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Properties

Label 27.27.1410763180...5561.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 109^{26}$
Root discriminant $396.39$
Ramified primes $3, 109$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7092040020823, -13411789508832, 80431235416368, 115302813487550, -123198584898693, -153407836629198, 81336836424974, 90619763226495, -29277206700381, -29734911579460, 6362362120011, 5963472358983, -876995136787, -767119231971, 78260935833, 64778023437, -4519002660, -3613267269, 165292505, 131962812, -3647358, -3072165, 43818, 43164, -218, -327, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 327*x^25 - 218*x^24 + 43164*x^23 + 43818*x^22 - 3072165*x^21 - 3647358*x^20 + 131962812*x^19 + 165292505*x^18 - 3613267269*x^17 - 4519002660*x^16 + 64778023437*x^15 + 78260935833*x^14 - 767119231971*x^13 - 876995136787*x^12 + 5963472358983*x^11 + 6362362120011*x^10 - 29734911579460*x^9 - 29277206700381*x^8 + 90619763226495*x^7 + 81336836424974*x^6 - 153407836629198*x^5 - 123198584898693*x^4 + 115302813487550*x^3 + 80431235416368*x^2 - 13411789508832*x - 7092040020823)
 
gp: K = bnfinit(x^27 - 327*x^25 - 218*x^24 + 43164*x^23 + 43818*x^22 - 3072165*x^21 - 3647358*x^20 + 131962812*x^19 + 165292505*x^18 - 3613267269*x^17 - 4519002660*x^16 + 64778023437*x^15 + 78260935833*x^14 - 767119231971*x^13 - 876995136787*x^12 + 5963472358983*x^11 + 6362362120011*x^10 - 29734911579460*x^9 - 29277206700381*x^8 + 90619763226495*x^7 + 81336836424974*x^6 - 153407836629198*x^5 - 123198584898693*x^4 + 115302813487550*x^3 + 80431235416368*x^2 - 13411789508832*x - 7092040020823, 1)
 

Normalized defining polynomial

\( x^{27} - 327 x^{25} - 218 x^{24} + 43164 x^{23} + 43818 x^{22} - 3072165 x^{21} - 3647358 x^{20} + 131962812 x^{19} + 165292505 x^{18} - 3613267269 x^{17} - 4519002660 x^{16} + 64778023437 x^{15} + 78260935833 x^{14} - 767119231971 x^{13} - 876995136787 x^{12} + 5963472358983 x^{11} + 6362362120011 x^{10} - 29734911579460 x^{9} - 29277206700381 x^{8} + 90619763226495 x^{7} + 81336836424974 x^{6} - 153407836629198 x^{5} - 123198584898693 x^{4} + 115302813487550 x^{3} + 80431235416368 x^{2} - 13411789508832 x - 7092040020823 \)

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:  \(14107631800772057462614083047171090763038947897293819627757634951745561=3^{36}\cdot 109^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $396.39$
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:  $3, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(981=3^{2}\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{981}(1,·)$, $\chi_{981}(130,·)$, $\chi_{981}(580,·)$, $\chi_{981}(838,·)$, $\chi_{981}(7,·)$, $\chi_{981}(136,·)$, $\chi_{981}(841,·)$, $\chi_{981}(778,·)$, $\chi_{981}(961,·)$, $\chi_{981}(844,·)$, $\chi_{981}(898,·)$, $\chi_{981}(910,·)$, $\chi_{981}(400,·)$, $\chi_{981}(22,·)$, $\chi_{981}(343,·)$, $\chi_{981}(154,·)$, $\chi_{981}(541,·)$, $\chi_{981}(223,·)$, $\chi_{981}(97,·)$, $\chi_{981}(484,·)$, $\chi_{981}(679,·)$, $\chi_{981}(172,·)$, $\chi_{981}(829,·)$, $\chi_{981}(49,·)$, $\chi_{981}(439,·)$, $\chi_{981}(952,·)$, $\chi_{981}(445,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{41} a^{22} + \frac{9}{41} a^{21} - \frac{11}{41} a^{20} + \frac{11}{41} a^{19} - \frac{2}{41} a^{18} + \frac{8}{41} a^{17} - \frac{18}{41} a^{15} - \frac{12}{41} a^{14} - \frac{9}{41} a^{13} - \frac{6}{41} a^{12} - \frac{11}{41} a^{11} + \frac{5}{41} a^{10} - \frac{7}{41} a^{9} + \frac{12}{41} a^{8} - \frac{10}{41} a^{7} - \frac{9}{41} a^{6} - \frac{14}{41} a^{5} + \frac{12}{41} a^{4} - \frac{17}{41} a^{3} - \frac{20}{41} a^{2} + \frac{9}{41} a - \frac{3}{41}$, $\frac{1}{4141} a^{23} + \frac{22}{4141} a^{22} + \frac{803}{4141} a^{21} + \frac{729}{4141} a^{20} + \frac{2068}{4141} a^{19} - \frac{1658}{4141} a^{18} - \frac{1290}{4141} a^{17} + \frac{1048}{4141} a^{16} - \frac{47}{101} a^{15} + \frac{2008}{4141} a^{14} + \frac{6}{101} a^{13} - \frac{663}{4141} a^{12} + \frac{477}{4141} a^{11} + \frac{1247}{4141} a^{10} - \frac{120}{4141} a^{9} + \frac{2032}{4141} a^{8} + \frac{1378}{4141} a^{7} - \frac{1197}{4141} a^{6} - \frac{1236}{4141} a^{5} + \frac{1656}{4141} a^{4} + \frac{1809}{4141} a^{3} + \frac{446}{4141} a^{2} - \frac{1854}{4141} a + \frac{781}{4141}$, $\frac{1}{4141} a^{24} + \frac{16}{4141} a^{22} + \frac{1041}{4141} a^{21} + \frac{1786}{4141} a^{20} - \frac{795}{4141} a^{19} - \frac{1477}{4141} a^{18} - \frac{1983}{4141} a^{17} - \frac{137}{4141} a^{16} + \frac{4}{101} a^{15} + \frac{1116}{4141} a^{14} + \frac{793}{4141} a^{13} + \frac{317}{4141} a^{12} - \frac{1773}{4141} a^{11} - \frac{2}{101} a^{10} - \frac{1489}{4141} a^{9} - \frac{1411}{4141} a^{8} + \frac{504}{4141} a^{7} - \frac{1162}{4141} a^{6} - \frac{38}{4141} a^{5} - \frac{990}{4141} a^{4} - \frac{1073}{4141} a^{3} - \frac{1465}{4141} a^{2} + \frac{1573}{4141} a + \frac{291}{4141}$, $\frac{1}{1039391} a^{25} + \frac{66}{1039391} a^{24} + \frac{102}{1039391} a^{23} + \frac{6615}{1039391} a^{22} + \frac{366093}{1039391} a^{21} + \frac{196440}{1039391} a^{20} + \frac{152787}{1039391} a^{19} - \frac{379817}{1039391} a^{18} + \frac{209717}{1039391} a^{17} - \frac{403247}{1039391} a^{16} - \frac{403959}{1039391} a^{15} + \frac{248753}{1039391} a^{14} + \frac{178548}{1039391} a^{13} + \frac{194835}{1039391} a^{12} + \frac{222175}{1039391} a^{11} + \frac{465456}{1039391} a^{10} + \frac{165624}{1039391} a^{9} + \frac{179898}{1039391} a^{8} - \frac{28867}{1039391} a^{7} - \frac{306831}{1039391} a^{6} - \frac{134135}{1039391} a^{5} - \frac{517776}{1039391} a^{4} - \frac{189106}{1039391} a^{3} + \frac{354511}{1039391} a^{2} + \frac{278874}{1039391} a + \frac{471889}{1039391}$, $\frac{1}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{26} + \frac{54406231017505085197349553655889708568017812500850501946570768838182789267593967839393887913488918810116852787091457690975144402312942785372902688059873614534}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{25} - \frac{190026779967331364851769657952124298196329367026417914455019189101031667297987726881517426790392218010861459699079935256901346311313053816221128307444805523301294}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{24} + \frac{612690884892655128710222063256883261627803801165816304774870790466695297355558577411433841887921466678235778153252215178831684502866899890766103772632977687810534}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{23} - \frac{40270311371992702625561876630210293737101667139871745259781334809457028347281269179715131098625227854753204883343135803019903242391380397131834134309392846776456862}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{22} + \frac{1080275763996339396256786786235272371339265943500003115714089396554582721043251742420471945106565246404219116004197759139981184686986190354253650145034965088244020076}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{21} + \frac{2263006841524710511504612258313567885485636845395588582914496263675099062700756733807817578585008393064281729097422449340029347404469953516307589660783287738680420809}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{20} - \frac{727319153910377904009524663493540002042026443864824163274104830414019780312331084997451961710404850479809733945695856287181319616937095070565241323156111871090857055}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{19} + \frac{1617137915695825849402649095240017067747274017259980082980366506061728238605088511200670330427989889236778524824301064254682912072356673228323499791411731743048661319}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{18} + \frac{1509364978864029654308969564292736174630064110629300361778197346290574830130846115556381475302493149056829571505074196162349595609405162268260086462341131736747388244}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{17} + \frac{2336129456359934125067289691716333779468466113083564550932884023417197816484273669961318933120108540054691215123180495376196119155843956514211360510307944598549562724}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{16} + \frac{2724172221558257435846548885073573772247288109282150191084568584360213116707013536423895798058839750666808422294978873539133633682261851822726027492724743584346957210}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{15} + \frac{76545807981216152101144633311485757293931545913220527748451547071379199450614274773405663162386153283639027221423157329760097758273830831430809261044541030950585484}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{14} + \frac{691017964803415216016283685081526264554924058154189718523773032951058320899767850201743675074106797272728776682691032070676624077550574803304585047223623986236437819}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{13} - \frac{2206475649772034564491814763582093043721072187888530506503042284321540037299923942504423826109492380734081433721257334029237645972614835655277554962328520242023927253}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{12} - \frac{1512468003047462359925410539866863658112063009712461743582511824219027462051288555250918746117910353478541613685771201908433058203702258390453250541148045667611853471}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{11} + \frac{1453885222182794423143989980455463883560952808428538614713658653979830268972252259218887115338529525243879500319910705006113318085257835514494388637435273626327862365}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{10} + \frac{442316911599240722153052741410589091331189354354465581708353333751679803085082049123666304649813023417819654365239634115656458490190914844801847509677552286217386867}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{9} + \frac{7229800870701760894910241326158556692690777879487193058417199589131828484930270352227750736142946015898583220710229138920470928343919642895453039875400732953550357}{62161305219457934544405297029715488303287399008315365747309630530469458631042184419330597685738914957757813871608607649565850153952987512654543471964149484449424593} a^{8} - \frac{52722684287452035569600870173168723061018988653499056680709812986640034483812878027536571357989745450543556960791095140018454065726321998036115657398636301807877554}{153129068955250033877681341463445471186147007313167120499470065453107690774030746984204643079503180749598517098352911526979289403640286311661192455326319461692484973} a^{7} + \frac{121217458907241893737485467943638231431117283215321316975998309836779395697670474218962482825372715751674275357684798742282446106106234172822066198424914721757172559}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{6} - \frac{2507083422904526777317380764757736214935851482241541245504575722990687873505380644314255697032428983778586528317594826208583411373249942758006199327265010387578332513}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{5} + \frac{812489208891700620352384634669206094407884119244256963437705751993579656337759282826994952472995631495933643498930378026032597307100993506638146023137463297986027475}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{4} - \frac{1819263264902305602476569579820822692635033518453339381242463442459124219668920554205949290327110895733413436778794803246576706707345969427062517898395034381675341972}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{3} + \frac{1514455772464306595024364523150918050554158569063959016145519079085882434837812686436976584770330608245554510819037891725054150465175300427085942620336681764220314816}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a^{2} + \frac{1871078939605827432894960637579344859078541798624729018591479202934381953160476160290076538154892450458256342483861654313986604137036005287059862195911231262112757009}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893} a + \frac{456015785895562276113269594542607864324970089485938075410574187324820161509858012282588647817250675160840791691643143407391887668197667143909856397635074283060171355}{6278291827165251388984935000001264318632027299839851940478272683577415321735260626352390366259630410733539201032469372606150865549251738778108890668379097929391883893}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 158126812103398100000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

3.3.11881.1, 9.9.19925626416901921.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{27}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $27$ $27$ $27$

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
3Data not computed
109Data not computed