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Properties

Label 27.27.3424497960...8961.1
Degree $27$
Signature $[27, 0]$
Discriminant $19^{24}\cdot 31^{18}$
Root discriminant $135.18$
Ramified primes $19, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16988281, 1023554967, -134762902, -6246677485, 1237075384, 13978784663, -4124731002, -15899447063, 6199663224, 10254915272, -4971409308, -3871097360, 2305955193, 838327648, -644047317, -93223059, 110719275, 2312179, -11833279, 631294, 781658, -78250, -30862, 4039, 665, -101, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 101*x^25 + 665*x^24 + 4039*x^23 - 30862*x^22 - 78250*x^21 + 781658*x^20 + 631294*x^19 - 11833279*x^18 + 2312179*x^17 + 110719275*x^16 - 93223059*x^15 - 644047317*x^14 + 838327648*x^13 + 2305955193*x^12 - 3871097360*x^11 - 4971409308*x^10 + 10254915272*x^9 + 6199663224*x^8 - 15899447063*x^7 - 4124731002*x^6 + 13978784663*x^5 + 1237075384*x^4 - 6246677485*x^3 - 134762902*x^2 + 1023554967*x + 16988281)
 
gp: K = bnfinit(x^27 - 6*x^26 - 101*x^25 + 665*x^24 + 4039*x^23 - 30862*x^22 - 78250*x^21 + 781658*x^20 + 631294*x^19 - 11833279*x^18 + 2312179*x^17 + 110719275*x^16 - 93223059*x^15 - 644047317*x^14 + 838327648*x^13 + 2305955193*x^12 - 3871097360*x^11 - 4971409308*x^10 + 10254915272*x^9 + 6199663224*x^8 - 15899447063*x^7 - 4124731002*x^6 + 13978784663*x^5 + 1237075384*x^4 - 6246677485*x^3 - 134762902*x^2 + 1023554967*x + 16988281, 1)
 

Normalized defining polynomial

\( x^{27} - 6 x^{26} - 101 x^{25} + 665 x^{24} + 4039 x^{23} - 30862 x^{22} - 78250 x^{21} + 781658 x^{20} + 631294 x^{19} - 11833279 x^{18} + 2312179 x^{17} + 110719275 x^{16} - 93223059 x^{15} - 644047317 x^{14} + 838327648 x^{13} + 2305955193 x^{12} - 3871097360 x^{11} - 4971409308 x^{10} + 10254915272 x^{9} + 6199663224 x^{8} - 15899447063 x^{7} - 4124731002 x^{6} + 13978784663 x^{5} + 1237075384 x^{4} - 6246677485 x^{3} - 134762902 x^{2} + 1023554967 x + 16988281 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3424497960405311916385247304908867275520984000621827548961=19^{24}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.18$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $19, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(589=19\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(5,·)$, $\chi_{589}(397,·)$, $\chi_{589}(273,·)$, $\chi_{589}(404,·)$, $\chi_{589}(149,·)$, $\chi_{589}(87,·)$, $\chi_{589}(408,·)$, $\chi_{589}(25,·)$, $\chi_{589}(346,·)$, $\chi_{589}(156,·)$, $\chi_{589}(218,·)$, $\chi_{589}(315,·)$, $\chi_{589}(36,·)$, $\chi_{589}(552,·)$, $\chi_{589}(63,·)$, $\chi_{589}(366,·)$, $\chi_{589}(125,·)$, $\chi_{589}(435,·)$, $\chi_{589}(180,·)$, $\chi_{589}(501,·)$, $\chi_{589}(118,·)$, $\chi_{589}(311,·)$, $\chi_{589}(377,·)$, $\chi_{589}(187,·)$, $\chi_{589}(253,·)$, $\chi_{589}(191,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{26} - \frac{1852116635055632806352788650687112785902163941305169454084341983968382245713239058556801192494358193978865487}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{25} + \frac{552476144181079465235229575362225599109944754367184822392440613640946588018970665348450082199097441062369326}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{24} - \frac{811442244663267085317848834129392639537074666121377624918106735697115826796474594795326580278354335577244169}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{23} + \frac{3538229970736831282870644654234023971218047605471461427596359561913627944852397025580670687157188196178567939}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{22} + \frac{1443291477468143512386362234378552615647685800206685650805579246328352707797802189491204039199624465013188959}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{21} - \frac{2800989538614700711558584779331363874281704905530872679883906911123721865446413073978334674793378574895689275}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{20} + \frac{1387152566618333728680741460004945027642062346181161836755615692724751299159793481766889981126971856664787855}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{19} + \frac{195343297845232707726920804095763757399447718354665384307405195854273035294626541520417575778385970532372724}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{18} + \frac{1599691601991489260133513143132662196819332244903437479053771412650852068039908902267629339127142655165185695}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{17} + \frac{3491907434754594345616473362726576967707164755229681161434064993421911579557342301207107480937946999010242185}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{16} + \frac{6321662647010084619655928754036545594840398291083533732472371021680919563557460651955992374312107878288607161}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{15} + \frac{3533567726866662995256127419289403400465540224764599894679044357299037188961943705558820302001546842317292363}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{14} - \frac{3242181828496091487517296023546286390923791867238198686872984913197702352275155107143554821843100941715560612}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{13} + \frac{3278621976588710291054680121866581281057359370560740509771792165667339768511416451333007940108056872616488089}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{12} + \frac{441882611054543217338360469007227997303687936144299750911257193359117959727670346245391915716137987361835269}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{11} - \frac{4150501714191356290148588545523384607469558757989615659258435873868682406856296812433968306600711190412892379}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{10} - \frac{4332214659519451950732157602284046116501660243994223279626386484579764923380280898352254648279382991602755309}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{9} + \frac{124965205737382932388298537379676696004760051468119392093989968438242603118435706846559227058732478298514111}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{8} + \frac{2015323695401583782464542882464374679966836041916683294117356517620964339935415471212988860628951188199565858}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{7} - \frac{1201005470262809622852730479675319742610160608169004976809179462213387575751931974859434727987076139105561095}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{6} + \frac{2462868587898647116676927001358774399461229364298292277448479071911441248439891676098599454056604950198247532}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{5} - \frac{494757569120943151052284888272088830219793310143656910535193814361455376675076199799297497906114619310106902}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{4} - \frac{5307517050156863123014869907424245842766436166926757464970588156914946147445730280779428683242850204834755811}{14293955033054146479807063760033550954790993215344444421029667345358185060124220784665880962412290141305146062} a^{3} - \frac{331178003415630378680885706534146548218663528037847618130133454572627430764478045645161435213349568887687935}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a^{2} - \frac{1288885857467645978259530428116290990542328095925242092147091614115877536115265651696234663330714012252110893}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031} a - \frac{531641862288105815373518081981543414376809262508645036405581204991080021827167577524760286864326406669394332}{7146977516527073239903531880016775477395496607672222210514833672679092530062110392332940481206145070652573031}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $26$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 104291368915896570000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.961.1, 3.3.361.1, 3.3.346921.1, 3.3.346921.2, 9.9.41753392563387961.1, \(\Q(\zeta_{19})^+\), 9.9.15072974715383053921.2, 9.9.15072974715383053921.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
19Data not computed
$31$31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$