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Properties

Label 27.27.1808013057...9521.1
Degree $27$
Signature $[27, 0]$
Discriminant $271^{26}$
Root discriminant $220.22$
Ramified prime $271$
Class number $1$ (GRH)
Class group Trivial (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![-1762213, -208051850, 601104909, 2109337058, -4024809305, -6439483933, 8336484081, 9927852445, -7794039480, -8563493173, 3876101969, 4408831829, -1094616402, -1416926864, 176273890, 292752760, -14938296, -39562422, 387789, 3502857, 36137, -199218, -3280, 6904, 97, -130, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 - 130*x^25 + 97*x^24 + 6904*x^23 - 3280*x^22 - 199218*x^21 + 36137*x^20 + 3502857*x^19 + 387789*x^18 - 39562422*x^17 - 14938296*x^16 + 292752760*x^15 + 176273890*x^14 - 1416926864*x^13 - 1094616402*x^12 + 4408831829*x^11 + 3876101969*x^10 - 8563493173*x^9 - 7794039480*x^8 + 9927852445*x^7 + 8336484081*x^6 - 6439483933*x^5 - 4024809305*x^4 + 2109337058*x^3 + 601104909*x^2 - 208051850*x - 1762213)
 
gp: K = bnfinit(x^27 - x^26 - 130*x^25 + 97*x^24 + 6904*x^23 - 3280*x^22 - 199218*x^21 + 36137*x^20 + 3502857*x^19 + 387789*x^18 - 39562422*x^17 - 14938296*x^16 + 292752760*x^15 + 176273890*x^14 - 1416926864*x^13 - 1094616402*x^12 + 4408831829*x^11 + 3876101969*x^10 - 8563493173*x^9 - 7794039480*x^8 + 9927852445*x^7 + 8336484081*x^6 - 6439483933*x^5 - 4024809305*x^4 + 2109337058*x^3 + 601104909*x^2 - 208051850*x - 1762213, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 130 x^{25} + 97 x^{24} + 6904 x^{23} - 3280 x^{22} - 199218 x^{21} + 36137 x^{20} + 3502857 x^{19} + 387789 x^{18} - 39562422 x^{17} - 14938296 x^{16} + 292752760 x^{15} + 176273890 x^{14} - 1416926864 x^{13} - 1094616402 x^{12} + 4408831829 x^{11} + 3876101969 x^{10} - 8563493173 x^{9} - 7794039480 x^{8} + 9927852445 x^{7} + 8336484081 x^{6} - 6439483933 x^{5} - 4024809305 x^{4} + 2109337058 x^{3} + 601104909 x^{2} - 208051850 x - 1762213 \)

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:  \(1808013057924621803706133777735157383417426695926299555844999521=271^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $220.22$
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:  $271$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(271\)
Dirichlet character group:    $\lbrace$$\chi_{271}(1,·)$, $\chi_{271}(258,·)$, $\chi_{271}(259,·)$, $\chi_{271}(5,·)$, $\chi_{271}(32,·)$, $\chi_{271}(140,·)$, $\chi_{271}(77,·)$, $\chi_{271}(206,·)$, $\chi_{271}(144,·)$, $\chi_{271}(211,·)$, $\chi_{271}(217,·)$, $\chi_{271}(88,·)$, $\chi_{271}(25,·)$, $\chi_{271}(156,·)$, $\chi_{271}(158,·)$, $\chi_{271}(160,·)$, $\chi_{271}(242,·)$, $\chi_{271}(28,·)$, $\chi_{271}(106,·)$, $\chi_{271}(178,·)$, $\chi_{271}(238,·)$, $\chi_{271}(114,·)$, $\chi_{271}(83,·)$, $\chi_{271}(169,·)$, $\chi_{271}(248,·)$, $\chi_{271}(125,·)$, $\chi_{271}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{20} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{21} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{22} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{23} - \frac{1}{9} a^{22} - \frac{1}{9} a^{21} - \frac{1}{9} a^{20} - \frac{1}{9} a^{19} + \frac{2}{9} a^{18} + \frac{1}{3} a^{17} - \frac{1}{9} a^{16} - \frac{1}{3} a^{15} - \frac{2}{9} a^{14} - \frac{4}{9} a^{13} - \frac{2}{9} a^{12} - \frac{4}{9} a^{11} + \frac{2}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{4}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{9} a^{24} + \frac{1}{9} a^{22} + \frac{1}{9} a^{21} + \frac{1}{9} a^{20} + \frac{1}{9} a^{19} + \frac{2}{9} a^{18} - \frac{1}{9} a^{17} + \frac{2}{9} a^{16} + \frac{4}{9} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{7407} a^{25} + \frac{295}{7407} a^{24} - \frac{320}{7407} a^{23} - \frac{1210}{7407} a^{22} - \frac{856}{7407} a^{21} + \frac{863}{7407} a^{20} + \frac{21}{823} a^{19} + \frac{3016}{7407} a^{18} - \frac{2078}{7407} a^{17} - \frac{1088}{2469} a^{16} + \frac{2353}{7407} a^{15} - \frac{275}{2469} a^{14} + \frac{490}{2469} a^{13} - \frac{68}{7407} a^{12} + \frac{790}{2469} a^{11} + \frac{72}{823} a^{10} + \frac{895}{7407} a^{9} - \frac{2749}{7407} a^{8} + \frac{1652}{7407} a^{7} - \frac{3404}{7407} a^{6} + \frac{2404}{7407} a^{5} - \frac{3364}{7407} a^{4} - \frac{604}{7407} a^{3} + \frac{2702}{7407} a^{2} - \frac{3365}{7407} a + \frac{2777}{7407}$, $\frac{1}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{26} + \frac{217343605965674813733539536069483791039484893104239188603791753452615931915406767972410375657237}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{25} + \frac{55588314304989478732024663221783615357203347517299596298479640690194096140723839545414600349110611}{2001281912875106316305197885466088926970794010401506727856580379239261042618062001507735096664892747} a^{24} - \frac{102455224111094043811247037998547470265681043969154410907508828943624363680987883553678869520403251}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{23} + \frac{325195561735362826757656673752770264379910433651110376427351545011236593071755503437106823001781176}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{22} + \frac{828807112937228845156224180496046116860425664457435522661175887950471868102867160199471483732515008}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{21} + \frac{80040599636664511741086587507058095055169157826695986998745325380836796609433683994875929455414411}{2001281912875106316305197885466088926970794010401506727856580379239261042618062001507735096664892747} a^{20} + \frac{247585798982869489237710885308379411873389491779715378485787224297752126863566592900580215284794701}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{19} - \frac{928249202105590061870216819827621699031064363851482293989248844205621937039138079537397541978694705}{2001281912875106316305197885466088926970794010401506727856580379239261042618062001507735096664892747} a^{18} + \frac{178922710511398986419994071872365755620199567087231514374852701413191823915061575492466390110525064}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{17} + \frac{560670278239427219529219373525002271821663007088577793334051870937802814299365995710190416607534263}{2001281912875106316305197885466088926970794010401506727856580379239261042618062001507735096664892747} a^{16} - \frac{2679497894797687333208231092942302110124777336891316628006487908907852126023676919108172698424265793}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{15} + \frac{987811648731194908293262800940954469161360898730706196771245228812095507050214333775930007783676609}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{14} + \frac{1939464826569668131985096461336273483895462845958677418989680604740217658205420575091857428879576805}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{13} - \frac{1880096367837494673807878977700604420273288666134806946072240763491737928993029070955440638452375493}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{12} - \frac{488506531263811640634397621356027991532438900407674796945571571591389502972630132949505553903450259}{2001281912875106316305197885466088926970794010401506727856580379239261042618062001507735096664892747} a^{11} - \frac{235271887272872322449238056291399459982913729644466223737596700874245401177301189987079505910664092}{2001281912875106316305197885466088926970794010401506727856580379239261042618062001507735096664892747} a^{10} + \frac{2001561423679249935999201565931296820827103893803029088055549646212125051844782072811067923210782163}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{9} + \frac{2487144520231485258346986574426823544055328539404962805246436269490444078186444013652853683331606489}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{8} - \frac{847167860341526913620755053183229520242244257533884682063879615790324925334062817278493506422944407}{2001281912875106316305197885466088926970794010401506727856580379239261042618062001507735096664892747} a^{7} + \frac{2782014616751161108241183804814434549476444476274636447646989108567505833180533180226081632197887378}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{6} - \frac{1093365585269894764766106218350450552578601909334433130353158330595741882904079872851817854432144575}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{5} - \frac{2940757815892995459696966388410239216114590699557184513473294509925497695907466612842562616628498450}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{4} - \frac{123135776582746504801632705774694315080520711432453696151988560822167937236997123333261339224313375}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{3} - \frac{856632128614681917974868239891342969722459651942722448540469776564231253637236590585829868730801718}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241} a^{2} - \frac{125919602060091243243830108014897361793709080497903250679360705928769002515410264433399174438303876}{667093970958368772101732628488696308990264670133835575952193459746420347539354000502578365554964249} a - \frac{2160176304481567064113929686051665469139401484466672211358763065275850013032079431631962199509194122}{6003845738625318948915593656398266780912382031204520183569741137717783127854186004523205289994678241}$

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:  \( 94216402766606270000000 \) (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

3.3.73441.1, 9.9.29090710405024191361.1

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 $27$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{9}$ $27$ $27$ $27$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ $27$ $27$

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
271Data not computed