x)=argument of x,such that -pi<arg(x)<=piasinh(x)=inverse hyperbolic sine of xassmat(x)=associated matrix to polynomial xatanh(x)=inverse hyperbolic tangent of xbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithmbasis2(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 2 algorithmbasistoalg(nf,x)=transforms the vertical vector x on the integral basis into an algebraic numberbernreal(x)=Bernoulli number B_x, as a real number with the current precisionbernvec(x)=Vector of rational Bernoulli numbers B_0, B_2,... up to B_(2x)bestappr(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=dbezoutres(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 polynomialsbigomega(x)=number of repeated prime divisors 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 pointsbin(x,y)=binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbinary(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 termsboundfact(x,lim)=partial factorization of the integer x (using primes up to lim)buchcertify(bnf)=certify the correctness (i.e. remove the GRH) of the bnf data output by buchinit or buchinitfubuchfu(bnf)=compute the fundamental units of the number field bnf output by buchinitGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,pbuchgen(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)buchgenforcefu(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)buchgenfu(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)buchimag(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)buchinit(P,...)=compute the necessary data for future use in ideal and unit group computations. See manual for detailsbuchinitforcefu(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 detailsbuchnarrow(bnf)=given a big number field as output by buchinitxx, gives as a 3-component vector the structure of the narrow class groupbuchray(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 modulebuchrayinit(bnf,ideal)=same as buchrayinitgen, except that the generators are not explicitly computedbuchrayinitgen(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 structurebuchreal(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)bytesize(x)=number of bytes occupied by the complete tree of the object xceil(x)=ceiling of x=smallest integer>=xcenterlift(x)=centered lift of x. Same as lift except for integermodscf(x)=continued fraction expansion of x (x rational,real or rational function)cf2(b,x)=continued fraction expansion of x (x rational,real or rational function), where b is the vector of numerators of the continued fractionchangevar(x,y)=change variables of x according to the vector ychar(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using the comatrixchar1(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using Lagrange interpolationchar2(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 ringchell(x,y)=change data on elliptic curve according to y=[u,r,s,t]chinese(x,y)=x,y being integers modulo mx and my,finds z such that z is congruent to x mod mx and y mod mychptell(x,y)=change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]classno(x)=class number of discriminant xclassno2(x)=class number of discriminant xcoeff(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[])compimag(x,y)=Gaussian composition of the binary quadratic forms x and y of negative discriminantcompo(x,s)=the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]compositum(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2compositum2(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 polynomialscomprealraw(x,y)=Gaussian composition without reduction of the binary quadratic forms x and y of positive discriminantconcat(x,y)=concatenation of x and yconductor(bnr,subgroup)=conductor of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupconductorofchar(bnr,chi)=conductor of the character chi on the ray class group bnrconj(x)=the algebraic conjugate of xconjvec(x)=conjugate vector of the algebraic number xcontent(x)=gcd of all the components of x, when this makes senseconvol(x,y)=convolution (or Hadamard product) of two power seriescore(n)=unique (positive of negative) squarefree integer d dividing n such that n/d is a squarecore2(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 squarecoredisc(n)=discriminant of the quadratic field Q(sqrt(n))coredisc2(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 integercosh(x)=hyperbolic cosine of xcvtoi(x)=truncation of x, without taking into account loss of integer part precisioncyclo(n)=n-th cyclotomic polynomialdecodefactor(fa)=given a factorisation fa, gives the factored object backdecodemodule(nf,fa)=given a coded module fa as in discrayabslist, gives the true moduledegree(x)=degree of the polynomial or rational function x. -1 if equal 0, 0 if non-zero scalardenom(x)=denominator of x (or lowest common denominator in case of an array)deplin(x)=finds a linear dependence between the columns of the matrix xderiv(x,y)=derivative of x with respect to the main variable of ydet(x)=determinant of the matrix xdet2(x)=determinant of the matrix x (better for integer entries)detint(x)=some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with hermitemoddiagonal(x)=creates the diagonal matrix whose diagonal entries are the entries of the vector xdirdiv(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)dirmul(x,y)=multiplication of the Dirichlet series x by the Dir. series ydirzetak(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 xdiscf(x)=discriminant of the number field defined by the polynomial x using round 4discf2(x)=discriminant of the number field defined by the polynomial x using round 2discrayabs(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayabscond(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 discrayabsconddiscrayabslistarch(bnf,arch,bound)=gives list of discrayabscond of all modules up to norm bound with archimedean places arch, in a longvector formatdiscrayabslistarchall(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 formatdiscrayabslistlong(bnf,bound)=gives list of discrayabscond of all modules up to norm bound without archimedean places, in a longvector formatdiscrayrel(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayrelcond(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 conductordivisors(x)=gives a vector formed by the divisors of x in increasing orderdivres(x,y)=euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainderdivsum(n,X,expr)=sum of expression expr, X running over the divisors of neigen(x)=eigenvectors of the matrix x given as columns of a matrixeint1(x)=exponential integral E1(x)erfc(x)=complementary error functioneta(x)=eta function without the q^(1/24)euler=euler()=euler's constant with current precisioneval(x)=evaluation of x, replacing variables by their valueextract(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 numberfactcantor(x,p)=factorization mod p of the polynomial x using Cantor-Zassenhausfactfq(x,p,a)=factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factmod(x,p)=factorization mod p of the polynomial x using Berlekampfactoredbasis(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 xfactoreddiscf(x,p)=discriminant of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant 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)factoredpolred2(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)factornf(x,t)=factorization of the polynomial x over the number field defined by the polynomial tfactorpadic(x,p,r)=p-adic factorization of the polynomial x to precision r, using the round 4 algorithmfactorpadic2(x,p,r)=p-adic factorization of the polynomial x to precision r, using Buchmann-Lenstrafactpol(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 hintfactpol2(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0) using root findingfibo(x)=fibonacci number of index x (x C-integer)floor(x)=floor of x=largest integer<=xfor(X=a,b,seq)=the sequence is evaluated, X going from a up to bfordiv(n,X,seq)=the sequence is evaluated, X running over the divisors of nforprime(X=a,b,seq)=the sequence is evaluated, X running over the primes between a and bforstep(X=a,b,s,seq)=the sequence is evaluated, X going from a to b in steps of sforvec(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,..,1fpn(p,n)=monic irreducible polynomial of degree n over F_p[x]frac(x)=fractional part of x=x-floor(x)galois(x)=Galois group of the polynomial x (see manual for group coding)galoisapply(nf,aut,x)=Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfgaloisconj(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)galoisconj1(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)galoisconjforce(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 loopgamh(x)=gamma of x+1/2 (x integer)gauss(a,b)=gaussian solution of ax=b (a matrix,b vector)gaussmodulo(M,D,Y)=(long)gen_1 solution of system of congruences MX=Y mod Dgaussmodulo2(M,D,Y)=all solutions of system of congruences MX=Y mod Dgcd(x,y)=greatest common divisor of x and ygetheap()=2-component vector giving the current number of objects in the heap and the space they occupygetrand()=current value of random number seedgetstack()=current value of stack pointer avmagettime()=time (in milliseconds) since last call to gettimeglobalred(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_pgoto(n)=THIS FUNCTION HAS BEEN SUPPRESSEDhclassno(x)=Hurwitz-Kronecker class number of x>0hell(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using theta-functionshell2(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using Tate's methodhermite(x)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, using a naive algorithmhermite2(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 algorithmhermitehavas(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 algorithmhermitemod(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 latticehermitemodid(x,d)=(upper triangular) Hermite normal form of x concatenated with d times the identity matrixhermiteperm(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 algorithmhilb(x,y,p)=Hilbert symbol at p of x,y (integers or fractions)hilbert(n)=Hilbert matrix of order n (n C-integer)hilbp(x,y)=Hilbert symbol of x,y (where x or y is integermod or p-adic)hvector(n,X,expr)=row vector with n components of expression expr, the variable X ranging from 1 to nhyperu(a,b,x)=U-confluent hypergeometric functionidealadd(nf,x,y)=sum of two ideals x and y in the number field defined by nfidealaddone(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=1idealaddmultone(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 1idealappr(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 pidealapprfact(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 pidealchinese(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 pidealcoprime(nf,x,y)=gives an element b in nf such that b.x is an integral ideal coprime to the integral ideal yidealdiv(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nfidealdivexact(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 idealidealfactor(nf,x)=factorization of the ideal x given in HNF into prime ideals in the number field nfidealhermite(nf,x)=hermite normal form of the ideal x in the number field nf, whatever form x may haveidealhermite2(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 elementsidealintersect(nf,x,y)=intersection of two ideals x and y in HNF in the number field defined by nfidealinv(nf,x)=inverse of the ideal x in the number field nf not using the differentidealinv2(nf,x)=inverse of the ideal x in the number field nf using the differentideallist(nf,bound)=vector of vectors of all ideals of norm<=bound in nfideallistarch(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, without generatorsideallistarchgen(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, with generatorsideallistunit(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 generatorsideallistunitarch(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitideallistunitarchgen(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitgenideallistunitgen(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 generatorsideallistzstar(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, without generatorsideallistzstargen(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, with generatorsideallllred(nf,x,vdir)=LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealmul(nf,x,y)=product of the two ideals x and y in the number field nfidealmulred(nf,x,y)=reduced product of the two ideals x and y in the number field nfidealnorm(nf,x)=norm of the ideal x in the number field nfidealpow(nf,x,n)=n-th power of the ideal x in HNF in the number field nfidealpowred(nf,x,n)=reduced n-th power of the ideal x in HNF in the number field nfidealtwoelt(nf,x)=(long)gen_2-element representation of an ideal x in the number field nfidealtwoelt2(nf,x,a)=(long)gen_2-element representation of an ideal x in the number field nf, with the first element equal to aidealval(nf,x,p)=valuation at p given in primedec format of the ideal x in the number field nfidmat(n)=identity matrix of order n (n C-integer)if(a,seq1,seq2)=if a is nonzero, seq1 is evaluated, otherwise seq2image(x)=basis of the image of the matrix ximage2(x)=basis of the image of the matrix ximagecompl(x)=vector of column indices not corresponding to the indices given by the function imageincgam(s,x)=incomplete gamma functionincgam1(s,x)=incomplete gamma function (for debugging only)incgam2(s,x)=incomplete gamma function (for debugging only)incgam3(s,x)=complementary incomplete gamma functionincgam4(s,x,y)=incomplete gamma function where y=gamma(s) is precomputedindexrank(x)=gives two extraction vectors (rows and columns) for the matrix x such that the exracted matrix is square of maximal rankindsort(x)=indirect sorting of the vector xinitalg(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]initalgred(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]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]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]initzeta(x)=compute number field information necessary to use zetak, where x is an irreducible polynomialinteg(x,y)=formal integration of x with respect to the main variable of yintersect(x,y)=intersection of the vector spaces whose bases are the columns of x and yintgen(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 signintnum(X=a,b,s)=numerical integration of s from a to b with respect to Xintopen(X=a,b,s)=numerical integration of s from a to b with respect to X, where s has only limits at a or binverseimage(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) otherwiseisfund(x)=true(1) if x is a fundamental discriminant (including 1), false(0) if notisideal(nf,x)=true(1) if x is an ideal in the number field nf, false(0) if notisincl(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 isomorphismsisinclfast(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 constantisisom(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 isomorphismsisisomfast(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 sureisoncurve(e,x)=true(1) if x is on elliptic curve e, false(0) if notisprime(x)=true(1) if x is a strong pseudoprime for 10 random bases, false(0) if notisprincipal(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 vectorisprincipalforce(bnf,x)=same as isprincipal, except that the precision is doubled until the result is obtainedisprincipalgen(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 vectorisprincipalgenforce(bnf,x)=same as isprincipalgen, except that the precision is doubled until the result is obtainedisprincipalray(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 vectorisprincipalraygen(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 vectorispsp(x)=true(1) if x is a strong pseudoprime, false(0) if notisqrt(x)=integer square root of x (x integer)isset(x)=true(1) if x is a set (row vector with strictly increasing entries), false(0) if notissqfree(x)=true(1) if x is squarefree, false(0) if notissquare(x)=true(1) if x is a square, false(0) if notisunit(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 otherwisejacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xjbesselh(n,x)=J-bessel function of index n+1/2 and argument x, where n is a non-negative integerjell(x)=elliptic j invariant of xkaramul(x,y,k)=THIS FUNCTION HAS BEEN SUPPRESSEDkbessel(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)kbessel2(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)ker(x)=basis of the kernel of the matrix xkeri(x)=basis of the kernel of the matrix x with integer entrieskerint(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkerint1(x)=LLL-reduced Z-basis of the kernel of the matrix x with rational entries using matrixqz3 and the HNFkerint2(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkro(x,y)=kronecker symbol (x/y)label(n)=THIS FUNCTION HAS BEEN SUPPRESSEDlambdak(nfz,s)=Dedekind lambda function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)laplace(x)=replaces the power series sum of a_n*x^n/n! by sum of a_n*x^nlcm(x,y)=least common multiple of x and y=x*y/gcd(x,y)legendre(n)=legendre polynomial of degree n (n C-integer)length(x)=number of non code words in xlex(x,y)=compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)lexsort(x)=sort the elements of the vector x in ascending lexicographic orderlift(x)=lifts every element of Z/nZ to Z or Z[x]/PZ[x] to Z[x]lindep(x)=Z-linear dependencies between components of x (Hastad et al)lindep2(x,dec)=Z-linear dependencies between components of x using LLL, where dec should be about one half the number of decimal digits of precisionlll(x)=lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lll1(x)=old version of 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)lllgram(x)=lll reduction of the lattice whose gram matrix is 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)lllgramgen(x)=lll reduction of the lattice whose gram matrix is x with polynomial coefficients (gives the unimodular transformation matrix)lllgramint(x)=lll reduction of the lattice whose gram matrix is the integral matrix x (gives the unimodular transformation matrix)lllgramkerim(x)=kernel and lll reduction of the lattice whose gram matrix is the integral matrix xlllgramkerimgen(x)=kernel and lll reduction of the lattice whose gram matrix is the matrix x with polynomial coefficientslllint(x)=lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllintpartial(x)=partial (hence faster) lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllkerim(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 coefficientslllrat(x)=lll reduction of the vectors forming the matrix x, computations done with rational numbers (gives the unimodular transformation matrix)ln(x)=log(x)=natural logarithm of xlngamma(x)=logarithm of the gamma function of xlocalred(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_plog(x)=ln(x)=natural logarithm of xlogagm(x)=natural logarithm of x, computed using agm (faster than log for more than a few hundred decimal digits)lseriesell(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 1makebigbnf(sbnf)=transforms small sbnf as output by smallbuchinit into a true big bnfmat(x)=transforms any GEN x into a matrixmatextract(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)mathell(e,x)=gives the height matrix for vector of points x on elliptic curve e using theta functionsmatrix(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 nmatrixqz(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 otherwisematrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by the columns of xmatrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsize(x)=number of rows and columns of the vector/matrix x as a 2-vectorminideal(nf,ix,vdir)=minimum of the ideal ix in the direction vdir in the number field nfminim(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=0minim2(x,bound)=looks for vectors of square norm <= bound, return the first one and its normmod(x,y)=creates the integer x modulo y on the PARI stackmodp(x,y)=creates the integer x modulo y as a permanent object (on the heap)modreverse(x)=reverse polymod of the polymod x, if it existsmodulargcd(x,y)=gcd of the polynomials x and y using the modular methodnewtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime pnextprime(x)=smallest prime number>=xnfdetint(nf,x)=multiple of the ideal determinant of the pseudo generating set xnfdiv(nf,a,b)=element a/b in nfnfdiveuc(nf,a,b)=gives algebraic integer q such that a-bq is smallnfdivres(nf,a,b)=gives [q,r] such that r=a-bq is smallnfhermite(nf,x)=if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhermitemod(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_jnfmod(nf,a,b)=gives r such that r=a-bq is small with q algebraic integernfmul(nf,a,b)=element a.b in nfnfpow(nf,a,k)=element a^k in nfnfreduce(nf,a,id)=gives r such that a-r is the ideal id and r is smallnfsmith(nf,x)=if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfval(nf,a,pr)=valuation of element a at the prime prnorml2(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 precomputednumdiv(x)=number of divisors of xnupow(x,n)=n-th power of primitive positive definite quadratic form x using nucomp and nuduplo(a^b)=O(a^b)=p-adic or power series zero with precision given by bomega(x)=number of unrepeated prime divisors of xordell(e,x)=y-coordinates corresponding to x-ordinate x on elliptic curve eorder(x)=order of the integermod x in (Z/nZ)*orderell(e,p)=order of the point p on the elliptic curve e over Q, 0 if non-torsionordred(x)=reduction of the polynomial x, staying in the same orderpadicprec(x,p)=absolute p-adic precision of object xpascal(n)=pascal triangle of order n (n C-integer)perf(a)=rank of matrix of xx~ for x minimal vectors of a gram matrix apermutation(n,k)=permutation number k (mod n!) of n letters (n C-integer)permutation2num(vect)=ordinal (between 1 and n!) of permutation vectpf(x,p)=returns the prime form whose first coefficient is p, of discriminant xphi(x)=Euler's totient function of xpi=pi()=the constant pi, with current precisionpnqn(x)=[p_n,p_{n-1};q_n,q_{n-1}] corresponding to the continued fraction xpointell(e,z)=coordinates of point on the curve e corresponding to the complex number zpolint(xa,ya,x)=polynomial interpolation at x according to data vectors xa, yapolred(x)=reduction of the polynomial x (gives minimal polynomials only)polred2(x)=reduction of the polynomial x (gives elements and minimal polynomials)polredabs(x)=a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 normpolredabs2(x)=gives [pol,a] where pol is as in polredabs, and alpha is the element whose characteristic polynomial is polpolredabsall(x)=complete list of the smallest generating polynomials of the number field for the T2 norm on the rootspolredabsfast(x)=a smallest generating polynomial of the number field for the T2 norm on the rootspolredabsnored(x)=a smallest generating polynomial of the number field for the T2 norm on the roots without initial polredpolsym(x,n)=vector of symmetric powers of the roots of x up to npolvar(x)=main variable of object x. Gives p for p-adic x, error for scalarspoly(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientpolylog(m,x)=m-th polylogarithm of xpolylogd(m,x)=D_m~-modified m-th polylog of xpolylogdold(m,x)=D_m-modified m-th polylog of xpolylogp(m,x)=P_m-modified m-th polylog of xpolyrev(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termpolzag(n,m)=Zagier's polynomials of index n,mpowell(e,x,n)=n times the point x on elliptic curve e (n in Z)powrealraw(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)precision(x)=real precision of object xprime(n)=returns the n-th prime (n C-integer)primedec(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 constantprimes(n)=returns the vector of the first n primes (n C-integer)primroot(n)=returns a primitive root of n when it existsprincipalideal(nf,x)=returns the principal ideal generated by the algebraic number x in the number field nfprincipalidele(nf,x)=returns the principal idele generated by the algebraic number x in the number field nfprod(x,X=a,b,expr)=x times the product (X runs from a to b) of expressionprodeuler(X=a,b,expr)=Euler product (X runs over the primes between a and b) of real or complex expressionprodinf(X=a,expr)=infinite product (X goes from a to infinity) of real or complex expressionprodinf1(X=a,expr)=infinite product (X goes from a to infinity) of real or complex 1+expressionqfi(a,b,c)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c<0qfr(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 dquaddisc(x)=discriminant of the quadratic field Q(sqrt(x))quadgen(x)=standard generator of quadratic order of discriminant xquadpoly(x)=quadratic polynomial corresponding to the discriminant xrandom()=random integer between 0 and 2^31-1rayclassno(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 wantedrayclassnolist(bnf,liste)=if listes is as output by idealisunit or similar, gives list of corresponding ray class numbersrecip(x)=reciprocal polynomial of xredimag(x)=reduction of the binary quadratic form x with D<0redreal(x)=reduction of the binary quadratic form x with D>0redrealnod(x,sq)=reduction of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]reduceddisc(f)=vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fregula(x)=regulator of the real quadratic field of discriminant xreorder(x)=reorder the variables for output according to the vector xresultant(x,y)=resultant of the polynomials x and y with exact entriesresultant2(x,y)=resultant of the polynomials x and yreverse(x)=reversion of the power series xrhoreal(x)=single reduction step of the binary quadratic form x of positive discriminantrhorealnod(x,sq)=single reduction step of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]rndtoi(x)=take the nearest integer to all the coefficients of x, without taking into account loss of integer part precisionrnfbasis(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 setrnfdiscf(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*^2rnfequation(nf,pol)=given a pol with coefficients in nf, gives the absolute equation of the number field defined by polrnfequation2(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 apolrnfhermitebasis(bnf,order)=given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfisfree(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 notrnflllgram(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 matrixrnfpolred(nf,pol)=given a pol with coefficients in nf, finds a list of polynomials defining some subfields, hopefully simplerrnfpseudobasis(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*^2rnfsteinitz(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 trivialrootmod(x,p)=roots mod p of the polynomial xrootmod2(x,p)=roots mod p of the polynomial x, when p is smallrootpadic(x,p,r)=p-adic roots of the polynomial x to precision rroots(x)=roots of the polynomial x using Schonhage's method modified by Gourdonrootsof1(nf)=number of roots of unity and primitive root of unity in the number field nfrootsold(x)=roots of the polynomial x using a modified Newton's methodround(x)=take the nearest integer to all the coefficients of 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 coefficientset(x)=convert x into a set, i.e. a row vector with strictly increasing coefficientssetintersect(x,y)=intersection of the sets x and ysetminus(x,y)=set of elements of x not belonging to ysetrand(n)=reset the seed of the random number generator to nsetsearch(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]setunion(x,y)=union of the sets x and yshift(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)sigma(x)=sum of the divisors of xsigmak(k,x)=sum of the k-th powers of the divisors of x (k C-integer)sign(x)=sign of x, of type integer, real or fractionsignat(x)=signature of the symmetric matrix xsignunit(bnf)=matrix of signs of the real embeddings of the system of fundamental units found by buchinitsimplefactmod(x,p)=same as factmod except that only the degrees of the irreducible factors are givensimplify(x)=simplify the object x as much as possiblesize(x)=maximum number of decimal digits minus one of (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 xsmallbuchinit(pol)=small buchinit, which can be converted to a big one using makebigbnfsmalldiscf(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 xsmallfact(x)=partial factorization of the integer x (using only the stored primes)smallinitell(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]smallpolred(x)=partial reduction of the polynomial x (gives minimal polynomials only)smallpolred2(x)=partial reduction of the polynomial x (gives elements and minimal polynomials)smith(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vectorsmith2(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))smithclean(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 vectorsolve(X=a,b,expr)=real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sort(x)=sort in ascending order of the vector xsqr(x)=square of x. NOT identical to x*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)srgcd(x,y)=polynomial gcd of x and y using the subresultant algorithmsturm(x)=number of real roots of the polynomial xsturmpart(x,a,b)=number of real roots of the polynomial x in the interval (a,b]subcyclo(p,d)=finds an equation for the d-th degree subfield of Q(zeta_p), where p must be a prime powersubell(e,z1,z2)=difference of the points z1 and z2 on elliptic curve esubst(x,y,z)=in expression x, replace the variable y by the expression zsum(x,X=a,b,expr)=x plus the sum (X goes from a to b) of expression exprsumalt(X=a,expr)=Villegas-Zagier's acceleration of alternating series expr, X starting at asumalt2(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 exprsumpos(X=a,expr)=sum of positive series expr, the formal variable X starting at asumpos2(X=a,expr)=sum of positive series expr, the formal variable X starting at a, using Zagier's polynomialssupplement(x)=supplement the columns of the matrix x to an invertible matrixsylvestermatrix(x,y)=forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowstanh(x)=hyperbolic tangent of xtaniyama(e)=modular parametrization of elliptic curve etaylor(x,y)=taylor expansion of x with respect to the main variable of ytchebi(n)=Tchebitcheff polynomial of degree n (n C-integer)teich(x)=teichmuller character of p-adic number xtheta(q,z)=Jacobi sine theta-functionthetanullk(q,k)=k'th derivative at z=0 of theta(q,z)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]threetotwo2(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]torsell(e)=torsion subgroup of elliptic curve e: order, structure, generatorstrunc(x)=truncation of x;when x is a power series,take away the O(X^)tschirnhaus(x)=random Tschirnhausen transformation of the polynomial xtwototwo(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 smallerunit(x)=fundamental unit of the quadratic field of discriminant x where x must be positiveuntil(a,seq)=evaluate the expression sequence seq until a is nonzerovaluation(x,p)=valuation of x with respect to pvec(x)=transforms the object x into a vector. Used mainly if x is a polynomial or a power seriesvecindexsort(x): indirect sorting of the vector xveclexsort(x): sort the elements of the vector x in ascending lexicographic ordervecmax(x)=maximum of the elements of the vector/matrix xvecmin(x)=minimum of the elements of the vector/matrix xvecsort(x,k)=sorts the vector of vector (or matrix) x according to the value of its k-th componentvector(n,X,expr)=row vector with n components of expression expr (X ranges from 1 to n)vvector(n,X,expr)=column vector with n components of expression expr (X ranges from 1 to n)weipell(e)=formal expansion in x=z of Weierstrass P functionweberf(x)=Weber's f function of x (j=(f^24-16)^3/f^24)weberf2(x)=Weber's f2 function of x (j=(f2^24+16)^3/f2^24)while(a,seq)=while a is nonzero evaluate the expression sequence seq. Otherwise 0zell(e,z)=In the complex case, lattice point corresponding to the point z on the elliptic curve ezeta(s)=Riemann zeta function at szetak(nfz,s)=Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)zideallog(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)zidealstar(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 generatorszidealstarinit(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)^*zidealstarinitgen(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)^*znstar(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 generatorsacos(x)=inverse cosine of xacoshaddelladj(x)=adjoint matrix of xakellalgdepGLpalgdep2GLLpanellapell2asin(x)=inverse sine of xasinhatan(x)=inverse tangent of xbasis2bernrealbernvecbezoutbezoutresbilhellbittestboundcfboundfactbuchcertifylGbuchfubuchgenbuchgenforcefubuchgenfubuchimagGD0.1,G,D0.1,G,D5,G,buchinitforcefubuchinitfubuchnarrowbuchraybuchrayinitbuchrayinitgenbuchrealGD0,G,D0.1,G,D0.1,G,D5,G,pbytesizechangevarGnchar1char2chellchptellcompimagcompositum2comprealrawGDGDGD1,G,corecore2coredisccoredisc2cos(x)=cosine of xcvtoidecodefactordilogdilog(x)=dilogarithm of xdirdivdireulerV=GGIDGdiscf2discrayabsGD0,G,D0,G,D0,L,discrayabscondGD0,G,D0,G,D2,L,discrayabslistarchalldiscrayabslistlongdiscrayrelGD0,G,D0,G,D1,L,discrayrelcondGD0,G,D0,G,D3,L,divsumexp(x)=exponential of xfactcantorfactfqfactor(x)=factorization of xfactoredbasisGGffactoreddiscffactoredpolredfactoredpolred2factorpadic2GLLfactpol2fiboforvV=GGIfordivvGVIforprimevV=GGGIvV=GID0,L,fpnGLDngaloisconj1galoisconjforcegamhgamma(x)=gamma function at xgaussmodulo2getheapgetrandgetstackgettimegotohell2hermitehavashermitemodidhermitepermhess(x)=Hessenberg form of xhilblGGGhilbplGGhvectorhyperui=i()=square root of -1idealaddidealaddmultoneidealaddoneidealappridealapprfactidealchineseidealcoprimeidealdivexactidealhermite2idealintersectidealinv2ideallistarchgenideallistunitideallistunitarchideallistunitarchgenideallistunitgenideallistzstarideallistzstargenideallllredidealmulredidealpowidealpowredidealtwoelt2idmatimag(x)=imaginary part of xincgamincgam1incgam3incgam4indsortinitalgredinitalgred2initzetaintgenV=GGID1,L,pintinfV=GGID2,L,pV=GGID0,L,pintopenV=GGID3,L,pisfundisinclfastisisomfastiGGisprincipalforceisprincipalgenisprincipalgenforceisprincipalraygenispspisqrtissqfreejellkaramulkbessel2kerint1kerint2krolabellambdaklcmlengthlll1lllgenlllgram1lllgramgenlllgramintlllgramkerimlllgramkerimgenlllkerimlllkerimgenlllratlogagmlseriesellGGGGpmakebigbnfmathellGGVVImax(x,y)=maximum of x and ymin(x,y)=minimum of x and yminidealmodpmumu(x)=Moebius function of xnextprimenfdiveucnfdivresnfpownfreducenfvalnorm(x)=norm of xnumer(x)=numerator of xordellorderellLDGperfpermutationLGpermutation2numpfpointellGGGD&polredabs2polredabsallpolredabsfastpolredabsnoredLGppolylogdpolylogdoldpolylogppolyrevpolzagpowellprincipalideleprodGV=GGIprodeulerV=GGIpV=GID0,L,pprodinf1V=GID1,L,ppsi(x)=psi-function at xqfiqfrGGGGquaddiscquadgenrank(x)=rank of the matrix xrayclassnorayclassnolistreal(x)=real part of xredrealredrealnodreduceddiscreorderresultant2rhorealrhorealnodrnfdiscfrnfequation2rnfhermitebasisrnfpseudobasisrounderrorsetrandlLlGGD0,L,shiftmulsigmakiGsignatsimplefactmodsimplifysin(x)=sine of xsinh(x)=hyperbolic sine of xsmallbasissmallbuchinitsmalldiscfsmallfactsmallinitellsmallpolredsmallpolred2smith2smithpolsqredsqrt(x)=square root of xsturmpartLLDnsubellGnGsumalt2V=GIptan(x)=tangent of xGPtaylorGnPteichthreetotwothreetotwo2torselltrace(x)=trace of xtrans(x)=x~=transpose of xtwototwovaluationvecindexsortveclexsortvvectorweipellwfwf2zellzidealstarzidealstarinitzidealstarinitgenznstarGPHELPunknown default: %sinteger too largecomment>    prompt%s = "%s"
prettyprinteryesbroken prettyprinter: '%s'   prettyprinter = "%s"
   path = "%s"
none   datadir = "%s"
   help = "%s"
   %s = "%s"
psfilelogfiledarkbg1, 5, 3, 7, 6, 2, 3lightbg1, 6, 3, 4, 5, 2, 3boldfgexpected character: ']'[%ld,,%ld][%ld,%ld,%ld]   colors = "%s"
default: inexistent format%c%ld.%ld   format = %c%ld.%ld
   %s = 1 (on)
   %s = 0 (off)
new_galois_formatfactor_add_primestimerstrictmatchechosecurerealprecision (%ld digits displayed)   %s = %lu %s
   %s = %lu
parisize(raw)(prettymatrix)(prettyprint)(external prettyprint)TeXstyle   [logfile was "%s"]
PARIbreakPARIpromptSTART\egroup\bgroup\ttPARIpromptEND\egroupPARIinputENDPARIouthistsizelinesdebugmemdebugfilesreadlinedebug(no backward compatibility)compatibleuser functions re-initializedsignificant termsseriesprecisioncolorsdatadirpathpromptprompt_cont(off)(on)(on with colors)(TeX output)[secure mode]: can't modify '%s' default (to %s)I was expecting an integer hereget_sep: argument too long (< %ld chars)arguments must be positive integersdefault: incorrect value for %s [%lu-%lu]\ifx\%s\undefined
  \def\%s{%s}\fi
[1,,1], [5,,1], [3,,1], [7,,1], [6,,1], , [2,,1]default: incorrect value for %s [0:off / 1:on][secure mode]: Do you want to modify the 'secure' flag? (^C if not)
   realprecision = %ld significant digits\hskip 0pt plus \hsize\relax\discretionary{}{}{}}\vskip\medskipamount\bgroup\bf\vskip\smallskipamount$\displaystyle{\tt\%#1} = #2$\ifx\%s\undefined
  \def\%s#1#2{%s}\fi
(warn when using obsolete functions)(use old functions, don't ignore case)(use old functions, ignore case)tex2mail -TeX -noindent -ragged -by_par(bits 0x2/0x4 control output of \left/\PARIbreak)(bits 0x2/0x4 control matched-insert/arg-complete)can't deflateunexpected charactervariable name expectedobsolete functionerror opening invalid flagWarning:Warning: increasing precWarning: failed toaccuracy problemsbug insorry,collecting garbage inprecision too lowincorrect typeinconsistent dataimpossible assignment I-->Simpossible assignment I-->Ithe PARI stack overflows !length (lg) overflowexponent (expo) overflowvaluation (valp) overflowoverflow in R->dbl conversionnot a square matrixnot enough precomputed primesimpossible inverse modulo: constant polynomialnot a polynomialreducible polynomialzero polynomialimpossibledivision by zeronot enough memoryinfinite precisionnegative exponentwhat's going on ?unknown function or error in formal parameterssorry, not yet available on this systemnon invertible matrix in gaussunknown identifier valence, please reportnot an integer argument in an arithmetic functionnot enough precomputed primes, need primelimit ~ too many iterations for desired precision in integration routinenot a definite matrix in lllgrambad argument for an elliptic curve related functiontrying to overwrite a universal objectnon quadratic residue in gsqrt.:~:~/gpNULLI can't see into the futureTeX variable name too longt_STRt_VECSMALLt_INTt_REALt_INTMODt_FRACt_COMPLEXt_PADICt_QUADt_POLMODt_POLt_SERt_RFRACt_QFRt_VECt_COLt_MATt_LISTunknown type %ld#<%d>%c[0m%c[%ld;%ldm%c[%ld;%ld;%ldmrun-away string. Closing itrun-away comment. Closing itread failed	-%s is not a GP binary fileunexpected endianness in %swrite failedbinary outputembedded braces (in parser)unexpected closing braceGPTMPDIR/var/tmp.%ld.%ld%.8s%scan't expand ~unknown user %s%sLINESCOLUMNS[+++]Mod(mod( + O(Qfb(qfr(qfi(List([])[;]matrix(0,%ld)matrix(0,%ld,j,k,0)Mat(mat(%016lx  [;]
]

setting %s
unknown code in readobjI/O: checking output pipe...
                  
I/O: can't remove file %sI/O: removed file %s
closedeleteclose pipe[pipe:] '%s' failedskipping directory %s.Z.gz/usr/bin/gzip -dc%s "%s"%s.gp%s/%sinputtempfile %s already exists^{%ld}\*\PARIbreak  -  + mod \frac{}{\pmatrix{ \cr}
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 \cdot ( \left(\right) [&=%016lx]    next = %s no such function*** hashcode = %lu
%3ld:%3ld NULL
gen_0
,CLONE%s(lg=%ld%s):chars:(%c,lgefint=%ld):(%c,expo=%ld):(precp=%ld,valp=%ld):(%c,varn=%ld):(lgeflist=%ld):int = pol = mod = num = den = real = imag =   p : p^l :   I : coef of degree %ld = %ld%s component = %ld%s column = mat(%ld,%ld) = 0.0.E%ldVecsmall([List( / / []
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}9s}s9s66666-+*N&*)*-+[,m+N&|)N&))(:(_'N&N&O'this object uses debugging variablesout of range in integer -> character conversion (%ld)%s written by an incompatible version of GP%s not written for a %ld bit architecture%s is a GP binary file. Please use writebin---- (type RETURN to continue) ----%s is set (%s), but is not writeable%s is set (%s), but is not a directorycouldn't find a suitable name for a tempdir (%s)couldn't find a suitable name for a tempfile (%s)undefined environment variable: %s[secure mode]: about to write to '%s'. OK ? (^C if not)
malformed binary file (no name)I/O: closing file %s (code %d) 
I/O: leaked file descriptor (%d): %sI/O: new pariFILE %s (code %d) 
You never gave me anything to read!could not open requested file %sI/O: opening file %s (mode %s)

 Top : %lx   Bottom : %lx   Current stack : %lx
 Used :                         %ld  long words  (%ld K)
 Available :                    %ld  long words  (%ld K)
 Occupation of the PARI stack : %6.2f percent
 %ld objects on heap occupy %ld long words

 %ld variable names used out of %d

:
   hash = %3ld, valence = %3ld, menu = %2ld, code = %s
invalid range in print_functions_hash(%c,varn=%ld,prec=%ld,valp=%ld):gp_readvec_stream: reaching %ld entries
gp_readvec_stream: found %ld entries
%ld unnamed objects read. Returning then in a vectoryPDӿ,b7@Time bot=0x%lx	top=0x%lx
0x%p:	0x%lx	%lu
killing bloc (no %ld): %08lx
popping %s (bloc no %ld)
  *** %s: %s  ***   %s in %s; new prec = %ld
 %s: %s
bad object %ZResetting all trapsleaving recover()
use pari_warn for warningsuncaught error: %ld%s file  ###   user error:  %s is not yet implemented. in %s. %s, please reportadditionmultiplicationassignment-->division %s %s %s %s.%lu.broken pipeunknown signaluser interruptmallocing NULL object  ***   %s: can't trap memory errorsno such error number: %lderrpiletypeergdiverinvmoderaccurerarchersigintertalkeruserthis trap keywordpari.pspari.logGP_DATA_DIR/usr/local/share/parilbot>ltop in gerepileCannot initialize kernelColEulerI=I(): square root of -1ListMatPiPolPolrevGGGDGpSerSetStrStrchrStrexpandStrtexVecVecsmallacos(x): inverse cosine of xaddhelpvSsGLD0,L,paliasvrrallocatememvD0,L,asin(x): inverse sine of xatan(x): inverse tangent of xbernfracbesselh1besselh2besselibesseljbesseljhbesselnbitandbitnegGD-1,L,bitnegimplybitorbitxorGD0,L,DGpbnfclgpbnfdecodemodulebnfisintnormbnfisnormGGD1,L,pbnfisprincipalGGD1,L,bnfisunitbnfnarrowbnfregbnfsignunitbnfunitbnrconductorGDGDGDGbnrconductorofcharbnrdiscGDGDGD0,L,bnrdisclistbnrisconductorlGDGDGbnrisprincipalbnrrootnumbercontfraccontfracpnqncos(x): cosine of xcotan(x): cotangent of xdefaultD"",r,D"",s,D0,L,denominatordilog(x): dilogarithm of xV=GGEDGelladdellanellapellbilellchangecurveellchangepointellconvertnameelleisnumelletaellgeneratorsellglobalredellheightGGD2,L,pellheightmatrixellidentifyellisoncurveelljelllocalredelllseriesellminimalmodelellorderellordinateellpointtozellrootnolGDGellsearchellsigmaellsubelltaniyamaelltorsellwpGDGD0,L,pPellzetaellztopointvs*eulerphiexp(x): exponential of xfactorbackfactorcantorfactorialGGLD0,L,fibonacciforellvVLLIvV=GDGIGGD0,L,DngaloisidentifygaloisinitGDGD0,L,Dngaloissubfieldsgaloissubgroupsgamma(x): gamma function at xgammahlGGDGidealaddtooneGGGD0,L,GLD4,L,idealminGGGD0,L,pidealprimedecidealprincipalideleprincipalimag(x): imaginary part of xintcircV=GGEDGpintfouriercosV=GGGEDGpintfourierexpintfouriersinintfuncinitV=GGED0,L,D0,L,pintlaplaceinvintmellininvintmellininvshortintnuminitgenVGGED0,L,D0,L,pintnumrombV=GGED0,L,pintnumstepisfundamentallGDGD&ispseudoprimevSkroneckervGlistkill(list): kills listmatadjointmatcompanionmatdetintmatdiagonalmateigenmathessmathilbertmathnfmodmathnfmodidmatidmatimagecomplmatindexrankmatintersectmatinverseimagematisdiagonalmatkermatpascalmatrankGGDVDVDImatsolvematsolvemodmatsupplementmattransposemax(x,y): maximum of x and ymin(x,y): minimum of x and ynextGD0,L,DGnfeltdivnfeltdiveucnfeltdivmodprnfeltdivremnfeltmodnfeltmulnfeltmulmodprnfeltpownfeltpowmodprnfeltreducenfeltreducemodprnfeltvalnffactornffactormodnfgaloisapplylGGGDGnfhnfnfhnfmodnfisidealnfisinclnfisisomnfmodprinitnfrootsDGGnfrootsof1nfsnfnfsubfieldsnorm(x): norm of xnumbpartnumeratornumerator(x): numerator of xnumtopermpolcompositumpolcyclopoldegreelGDnpoldiscpolgaloispolhenselliftGGGLGDGDGD&polisirreduciblepollegendrepolredordGGDnD0,L,polrootspadicpolsturmpolsubcyclopolsylvestermatrixpoltchebipoltschirnhausLGD0,L,ppolzagierprecprimeprintprint1printpprintp1printtexV=GGEpV=GED0,L,ppsi(x): psi-function at xqfbcomprawqfbhclassnoqfbnucompqfbnupowqfbpowrawqfbprimeformGD0,L,DGDGDGqfgaussredqfjacobiGDGDGD0,L,pqfperfectionqfrepqfsignquadclassunitquadhilbertquadregulatorquadunitreadvecD"",s,real(x): real part of xremoveprimesreturnrnfalgtobasisrnfbasistoalgrnfcharpolyGGGDnrnfdiscrnfeltabstorelrnfeltdownrnfeltreltoabsrnfeltuprnfhnfbasisrnfidealdownrnfidealhnfrnfidealmulrnfidealnormabsrnfidealnormrelrnfidealreltoabsrnfidealtwoeltrnfidealuprnfinitGGD2,L,rnfkummerserconvolserlaplacesetissetsin(x): sine of xsinh(x): hyperbolic sine of xsizebytesizedigitsqrt(x): square root of xGGD&psubstpolGVEV=GEpV=GGEDGD0,L,psumnumaltsumnuminitGD0,L,D1,L,ptan(x): tangent of xteichmullerthueinittrace(x): trace of xtrapD"",r,DIDItruncateGDVDIvectorsmallvectorvwritevss*write1writebinvsDGwritetexzetakinitznlogznorder
For full compatibility with GP 1.39.15, type "default(compatible,3)", or set "compatible = 3" in your GPRC filebad component %ld in object %Zentering recover(), loc = %ld

  current stack size: %lu (%.3f Mbytes)
  [hint] you can increase GP stack with allocatemem()
not enough memory, new stack %ludoubling stack size; new stack = %lu (%.3f Mbytes)segmentation fault: bug in PARI or calling programbus error: bug in PARI or calling programfloating point exception: bug in PARI or calling programmallocing NULL object in newblocnew bloc, size %6lu (no %ld): %08lx
significant pointers lost in gerepile! (please report)  ***   Error in the PARI system. End of program.
variable out of range in reorderduplicate indeterminates in reorderCol({x=[]}): transforms the object x into a column vector. Empty vector if x is omittedEuler=Euler(): Euler's constant with current precisionList({x=[]}): transforms the vector or list x into a list. Empty list if x is omittedMat({x=[]}): transforms any GEN x into a matrix. Empty matrix if x is omittedMod(x,y): creates 'x modulo y'.O(a^b): p-adic or power series zero with precision given by bPi=Pi(): the constant pi, with current precisionPol(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientPolrev(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termQfb(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>0Ser(x,{v=x}): convert x (usually a vector) into a power series with variable v, starting with the constant coefficientSet({x=[]}): convert x into a set, i.e. a row vector with strictly increasing coefficients. Empty set if x is omittedStr({str}*): concatenates its (string) argument into a single stringStrchr(x): converts x to a string, translating each integer into a characterStrexpand({str}*): concatenates its (string) argument into a single string, performing tilde expansionStrtex({str}*): translates its (string) arguments to TeX format and returns the resulting stringVec({x=[]}): transforms the object x into a vector. Empty vector if x is omittedVecsmall({x=[]}): transforms the object x into a VECSMALL. Empty vector if x is omittedabs(x): absolute value (or modulus) of xacosh(x): inverse hyperbolic cosine of xaddhelp(symbol,"message"): add/change help message for a symboladdprimes({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 doneagm(x,y): arithmetic-geometric mean of x and yalgdep(x,n,{flag=0}): algebraic relations up to degree n of x, using lindep([1,x,...,x^(n-1)], flag).alias("new","old"): new is now an alias for oldallocatemem({s=0}): allocates a new stack of s bytes. doubles the stack if s is omittedarg(x): argument of x,such that -pi<arg(x)<=piasinh(x): inverse hyperbolic sine of xatanh(x): inverse hyperbolic tangent of xbernfrac(x): Bernoulli number B_x, as a rational numberbernreal(x): Bernoulli number B_x, as a real number with the current precisionbernvec(x): Vector of rational Bernoulli numbers B_0, B_2,...up to B_(2x)besselh1(nu,x): H^1-bessel function of index nu and argument xbesselh2(nu,x): H^2-bessel function of index nu and argument xbesseli(nu,x): I-bessel function of index nu and argument xbesselj(nu,x): J-bessel function of index nu and argument xbesseljh(n,x): J-bessel function of index n+1/2 and argument x, where n is a non-negative integerbesselk(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 hyperubesseln(nu,x): N-bessel function of index nu and argument xbestappr(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=dbezoutres(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 polynomialsbigomega(x): number of prime divisors of x, counted with multiplicitybinary(x): gives the vector formed by the binary digits of x (x integer)binomial(x,y): binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbitand(x,y): bitwise "and" of two integers x and y. Negative numbers behave as if modulo big power of 2bitneg(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 2bitnegimply(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 2bitor(x,y): bitwise "or" of two integers x and y. Negative numbers behave as if modulo big power of 2bittest(x,n): gives bit number n (coefficient of 2^n) of the integer x. Negative numbers behave as if modulo big power of 2bitxor(x,y): bitwise "exclusive 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 bnfinitbnfclassunit(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 orderbnfclgp(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 techbnfdecodemodule(nf,fa): given a coded module fa as in bnrdisclist, gives the true modulebnfinit(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 techbnfisintnorm(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)bnfisnorm(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 bnfbnfisprincipal(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 generatorsbnfissunit(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 otherwisebnfisunit(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 otherwisebnfmake(sbnf): transforms small sbnf as output by bnfinit with flag=3 into a true big bnfbnfnarrow(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 techbnfsignunit(bnf): matrix of signs of the real embeddings of the system of fundamental units found by bnfinitbnfsunit(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 detailsbnfunit(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)bnrL1(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 charactersbnrclass(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=0bnrclassno(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 wantedbnrclassnolist(bnf,list): if list is as output by ideallist or similar, gives list of corresponding ray class numbersbnrconductor(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]bnrconductorofchar(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 conductorbnrdisclist(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)bnrinit(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 structurebnrisconductor(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 resultbnrisprincipal(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 vbnrrootnumber(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 primitivebnrstark(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.break({n=1}): interrupt execution of current instruction sequence, and exit from the n innermost enclosing loopsceil(x): ceiling of x=smallest integer>=xcenterlift(x,{v}): centered lift of x. Same as lift except for integermodschangevar(x,y): change variables of x according to the vector ycharpoly(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 defaultchinese(x,{y}): x,y being both intmods (or polmods) computes z in the same residue classes as x and ycomponent(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 "."concat(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 concatenatedconj(x): the algebraic conjugate of xconjvec(x): conjugate vector of the algebraic number xcontent(x): gcd of all the components of x, when this makes sensecontfrac(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 expansioncontfracpnqn(x): [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the continued fraction xcore(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 squarecoredisc(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 integercosh(x): hyperbolic cosine of xdefault({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.denominator(x): denominator of x (or lowest common denominator in case of an array)deriv(x,{y}): derivative of x with respect to the main variable of y, or to the main variable of x if y is omitteddirdiv(x,y): division of the Dirichlet series x by the Dirichlet series ydireuler(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 termsdirmul(x,y): multiplication of the Dirichlet series x by the Dirichlet series ydirzetak(nf,b): Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1divisors(x): gives a vector formed by the divisors of x in increasing orderdivrem(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)eint1(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)elladd(e,z1,z2): sum of the points z1 and z2 on elliptic curve eellak(e,n): computes the n-th Fourier coefficient of the L-function of the elliptic curve eellan(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)ellap(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)ellbil(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 pointsellchangecurve(x,y): change data on elliptic curve according to y=[u,r,s,t]ellchangepoint(x,y): change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]ellconvertname(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.elleisnum(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 normalizationelleta(om): om=[om1,om2], returns the two-component row vector [eta1,eta2] of quasi-periods associated to [om1,om2]ellgenerators(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.ellglobalred(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_pellheight(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 AGMellheightmatrix(e,x): gives the height matrix for vector of points x on elliptic curve e using theta functionsellidentify(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).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.ellisoncurve(e,x): true(1) if x is on elliptic curve e, false(0) if notellj(x): elliptic j invariant of xelllocalred(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_pelllseries(e,s,{A=1}): L-series at s of the elliptic curve e, where A a cut-off point close to 1ellminimalmodel(e,{&v}): return the standard minimal integral model of the rational elliptic curve e. Sets v to the corresponding change of variablesellorder(e,p): order of the point p on the elliptic curve e over Q, 0 if non-torsionellordinate(e,x): y-coordinates corresponding to x-ordinate x on elliptic curve eellpointtoz(e,P): lattice point z corresponding to the point P on the elliptic curve eellpow(e,x,n): n times the point x on elliptic curve e (n in Z)ellrootno(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 pellsearch(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.ellsigma(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 seriesellsub(e,z1,z2): difference of the points z1 and z2 on elliptic curve eelltaniyama(e): modular parametrization of elliptic curve eelltors(e,{flag=0}): torsion subgroup of elliptic curve e: order, structure, generators. If flag = 0, use Doud's algorithm; if flag = 1, use Lutz-Nagellellwp(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 zellzeta(om,z): om=[om1,om2], value of the Weierstrass zeta function of the lattice generated by om at zellztopoint(e,z): coordinates of point P on the curve e corresponding to the complex number zerfc(x): complementary error functionerror("msg"): abort script with error message msgeta(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) transformationseulerphi(x): Euler's totient function of xeval(x): evaluation of x, replacing variables by their valuefactor(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)factorback(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]factorcantor(x,p): factorization mod p of the polynomial x using Cantor-Zassenhausfactorff(x,p,a): factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factorial(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)factormod(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 givenfactornf(x,t): factorization of the polynomial x over the number field defined by the polynomial tfactorpadic(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)ffinit(p,n,{v=x}): monic irreducible polynomial of degree n over F_p[v]fibonacci(x): fibonacci number of index x (x C-integer)floor(x): floor of x = largest integer<=xfor(X=a,b,seq): the sequence is evaluated, X going from a up to bfordiv(n,X,seq): the sequence is evaluated, X running over the divisors of nforell(E,a,b,seq): execute seq for each elliptic curves E of conductor between a and b in the elldata database.forprime(X=a,b,seq): the sequence is evaluated, X running over the primes between a and bforstep(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)forsubgroup(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 formforvec(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) sequencesfrac(x): fractional part of x = x-floor(x)galoisexport(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)galoisfixedfield(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 Pgaloisidentify(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 librarygaloisinit(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)galoisisabelian(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=2galoispermtopol(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 automorphismgaloissubcyclo(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.galoissubfields(G,{flags=0},{v}):Output all the subfields of G. flags have the same meaning as for galoisfixedfieldgaloissubgroups(G):Output all the subgroups of Ggammah(x): gamma of x+1/2 (x integer)gcd(x,{y}): greatest common divisor of x and y.getheap(): 2-component vector giving the current number of objects in the heap and the space they occupygetrand(): current value of random number seedgetstack(): current value of stack pointer avmagettime(): time (in milliseconds) since last call to gettimeglobal(x): declare x to be a global variablehilbert(x,y,{p}): Hilbert symbol at p of x,y.hyperu(a,b,x): U-confluent hypergeometric functionidealadd(nf,x,y): sum of two ideals x and y in the number field defined by nfidealaddtoone(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 1idealappr(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 exponentsidealchinese(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 pidealcoprime(nf,x,y): gives an element b in nf such that b. x is an integral ideal coprime to the integral ideal yidealdiv(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)idealfactor(nf,x): factorization of the ideal x given in HNF into prime ideals in the number field nfidealhnf(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 nfidealintersect(nf,x,y): intersection of two ideals x and y in the number field defined by nfidealinv(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 itideallist(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.ideallistarch(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.ideallog(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)idealmin(nf,ix,{vdir}): minimum of the ideal ix in the direction vdir in the number field nfidealmul(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 resultidealnorm(nf,x): norm of the ideal x in the number field nfidealpow(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 resultidealprimedec(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 constantidealprincipal(nf,x): returns the principal ideal generated by the algebraic number x in the number field nfidealred(nf,x,{vdir=0}): LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealstar(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 givenidealtwoelt(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 aidealval(nf,x,p): valuation at p given in idealprimedec format of the ideal x in the number field nfideleprincipal(nf,x): returns the principal idele generated by the algebraic number x in the number field nfif(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 alsoincgam(s,x,{y}): incomplete gamma function. y is optional and is the precomputed value of gamma(s)incgamc(s,x): complementary incomplete gamma functionintcirc(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.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 omittedintfouriercos(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.intfourierexp(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.intfouriersin(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.intfuncinit(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.intlaplaceinv(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.intmellininv(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.intmellininvshort(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.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).intnuminit(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,intnuminitgen(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.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 bintnumstep(): gives the default value of m used by all intnum and sumnum  routines, such that the integration step is 1/2^m.isfundamental(x): true(1) if x is a fundamental discriminant (including 1), false(0) if notispower(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.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.ispseudoprime(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 computedissquarefree(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 0kronecker(x,y): kronecker symbol (x/y)lcm(x,{y}): least common multiple of x and y, i.e. x*y / gcd(x,y)length(x): number of non code words in x, number of characters for a stringlex(x,y): compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)lift(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 intmodslindep(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 LLLlistcreate(n): creates an empty list of maximum length nlistinsert(list,x,n): insert x at index n in list, shifting the remaining elements to the rightlistput(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 xlistsort(list,{flag=0}): sort list in place. If flag is non-zero, suppress all but one occurence of each element in listlngamma(x): logarithm of the gamma function of xlog(x): natural logarithm of x.matadjoint(x): adjoint matrix of xmatalgtobasis(nf,x): nfalgtobasis applied to every element of the matrix xmatbasistoalg(nf,x): nfbasistoalg applied to every element of the matrix xmatcompanion(x): companion matrix to polynomial xmatdet(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)matdetint(x): some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with mathnfmodmatdiagonal(x): creates the diagonal matrix whose diagonal entries are the entries of the vector xmateigen(x): eigenvectors of the matrix x given as columns of a matrixmatfrobenius(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.mathess(x): Hessenberg form of xmathilbert(n): Hilbert matrix of order n (n C-integer)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 latticemathnfmodid(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)matimage(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 algorithmsmatimagecompl(x): vector of column indices not corresponding to the indices given by the function matimagematindexrank(x): gives two extraction vectors (rows and columns) for the matrix x such that the extracted matrix is square of maximal rankmatintersect(x,y): intersection of the vector spaces whose bases are the columns of x and ymatinverseimage(x,y): an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwisematisdiagonal(x): true(1) if x is a diagonal matrix, false(0) otherwisematker(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 entriesmatkerint(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 matrixqzmatmuldiagonal(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)matmultodiagonal(x,y): product of matrices x and y, knowing that the result will be a diagonal matrix. Much faster than general multiplication in that casematpascal(n,{q}): Pascal triangle of order n if q is omited. q-Pascal triangle otherwisematrank(x): rank of the matrix xmatrix(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 0smatrixqz(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 xmatsize(x): number of rows and columns of the vector/matrix x as a 2-vectormatsnf(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 dmatsolve(M,B): gaussian solution of MX=B (M matrix, B column vector)matsolvemod(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 solutionsmatsupplement(x): supplement the columns of the matrix x to an invertible matrixmattranspose(x): x~=transpose of xminpoly(A,{v=x}): minimal polynomial of the matrix or polmod A.modreverse(x): reverse polymod of the polymod x, if it existsmoebius(x): Moebius function of xnewtonpoly(x,p): Newton polygon of polynomial x with respect to the prime pnext({n=1}): interrupt execution of current instruction sequence, and start another iteration from the n-th innermost enclosing loopsnextprime(x): smallest pseudoprime >= xnfalgtobasis(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 onlynfbasistoalg(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 xnfdisc(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 nfbasisnfeltdiv(nf,a,b): element a/b in nfnfeltdiveuc(nf,a,b): gives algebraic integer q such that a-bq is smallnfeltdivmodpr(nf,a,b,pr): element a/b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltdivrem(nf,a,b): gives [q,r] such that r=a-bq is smallnfeltmod(nf,a,b): gives r such that r=a-bq is small with q algebraic integernfeltmul(nf,a,b): element a. b in nfnfeltmulmodpr(nf,a,b,pr): element a. b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltpow(nf,a,k): element a^k in nfnfeltpowmodpr(nf,a,k,pr): element a^k 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 smallnfeltreducemodpr(nf,a,pr): element a modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltval(nf,a,pr): valuation of element a at the prime pr as output by idealprimedecnffactor(nf,x): factor polynomial x in number field nfnffactormod(nf,pol,pr): factorize polynomial pol modulo prime ideal pr 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 nfnfgaloisconj(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)nfhilbert(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 pnfhnf(nf,x): if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhnfmod(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_jnfinit(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 2nfisideal(nf,x): true(1) if x is an ideal in the number field nf, false(0) if notnfisincl(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 usednfisisom(x,y): as nfisincl but tests whether x is isomorphic to ynfkermodpr(nf,x,pr): kernel of the matrix x in Z_K/pr, where pr is in modpr format (see nfmodprinit)nfmodprinit(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)nfnewprec(nf): transform the number field data nf into new data using the current (usually larger) precisionnfroots({nf},pol): roots of polynomial pol belonging to nf (Q if omitted) without multiplicitynfrootsof1(nf): number of roots of unity and primitive root of unity in the number field nfnfsnf(nf,x): if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfsolvemodpr(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)nfsubfields(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 nfnorml2(x): square of the L2-norm of the vector xnumbpart(x): number of partitions of xnumdiv(x): number of divisors of xnumtoperm(n,k): permutation number k (mod n!) of n letters (n C-integer)omega(x): number of distinct prime divisors of xpadicappr(x,a): p-adic roots of the polynomial x congruent to a mod ppadicprec(x,p): absolute p-adic precision of object xpermtonum(vect): ordinal (between 1 and n!) of permutation vectpolcoeff(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 otherwisepolcompositum(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 polpolcyclo(n,{v=x}): n-th cyclotomic polynomial (in variable v)poldegree(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 otherwisepoldisc(x,{v}): discriminant of the polynomial x, with respect to main variable if v is omitted, with respect to v otherwisepoldiscreduced(f): vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fpolgalois(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.polhensellift(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 ppolinterpolate(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 valuepolisirreducible(x): true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantpollead(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 vpolrecip(x): reciprocal polynomial of xpolred(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 discriminantpolredabs(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 reductionpolredord(x): reduction of the polynomial x, staying in the same orderpolresultant(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 algorithmpolroots(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 methodpolrootsmod(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 ppolrootspadic(x,p,r): p-adic roots of the polynomial x to precision rpolsturm(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.polsylvestermatrix(x,y): forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowspolsym(x,n): vector of symmetric powers of the roots of x up to npoltchebi(n,{v=x}): Tchebitcheff polynomial of degree n (n C-integer), in variable vpoltschirnhaus(x): random Tschirnhausen transformation of the polynomial xpolylog(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 xpolzagier(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 xprecprime(x): largest pseudoprime <= x, 0 if x<=1prime(n): returns the n-th prime (n C-integer)primepi(x): the prime counting function pi(x) = #{p <= x, p prime}.primes(n): returns the vector of the first n primes (n C-integer)print(a): outputs a (in raw format) ending with newlineprint1(a): outputs a (in raw format) without ending with newlineprintp(a): outputs a (in beautified format) ending with newlineprintp1(a): outputs a (in beautified format) without ending with newlineprinttex(a): outputs a in TeX formatprod(X=a,b,expr,{x=1}): x times the product (X runs from a to b) of expressionprodeuler(X=a,b,expr): Euler product (X runs over the primes between a and b) of real or complex expressionprodinf(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 insteadqfbclassno(x,{flag=0}): class number of discriminant x using Shanks's method by default. If (optional) flag is set to 1, use Euler productsqfbcompraw(x,y): Gaussian composition without reduction of the binary quadratic forms x and yqfbhclassno(x): Hurwitz-Kronecker class number of x>0qfbnucomp(x,y,l): composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedqfbnupow(x,n): n-th power of primitive positive definite quadratic form x using nucomp and nuduplqfbpowraw(x,n): n-th power without reduction of the binary quadratic form xqfbprimeform(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.qfbsolve(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.qfgaussred(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)qfjacobi(x): eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xqflll(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 coefficientsqflllgram(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 polynomialsqfminim(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 formsqfperfection(a): rank of matrix of xx~ for x minimal vectors of a gram matrix aqfrep(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_VECqfsign(x): signature of the symmetric matrix xquadclassunit(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 parametersquaddisc(x): discriminant of the quadratic field Q(sqrt(x))quadgen(x): standard generator of quadratic order of discriminant xquadhilbert(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.quadpoly(D,{v=x}): quadratic polynomial corresponding to the discriminant D, in variable vquadray(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.quadregulator(x): regulator of the real quadratic field of discriminant xquadunit(x): fundamental unit of the quadratic field of discriminant x where x must be positiverandom({N=2^31}): random integer between 0 and N-1readvec({filename}): create a vector whose components are the evaluation of all the expressions found in the input file filenameremoveprimes({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 omittedreorder({x=[]}): reorder the variables for output according to the vector x. If x is void or omitted, print the current list of variablesreturn({x=0}): return from current subroutine with result xrnfalgtobasis(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 setrnfbasistoalg(rnf,x): relative version of nfbasistoalg, where rnf is a relative numberfieldrnfcharpoly(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)rnfconductor(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 extensionrnfdedekind(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 discriminantrnfdet(nf,order): given a pseudomatrix, compute its pseudodeterminantrnfdisc(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*^2rnfeltabstorel(rnf,x): transforms the element x from absolute to relative representationrnfeltdown(rnf,x): expresses x on the base field if possible; returns an error otherwisernfeltreltoabs(rnf,x): transforms the element x from relative to absolute representationrnfeltup(rnf,x): expresses x (belonging to the base field) on the relative fieldrnfequation(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 apolrnfhnfbasis(bnf,order): given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfidealabstorel(rnf,x): transforms the ideal x from absolute to relative representationrnfidealdown(rnf,x): finds the intersection of the ideal x with the base fieldrnfidealhnf(rnf,x): relative version of idealhnf, where rnf is a relative numberfieldrnfidealmul(rnf,x,y): relative version of idealmul, where rnf is a relative numberfieldrnfidealnormabs(rnf,x): absolute norm of the ideal xrnfidealnormrel(rnf,x): relative norm of the ideal xrnfidealreltoabs(rnf,x): transforms the ideal x from relative to absolute representationrnfidealtwoelt(rnf,x): relative version of idealtwoelt, where rnf is a relative numberfieldrnfidealup(rnf,x): lifts the ideal x (of the base field) to the relative fieldrnfinit(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 detailsrnfisfree(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 notrnfisnorm(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/Krnfisnorminit(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 Galoisrnfkummer(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 bnrrnflllgram(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 matrixrnfnormgroup(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]rnfpolred(nf,pol): given a pol with coefficients in nf, finds a list of relative polynomials defining some subfields, hopefully simplerrnfpolredabs(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 reductionrnfpseudobasis(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*^2rnfsteinitz(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 trivialround(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 bitsserconvol(x,y): convolution (or Hadamard product) of two power seriesserlaplace(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))serreverse(x): reversion of the power series xsetintersect(x,y): intersection of the sets x and ysetisset(x): true(1) if x is a set (row vector with strictly increasing entries), false(0) if notsetminus(x,y): set of elements of x not belonging to ysetrand(n): reset the seed of the random number generator to nsetsearch(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 insertedsetunion(x,y): union of the sets x and yshift(x,n): shift x left n bits if n>=0, right -n bits if n<0.shiftmul(x,n): multiply x by 2^n (n>=0 or n<0)sigma(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 1sign(x): sign of x, of type integer, real or fractionsimplify(x): simplify the object x as much as possiblesizebyte(x): number of bytes occupied by the complete tree of the object xsizedigit(x): maximum number of decimal digits minus one of (the coefficients of) xsolve(X=a,b,expr): real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sqr(x): square of x. NOT identical to x*xsqrtint(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 zerosubgrouplist(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 conductorsubst(x,y,z): in expression x, replace the variable y by the expression zsubstpol(x,y,z): in expression x, replace the polynomial y by the expression z, using remainder decomposition of x.substvec(x,v,w): in expression x, make a best effort to replace the variables v1,...,vn by the expression w1,...,wnsum(X=a,b,expr,{x=0}): x plus the sum (X goes from a to b) of expression exprsumalt(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 polynomialssumdiv(n,X,expr): sum of expression expr, X running over the divisors of nsuminf(X=a,expr): infinite sum (X goes from a to infinity) of real or complex expression exprsumnum(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)).sumnumalt(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.sumnuminit(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.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 polynomialstanh(x): hyperbolic tangent of xtaylor(x,y): taylor expansion of x with respect to the main variable of yteichmuller(x): teichmuller character of p-adic number xtheta(q,z): Jacobi sine theta-functionthetanullk(q,k): k'th derivative at z=0 of theta(q,z)thue(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 unconditionalthueinit(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)trap({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 handlertruncate(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 bitstype(x): return the type of the GEN x.until(a,seq): evaluate the expression sequence seq until a is nonzerovaluation(x,p): valuation of x with respect to pvariable(x): main variable of object x. Gives p for p-adic x, error for scalarsvecextract(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.)vecmax(x): maximum of the elements of the vector/matrix xvecmin(x): minimum of the elements of the vector/matrix xvecsort(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 ordervector(n,{X},{expr=0}): row vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0svectorsmall(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 0sweber(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)while(a,seq): while a is nonzero evaluate the expression sequence seq. Otherwise 0write(filename,a): write the string expression a (same output as print) to filenamewrite1(filename,a): write the string expression a (same output as print1) to filenamewritebin(filename,{x}): write x as a binary object to file filename. If x is omitted, write all session variableswritetex(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 numberzetak(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 factorszetakinit(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)).znlog(x,g): g as output by znprimroot (modulo a prime). Return smallest non-negative n such that g^n = xznorder(x,{o}): order of the integermod x in (Z/nZ)*. Optional o is assumed to be a multiple of the order.znprimroot(n): returns a primitive root of n when it existsznstar(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 generatorsq7>incorrect a or b in intnumcode error in intnumx = 0 in Fourierincorrect abscissa in sumnumqrom2: iteration %ld: %Z
qrom3: iteration %ld: %Z
m too large in intnuminitintnuminit0integral transformsumnuminit0both nonzero real and imag. part in coding, real ignoredintegral from infty to infty or from -infty to -inftyincorrect table length in intnum initializationFourier transform of oscillating functionsneed exponential decrease in intinvmellinshortinfinities of the same sign in intnuminitgeninfinities of different type in intnuminitgenexponential increase in integral transformincorrect beginning value in sumnumttNNƼƼܾzļļtJ$??3333333@curve not defined over Rray regulatorray torsion unitsray units.fucurve not defined over a p-adic fieldnon integral index in sumstep equal to zero in forstepforpariprime_loop_initnon integral index in suminfnon integral index in sumalttoo many iterations in solvenon integral index in sumposnon integral index in prodinfnon integral index in sumpos2not a vector in forvecroots must be bracketed in solvenon integral index in prodinf1identical index variables in matrixnegative number of columns in matrixnegative number of rows in matrixnegative number of components in vectorconstant term != 1 in direulernot a vector of two-component vectors in forvec?ףp=
?Choosing t = %ld
aprcl: e(t) too smallIndividual Fermat powerings:
  %-3ld: %3ld
Jacobi sums and tables computed
Step4: q-values (# = %ld, largest = %ld): 
Step5: testing conditions lp
aprcl test fails! this is highly improbableStep6: testing potential divisors
Number of Fermat powerings = %lu
Maximal number of nondeterministic steps = %lu
No such elliptic curve%s/elldata/ell%ldElliptic curves files not available for conductor %ld
[missing %s]Elliptic files %s not compatible
No such elliptic curve in databaseIncorrect curve name in ellsearchIncomplete curve name in ellsearchIncorrect curve name in ellconvertnameIncorrect vector in ellconvertnamesingular curve in ellinitnot a prime in localredreduction mod SL2 (reduce_z)localred (p | c6)localred (nu_D - nu_j != 0,6)[apell1] baby steps, s = %ld[apell1] sorting[apell1] giant steps, i = %ldinitell for 2-adic numbersCM_ellpownorm too large in CM%Z - (%Z)
weipellnum  z  = %Z
  z1 = %Z
  z2 = %Z
badgoodellpointtoz: %s square root
point not on elliptic curve%lu is not prime, use ellakapell (f^(i*s) = 1)torsell (bug2)torsell (bug1)torsell (bug3)not a prime in apellnot an integral modelanell for n >= %luellinit data not accurate enough. Increase precisionw1 and w2 R-linearly dependent in elliptic functionk not a positive even integer in elleisnumprime too large in apell2, use apellnot a rational curve in ellintegralmodelnot an integral curve in elllocalredcan't evaluate log(ellsigma) at lattice pointcan't evaluate ellzeta at a poleincompatible p-adic numbers in initellvaluation of j must be negative in p-adic ellinitpowell for nonintegral CM exponentnot a complex multiplication in powellpowell for non integral, non CM, exponentsorderell for nonrational elliptic curvesexpecting a simple variable in ellwptwo vector/matrix types in bilhellnot an integral model in akellcut-off point must be positive in lseriesellm9N9/99688d788n8[8:9b:F:F:9:9:=)=<=Y><S=><<; ><<>>>>?UNJ<N#NMJKJKxK^KX[#!"@zG!"@	# rational integer roots = %ld:galois files not available
[missing %s]Galois names files not available, please upgrade galdata
[missing %s]galois files %s not compatible
too large precision in preci()partitions( %ld ) is meaninglessindefinite invariant polynomial in gpoly()more than %ld rational integer roots
        all integer roots are double roots
      Working with polynomial #%ld:

$$$$$ Tschirnhaus transformation of degree %ld: $$$$$

*** Entering isin_%ld_G_H_(%ld,%ld)

    Output of isin_%ld_G_H(%ld,%ld): %ld
    Reordering of the roots:     Output of isin_%ld_G_H(%ld,%ld): not included.
Galoisbig: reduced polynomial #1 = %Z
 %ld^%ld	%2ld: %Z
incorrect value in bin()%s/galdata/%s%ld_%ld_%ldopening %sread_object%s/galdata/NAM%ldPartitions of %ld (%ld)
i = %ld: %Z
$$$$$ New prec = %ld
    ----> Group # %ld/%ld:
tschirnCOSRES %d )
discriminant = %Z
EVENODD%s group
galois in degree > 11
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GaloisIndex: Using hash value u=%ld
GaloisIndex: Using hash value w=%ld
galoisindex for groups of order >127Classification of transitive groups of order > 30 is not knownNot a group in group_ident	


	

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	AY9bug%d in kummerreducing beta = %Z
reducebetabeta reduced = %Z
Computing Newton sums: %ld(%ld) [rnfkummer] conductorStep 1
polred(compositum) = %Z
Step 2
[rnfkummer] compositumStep 3
[rnfkummer] bnfinit(Kz)[rnfkummer] Selmer groupStep 4
Step 5
isvirtualunitStep 6
Step 8
Step 9, 10 and 11
Step 12
Step 13
Step 14, 15 and 17
Step 16
Step 18
[rnfkummer] candidate listpolrel(beta) = %Z
not an n-th power in idealsqrtnbeta reduced via ell-th root = %Z
beta LLL-reduced mod U^l = %Z
naive reduction mod U^l: unit exp. = %Z
main variable in kummer must not be xkummer for composite relative degree
[1, 0,  %lu %luerror whilst flushing file %scannot rename file %s to %sMPQS: renamed file %s to %s
MPQS: done sorting one file.
MPQS: decomposed a square
cube5th power7th powerMPQS: decomposed a %s
 %ld %ldMPQS: combining
    {%ld @ %s : %s}
  * {%ld @ %s : %s}
 == {%s}
MPQS: chose primes for A FB[%ld]=%ld%slongershorterMPQS panicking\\ MATRIX READ BY MPQS
FREL=cannot seek FREL fileFREL file truncated?![1]: mpqs_solve_linear_system[2]: mpqs_solve_linear_systemMPQS: X^2 - Y^2 != 0 mod N
	index i = %ld
, looking for more...MPQS: got %ld factors%s
[3]: mpqs_solve_linear_systemcomp.unknown	packaging %ld: %Z ^%ld (%s)
manyseveralMPQS: kN = %Z
MPQS: FB [-1,2,<%lu>,%lu...] Wait a second --
,%luMPQS: sieve threshold = %u
MPQS: starting main loop
MPQSLPTMPFRELFNEWLPRELLPNEWCOMBMPQS: found %lu candidate%s
%s @ %s :%s
MPQS: found factor = %Z
 and combiningMPQS: found %ld factors =
	%Z%s
MPQS: no factors found.

MPQS: giving up.
error whilst writing to file %sMQPS: short of space -- another buffer for sorting
MQPS: line wrap -- another buffer for sorting
MPQS: relations file truncated?!
MPQS: combined %ld full relation%s
MPQS: wrapping, more primes for A now chosen near FB[%ld] = %ld
MPQS: new bit pattern for primes for A: 0x%lX
MPQS: chose Q_%ld(x) = %Z x^2 %c %Z x + C
ftell error on full relations fileMPQS: full relations file %s than expected\\ KERNEL COMPUTED BY MPQS
KERNEL=MPQS: Gauss done: kernel has rank %ld, taking gcds...
MPQS: no solutions found from linear system solverMPQS (relation is a nonsquare)MPQS: wrong relation found after GaussMPQS: splitting N after %ld kernel vector%s
MPQS: got two factors, looking for more...
MPQS: resplitting a factor after %ld kernel vectors
MPQS: wrapping up vector of %ld factors
MPQS: number to factor N = %Z
MPQS: number too big to be factored with MPQS,
	giving upMPQS: factoring number of %ld decimal digits
MPQS: found multiplier %ld for N
MPQS: factoring this number will take %s hours:
N = %ZMPQS: kN has %ld decimal digits
MPQS: Gauss elimination will require more than
	128MBy of memory	(estimated memory needed: %4.1fMBy)
MPQS: creating factor base and allocating arrays...
MPQS: precomputing auxiliary primes up to %ld

MPQS: found factor = %ld whilst creating factor base
MPQS: sizing out of tune, FB size or tolerance
	too largeMPQS: computing logarithm approximations for p_i in FB
MPQS: sizing out of tune, FB too small or
	way too few primes in AMPQS: sieving interval = [%ld, %ld]
MPQS: size of factor base = %ld
MPQS: striving for %ld relations
MPQS: coefficients A will be built from %ld primes each
MPQS: primes for A to be chosen near FB[%ld] = %ld
MPQS: smallest prime used for sieving FB[%ld] = %ld
MPQS: largest prime in FB = %ld
MPQS: bound for `large primes' = %ld
MPQS: first sorting at %ld%%, then every %3.1f%% / %3.1f%%
MPQS: Ran out of primes for A, giving up.

MPQS: passing the %3.1f%% sort point, time = %ld ms

MPQS: passing the %3.1f%% sort point

MPQS: split N whilst combining, time = %ld ms
MPQS: done sorting%s, time = %ld ms
MPQS: found %3.1f%% of the required relations
MPQS: found %ld full relations
MPQS:   (%ld of these from partial relations)
MPQS: Net yield: %4.3g full relations per 100 candidates
MPQS:            %4.3g full relations per 100 polynomials
MPQS: %4.1f%% of the polynomials yielded no candidates
MPQS: next sort point at %3.1f%%

MPQS: starting Gauss over F_2 on %ld relations

MPQS: time in Gauss and gcds = %ld ms
MPQS: found factors = %Z
	and %Z

MPQS: restarting sieving ...
zgTzzAz1
rjrC;rjbrZr;j;CrjrP;rjrKr{s"ZGZ4!ZGZZ!G!4ZGZ!ZGZZbiUA-@ @ @ @ @ @ @ @ @ffffffuYLl>.??>`@
0*?333333?m@@nf_factor_boundexponent: %ld
for this tracenf_LLL_cmbfHensel lift@to find factor %ZnfsqffUsing Trager's method
choice of a prime idealPrime ideal chosen: %Z
bound computation  1) T_2 bound for %s: %Z
  3) Final bound: %Z
splitting mod %Z
Entering nffactor:
squarefree testnfissplit%3ld %s at prime
  %Z
Time: %ld
polynomial variable must have highest priority in nffactormodincorrect variables in rnfcharpolyfor this exponent, GSmin = %Z
Time reduction: %ld
nf_LLL_cmbf: checking factor %ld (avma - bot = %lu)
... mod p^k (avma - bot = %lu)
... lifted (avma - bot = %lu)
remaining modular factor(s): %ld
  2) Conversion from T_2 --> | |^2 bound : %Z
polynomial variable must have highest priority in nffactornumber of factor(s) found: %ld
polynomial variable must have highest priority in nfrootstest if polynomial is square-free
BA?partition functionarg to partition function must be < 10^15Q?diff(CHI) = %ZN0 = %ld
	character no: %ld (%ld/%ld)
S & TArtinNumberRecCoeff (cf = %ld, B = %Z)
RecCoeffS&Tincorrect subgroup in bnrL1Compute WN0 in QuickPol: %ld 
zetavalues = %Z
polrelnum = %Z
quickpolCompute %sIt's not a square...
Compute polrelnumAllStarkpolrel = %Z
RecpolnumCplxModuluscpl = 2^%ld
incorrect subgrp in bnrstarknew precision: %ld
Compute Cl(k)quadhilbertrealFindModulusNot enough precomputed primes (need all p <= %ld)* conductor no %ld/%ld (N = %ld)
	Init: conductor too large in ArtinNumber* Root Number: cond. no %ld/%ld (%ld chars)
RecCoeff3: no solution found!
incorrect character in bnrrootnumberthe ground field must be distinct from Qno non-trivial character in bnrL1Checking the square-root of the Stark unit...
stark (computation impossible)Looking for a modulus of norm: 
Trying modulus = %Z and subgroup = %Z
Trying to find another modulus...No, we're done!
Modulus = %Z and subgroup = %Z
main variable in bnrstark must not be xbase field not totally real in bnrstarkclass field not totally real in bnrstarkffffff?,@)\(4@TR_POL(1), i = %ld/%ldTR_POL(-1), i = %ld/%ldTR_POL, i = %ld/%ldEntering compute_data()

f = %Z
p = %Z, lift to p^%ld
2 * M = %Z
Chosen prime: p = %ld
delta[%ld] = %Z
d-1 test failed
pol. found = %Z
coeff too big for pol g(x)
candidate = %Z
coeff too big for embedding
embedding = %Z
lg(Z) = %ld, lg(Y) = %ld
Z = %Z
Y = %Z

ns = %ld
overflow in calc_block
Subfields of degree %ld: %Z

***** Leaving subfields

2 * Hadamard bound * ind = %Z
sorry, too many block systems in nfsubfieldsp = %ld,	lcm = %ld,	orbits: %Z
changing f(x): p divides disc(g)
lifting embedding mod p^k = %Z^%ld

* Look for subfields of degree %ld


***** Entering subfields

pol = %Z
invalid polynomial in thue (need n>2)expected an integer in bnfisintnormlooking for a fundamental unit of norm -1
%Z eliminated because of sign
invalid polynomial in thue (need deg>2)Semirat. reduction: B0 -> %Z x <= %Z
Non trivial conditional class group.
  *** May miss solutions of the norm equation* Checking for small solutions
  B0  = %Z
  Baker = %Z
  errdelta = %Z
get_embsol = %Z
Partial = %Z
non-monic polynomial in thueinithuec1 = %Z
c2 = %Z
Indice <= %Z
epsilon_3 -> %Z
gcd f_P  does not divide n_p
prec = %d
Checking (\pm %Z, \pm %Z)
Not enough precision in thue  Entering LLL...
C (bitsize) : %d
LLL_First_Pass successful !!
x <= %Z
LLL failed. Increasing kappa
thue (totally rational case)x1 -> %Z
x2 -> %Z
c14 = %Z
* real root no %ld/%ld
  c10 = %Z
  c13 = %Z
  - norm sol. no %ld/%ld
  c6  = %Z
  c8  = %Z
  c11 = %Z
  c15 = %Z
  Entering CF...
    B0 -> %Z
CF failed. Increasing kappa
Semirat. reduction: B0 -> %Z
not a tnf in thueAll solutions are <= %Z
SmallSolsII
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Zerion Mini Shell 1.0