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Properties

Label 27.27.5659106580...6209.7
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 7^{18}\cdot 13^{18}$
Root discriminant $296.70$
Ramified primes $3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (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![10071022217437, 3190222769922, -28679264186472, -7255493877603, 34657732013436, 6532980509736, -23310660585549, -2952136236060, 9640483374987, 683906165094, -2565990000360, -61101574695, 449758945140, -6381215334, -52467778476, 2286309906, 4084396767, -267294411, -211111831, 17169588, 7112223, -659574, -149400, 15075, 1770, -189, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 189*x^25 + 1770*x^24 + 15075*x^23 - 149400*x^22 - 659574*x^21 + 7112223*x^20 + 17169588*x^19 - 211111831*x^18 - 267294411*x^17 + 4084396767*x^16 + 2286309906*x^15 - 52467778476*x^14 - 6381215334*x^13 + 449758945140*x^12 - 61101574695*x^11 - 2565990000360*x^10 + 683906165094*x^9 + 9640483374987*x^8 - 2952136236060*x^7 - 23310660585549*x^6 + 6532980509736*x^5 + 34657732013436*x^4 - 7255493877603*x^3 - 28679264186472*x^2 + 3190222769922*x + 10071022217437)
 
gp: K = bnfinit(x^27 - 9*x^26 - 189*x^25 + 1770*x^24 + 15075*x^23 - 149400*x^22 - 659574*x^21 + 7112223*x^20 + 17169588*x^19 - 211111831*x^18 - 267294411*x^17 + 4084396767*x^16 + 2286309906*x^15 - 52467778476*x^14 - 6381215334*x^13 + 449758945140*x^12 - 61101574695*x^11 - 2565990000360*x^10 + 683906165094*x^9 + 9640483374987*x^8 - 2952136236060*x^7 - 23310660585549*x^6 + 6532980509736*x^5 + 34657732013436*x^4 - 7255493877603*x^3 - 28679264186472*x^2 + 3190222769922*x + 10071022217437, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 189 x^{25} + 1770 x^{24} + 15075 x^{23} - 149400 x^{22} - 659574 x^{21} + 7112223 x^{20} + 17169588 x^{19} - 211111831 x^{18} - 267294411 x^{17} + 4084396767 x^{16} + 2286309906 x^{15} - 52467778476 x^{14} - 6381215334 x^{13} + 449758945140 x^{12} - 61101574695 x^{11} - 2565990000360 x^{10} + 683906165094 x^{9} + 9640483374987 x^{8} - 2952136236060 x^{7} - 23310660585549 x^{6} + 6532980509736 x^{5} + 34657732013436 x^{4} - 7255493877603 x^{3} - 28679264186472 x^{2} + 3190222769922 x + 10071022217437 \)

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:  \(5659106580522258442740055894009895600932750567734671570320232266209=3^{66}\cdot 7^{18}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $296.70$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2457=3^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2457}(256,·)$, $\chi_{2457}(1,·)$, $\chi_{2457}(898,·)$, $\chi_{2457}(835,·)$, $\chi_{2457}(646,·)$, $\chi_{2457}(79,·)$, $\chi_{2457}(16,·)$, $\chi_{2457}(2395,·)$, $\chi_{2457}(2206,·)$, $\chi_{2457}(2146,·)$, $\chi_{2457}(2083,·)$, $\chi_{2457}(1894,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(1576,·)$, $\chi_{2457}(1387,·)$, $\chi_{2457}(2284,·)$, $\chi_{2457}(1327,·)$, $\chi_{2457}(1264,·)$, $\chi_{2457}(1075,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(1717,·)$, $\chi_{2457}(1654,·)$, $\chi_{2457}(568,·)$, $\chi_{2457}(1465,·)$, $\chi_{2457}(508,·)$, $\chi_{2457}(445,·)$, $\chi_{2457}(757,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{8} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{11} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{12} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{16} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7}$, $\frac{1}{7} a^{17} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{49} a^{18} + \frac{1}{49} a^{17} - \frac{2}{49} a^{16} + \frac{3}{49} a^{15} + \frac{3}{49} a^{14} - \frac{1}{49} a^{13} - \frac{2}{49} a^{11} - \frac{3}{49} a^{10} + \frac{3}{49} a^{9} - \frac{4}{49} a^{8} + \frac{5}{49} a^{7} - \frac{1}{49} a^{6} - \frac{6}{49} a^{5} - \frac{16}{49} a^{4} + \frac{8}{49} a^{3} + \frac{22}{49} a^{2} - \frac{18}{49} a + \frac{1}{49}$, $\frac{1}{49} a^{19} - \frac{3}{49} a^{17} - \frac{2}{49} a^{16} + \frac{3}{49} a^{14} + \frac{1}{49} a^{13} - \frac{2}{49} a^{12} - \frac{1}{49} a^{11} - \frac{1}{49} a^{10} - \frac{5}{49} a^{8} - \frac{6}{49} a^{7} - \frac{19}{49} a^{6} - \frac{3}{49} a^{5} - \frac{18}{49} a^{4} + \frac{1}{7} a^{3} + \frac{16}{49} a^{2} + \frac{12}{49} a - \frac{15}{49}$, $\frac{1}{49} a^{20} + \frac{1}{49} a^{17} + \frac{1}{49} a^{16} - \frac{2}{49} a^{15} + \frac{3}{49} a^{14} + \frac{2}{49} a^{13} - \frac{1}{49} a^{12} - \frac{2}{49} a^{10} - \frac{3}{49} a^{9} + \frac{3}{49} a^{8} - \frac{11}{49} a^{7} + \frac{22}{49} a^{6} - \frac{8}{49} a^{5} - \frac{6}{49} a^{4} - \frac{16}{49} a^{3} + \frac{15}{49} a^{2} - \frac{13}{49} a - \frac{11}{49}$, $\frac{1}{49} a^{21} - \frac{1}{49} a^{14} - \frac{1}{7} a^{8} + \frac{17}{49} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{1}{49}$, $\frac{1}{49} a^{22} - \frac{1}{49} a^{15} - \frac{4}{49} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{13}{49} a - \frac{1}{7}$, $\frac{1}{3577} a^{23} + \frac{25}{3577} a^{22} + \frac{5}{511} a^{21} - \frac{6}{3577} a^{20} + \frac{12}{3577} a^{19} - \frac{24}{3577} a^{18} - \frac{255}{3577} a^{17} - \frac{53}{3577} a^{16} - \frac{232}{3577} a^{15} - \frac{194}{3577} a^{14} - \frac{88}{3577} a^{13} - \frac{179}{3577} a^{12} + \frac{225}{3577} a^{11} - \frac{229}{3577} a^{10} + \frac{222}{3577} a^{9} - \frac{432}{3577} a^{8} - \frac{78}{511} a^{7} + \frac{113}{511} a^{6} - \frac{47}{3577} a^{5} - \frac{447}{3577} a^{4} - \frac{390}{3577} a^{3} + \frac{34}{73} a^{2} - \frac{792}{3577} a - \frac{1034}{3577}$, $\frac{1}{3577} a^{24} - \frac{6}{3577} a^{22} - \frac{5}{3577} a^{21} + \frac{16}{3577} a^{20} - \frac{32}{3577} a^{19} - \frac{20}{3577} a^{18} - \frac{25}{511} a^{17} + \frac{71}{3577} a^{16} + \frac{131}{3577} a^{15} + \frac{163}{3577} a^{14} - \frac{169}{3577} a^{13} + \frac{174}{3577} a^{12} + \frac{205}{3577} a^{11} - \frac{16}{511} a^{10} + \frac{4}{3577} a^{9} + \frac{1348}{3577} a^{8} + \frac{279}{3577} a^{7} + \frac{29}{73} a^{6} - \frac{367}{3577} a^{5} - \frac{1552}{3577} a^{4} + \frac{1123}{3577} a^{3} + \frac{44}{3577} a^{2} - \frac{506}{3577} a + \frac{373}{3577}$, $\frac{1}{3577} a^{25} - \frac{1}{3577} a^{22} + \frac{1}{511} a^{21} + \frac{5}{3577} a^{20} - \frac{3}{511} a^{19} - \frac{27}{3577} a^{18} + \frac{3}{73} a^{17} - \frac{41}{3577} a^{16} + \frac{158}{3577} a^{15} - \frac{34}{511} a^{14} - \frac{62}{3577} a^{13} + \frac{226}{3577} a^{12} + \frac{216}{3577} a^{11} + \frac{236}{3577} a^{10} - \frac{240}{3577} a^{9} + \frac{1775}{3577} a^{8} - \frac{395}{3577} a^{7} - \frac{585}{3577} a^{6} + \frac{137}{3577} a^{5} - \frac{5}{73} a^{4} - \frac{1639}{3577} a^{3} + \frac{1}{7} a^{2} - \frac{1094}{3577} a - \frac{291}{3577}$, $\frac{1}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{26} + \frac{19071981199891293981528674090800373254829176117278747837769310153793460556478023337826143271837166330868722248060}{581972689789251095349143802461308315134868396361918023109871094212376026256027776205288242965851734024218894729852293} a^{25} + \frac{6328566703497012022601768419224239181257282174943662890425920862661585204556048803343034280628699107759071961260}{390639202735250735234356798912385033446692485229232645649091556389129113514320014165193478155160752975160627969352909} a^{24} - \frac{889092227993383160608403985908672815812968397787866384066174550750406074719607880034092484947406461790537993187282}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{23} + \frac{37378979760973212427474376397679763587275356166471898011329998909203456864103025443598771751209466359608318660291210}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{22} + \frac{68047807213291314950719785660287757512347659867325057348273522631710305268863633336175968071201095415210997990521473}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{21} + \frac{216554727606289175918878469393122846116701298975288313223595939616509304547196871209199299953232839200359591377816224}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{20} - \frac{2445892872199067988831408435481985476641972901028503173477981952058395047578855086856820446058991719541219933062936}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{19} + \frac{151997512290010208788140960761274988072466537239151711646710455698712023600080265095441912346000868650332419754503280}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{18} + \frac{1953103136282655009243146634153741675326988822350562583049074648502716798954833459607404620097329164083737764279542343}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{17} - \frac{1207799382273756945749364042707969314933965896509888004387894548657583965806940125197279597568203090132772098238145332}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{16} - \frac{28475107985121324207254916500731939942339380150096131605802418321144721954438234996064999007000836316865261438408712}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{15} + \frac{561476391133933084120967786191743926474305781355680068914856936729484390749829614040305682659631761022550510133052170}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{14} + \frac{5344221749011398757857688974775215513803264227307014920273515188849138135890765384195007082072984415218585370604551}{4073808828524757667444006617229158205944078774533426161769097659486632183792194433437017700760962138169532263108966051} a^{13} - \frac{1527009656245210314294484183568965998945098648193779782698965601913312714748454429708859829049511768521114413436175282}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{12} + \frac{332617837003401398921058192896257784165747408226030241231569213319266919184198011296663343372382803781945434274711271}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{11} - \frac{391474077766477156444469258300475375693165188618839849816448284022270297285163060760848687418944901152937765478778395}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{10} + \frac{833684066218210899380983271299694331177932838795227980759663503813387205601219215974711480108256265424340176332967424}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{9} - \frac{11479363522743766461384611405784005973179944968976551121295675545520777037432012880334272915052749884179035592397535842}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{8} + \frac{8684416917045382791334532599948875256528036946825875188439952798706074014932249179489270597940366179514336864067838010}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{7} + \frac{2597237898209791526925425061616466847520318274165365437108108438307372886049135108239071411598485002053601632427908118}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{6} - \frac{788437179852726313332305677887955403493246890541789439699938071803256191085703155311663050687316516660761262667567595}{4073808828524757667444006617229158205944078774533426161769097659486632183792194433437017700760962138169532263108966051} a^{5} - \frac{12738724211173278893696139200521274499472478908978287316575803960883687628926606142433378391041175840668827486980542235}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{4} - \frac{13734760954627136194392158734844661520605481112984543435389637808013365910958421030431714975064364461127170409660394507}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{3} - \frac{12779511517923211692378486092517993120418260968005494316295673590887073283799617401938357624083080782430963133868666532}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a^{2} - \frac{12183042627644612091831010733390886789761910241297235552347157044127681771527795263673132705741686194480150645733356549}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357} a - \frac{6897434197769894664643195531366750192211900983652951538255154636516312837490559252764350464972841352820470428586002432}{28516661799673303672108046320604107441608551421733983132383683616406425286545361034059123905326734967186725841762762357}$

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:  \( 1837174452627169300000000 \) (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

3.3.13689.2, \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 9.9.2565164201769.1, 9.9.17820338848416865911969.3, 9.9.17820338848416865911969.2, 9.9.3691950281939241.1

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
3Data not computed
$7$7.9.6.2$x^{9} - 49 x^{3} + 686$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.2$x^{9} - 49 x^{3} + 686$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.2$x^{9} - 49 x^{3} + 686$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
13Data not computed