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Properties

Label 27.27.2160296194...0489.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 31^{18}$
Root discriminant $144.72$
Ramified primes $3, 31$
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![-273101111, 1093125672, 2434189824, -6523888629, -7014480921, 15172084575, 8789278983, -18850288581, -5005127394, 13853219952, 686308842, -6222060495, 642477579, 1710171315, -371494179, -282749058, 89638254, 26844534, -11822536, -1262637, 904248, 6678, -39807, 2007, 933, -81, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 81*x^25 + 933*x^24 + 2007*x^23 - 39807*x^22 + 6678*x^21 + 904248*x^20 - 1262637*x^19 - 11822536*x^18 + 26844534*x^17 + 89638254*x^16 - 282749058*x^15 - 371494179*x^14 + 1710171315*x^13 + 642477579*x^12 - 6222060495*x^11 + 686308842*x^10 + 13853219952*x^9 - 5005127394*x^8 - 18850288581*x^7 + 8789278983*x^6 + 15172084575*x^5 - 7014480921*x^4 - 6523888629*x^3 + 2434189824*x^2 + 1093125672*x - 273101111)
 
gp: K = bnfinit(x^27 - 9*x^26 - 81*x^25 + 933*x^24 + 2007*x^23 - 39807*x^22 + 6678*x^21 + 904248*x^20 - 1262637*x^19 - 11822536*x^18 + 26844534*x^17 + 89638254*x^16 - 282749058*x^15 - 371494179*x^14 + 1710171315*x^13 + 642477579*x^12 - 6222060495*x^11 + 686308842*x^10 + 13853219952*x^9 - 5005127394*x^8 - 18850288581*x^7 + 8789278983*x^6 + 15172084575*x^5 - 7014480921*x^4 - 6523888629*x^3 + 2434189824*x^2 + 1093125672*x - 273101111, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 81 x^{25} + 933 x^{24} + 2007 x^{23} - 39807 x^{22} + 6678 x^{21} + 904248 x^{20} - 1262637 x^{19} - 11822536 x^{18} + 26844534 x^{17} + 89638254 x^{16} - 282749058 x^{15} - 371494179 x^{14} + 1710171315 x^{13} + 642477579 x^{12} - 6222060495 x^{11} + 686308842 x^{10} + 13853219952 x^{9} - 5005127394 x^{8} - 18850288581 x^{7} + 8789278983 x^{6} + 15172084575 x^{5} - 7014480921 x^{4} - 6523888629 x^{3} + 2434189824 x^{2} + 1093125672 x - 273101111 \)

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:  \(21602961940568681480203888396858143586678350003185405580489=3^{66}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.72$
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:  $3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(837=3^{3}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{837}(1,·)$, $\chi_{837}(67,·)$, $\chi_{837}(769,·)$, $\chi_{837}(652,·)$, $\chi_{837}(397,·)$, $\chi_{837}(718,·)$, $\chi_{837}(466,·)$, $\chi_{837}(211,·)$, $\chi_{837}(532,·)$, $\chi_{837}(280,·)$, $\chi_{837}(25,·)$, $\chi_{837}(346,·)$, $\chi_{837}(94,·)$, $\chi_{837}(160,·)$, $\chi_{837}(676,·)$, $\chi_{837}(745,·)$, $\chi_{837}(583,·)$, $\chi_{837}(811,·)$, $\chi_{837}(559,·)$, $\chi_{837}(304,·)$, $\chi_{837}(625,·)$, $\chi_{837}(253,·)$, $\chi_{837}(373,·)$, $\chi_{837}(118,·)$, $\chi_{837}(439,·)$, $\chi_{837}(187,·)$, $\chi_{837}(490,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{218} a^{24} + \frac{16}{109} a^{23} - \frac{11}{109} a^{22} + \frac{22}{109} a^{21} - \frac{1}{109} a^{20} - \frac{8}{109} a^{19} + \frac{3}{218} a^{18} - \frac{2}{109} a^{17} + \frac{34}{109} a^{16} + \frac{46}{109} a^{15} + \frac{17}{218} a^{14} + \frac{27}{218} a^{13} + \frac{8}{109} a^{12} - \frac{33}{218} a^{11} - \frac{54}{109} a^{10} + \frac{33}{109} a^{8} + \frac{21}{218} a^{7} + \frac{25}{218} a^{6} + \frac{13}{109} a^{5} + \frac{51}{218} a^{4} + \frac{67}{218} a^{3} - \frac{17}{109} a^{2} - \frac{39}{218} a + \frac{4}{109}$, $\frac{1}{35534} a^{25} + \frac{28}{17767} a^{24} - \frac{4813}{35534} a^{23} + \frac{6383}{35534} a^{22} + \frac{1599}{35534} a^{21} + \frac{3751}{35534} a^{20} - \frac{817}{35534} a^{19} - \frac{3747}{35534} a^{18} + \frac{6403}{35534} a^{17} + \frac{4786}{17767} a^{16} + \frac{10073}{35534} a^{15} + \frac{3923}{35534} a^{14} - \frac{10345}{35534} a^{13} - \frac{8587}{35534} a^{12} - \frac{668}{17767} a^{11} - \frac{8369}{35534} a^{10} + \frac{6355}{17767} a^{9} + \frac{11633}{35534} a^{8} - \frac{3613}{35534} a^{7} + \frac{8597}{17767} a^{6} - \frac{1788}{17767} a^{5} + \frac{6087}{35534} a^{4} - \frac{6165}{35534} a^{3} - \frac{2444}{17767} a^{2} - \frac{16515}{35534} a + \frac{14253}{35534}$, $\frac{1}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{26} + \frac{1613024432059029922260466685535903559325444081285011371852832999567659608046633077376859932770377}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a^{25} - \frac{449607750727370238739433575000081587220516401520083818219177938235042219610488410158776095380547233}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{24} - \frac{89534648660339376288556127865499045062105156503227189754397866990473716670487321277765004908482469145}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{23} - \frac{59385475632362440257440989572984024071377123251120396093698794446704510817347543578190480737379992027}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{22} + \frac{62114116704153225791106051392690950455793964289021276154975031208559806759164587142300101790762098457}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{21} - \frac{6553808839970849382573548890063121838284165369464519945548548494782272170366566401813024916292939379}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{20} + \frac{34534348273194249029289699177041892807014488040148182098170050993531336438844258021112603113327208888}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a^{19} - \frac{30904713209460523522307135402088691353814936916686139547153760349258368695364445646165375065127279449}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a^{18} + \frac{111417817846860230121752059227162101409558893839005388132102449364670882564535219810528276560120761639}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a^{17} - \frac{182968312436103459835798616278213339643881217041501228381680965900000835862650689200157687079696584859}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{16} - \frac{218437564222756898676330530750962451831670899029375667181905135600849863085886385181459240391609544049}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{15} + \frac{41829072139295876141292447761178056692776542079237397908839307203158046390239447548726204506105710201}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{14} + \frac{63035641593294271382589593905561773595502968360391621464583453201212832820260989137251029955002322431}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{13} + \frac{45072722619623130306466263605890248989952238700339059353802751916306085541787085667513633915582647899}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a^{12} + \frac{98230862179187964825657582463069506365272566199688612340544797355181271974492580266862038219532722815}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{11} + \frac{49874437705286559342874575658737499222830742814730015664316370538770400877339375612905014231932103639}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a^{10} + \frac{35258031629989440524465567344710469075432276541743516767301374199875783639686117107803248853843427942}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a^{9} - \frac{65889498045120447964469289933794344288120726692580254423864016037725085665642372873919459404758245793}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{8} - \frac{69062670771478165497544596466706128118475952176644351846392808054725613345129776230221300532179800109}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{7} - \frac{185873860366418557878972509535768858869791351487368137268558415060161163350532799017249898668496408791}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{6} - \frac{68376873303269583134111116134748557145595997146372710158576240415109046768029162191832886582497627118}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a^{5} + \frac{234306478288601605442036859223498945568437790949994593863931412014236094421235576821552099262456587395}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{4} - \frac{72058298184525204498571324789478347388144680829598213783832408831502987712895775420320167147441080961}{519288556849520794439910280544612972418686550499739819404767010269083617781079564643357979786760521126} a^{3} + \frac{845663884801450570805026436918940148987692216396077589491104623712604518346578613989761525248293437}{2382057600227159607522524222681710882654525461007980822957646836096713843032475067171366879755782207} a^{2} + \frac{34736870346392550179749793646010089961371322375298690106086267384794669617905808151780905579078991}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563} a - \frac{62356404919309768705113224020168326319135674368252815723233064679588578674260439124074167617034035429}{259644278424760397219955140272306486209343275249869909702383505134541808890539782321678989893380260563}$

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:  \( 349009037595060140000 \) (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, \(\Q(\zeta_{9})^+\), 3.3.77841.1, 3.3.77841.2, 9.9.471655843734321.1, \(\Q(\zeta_{27})^+\), 9.9.27850805916667920729.2, 9.9.27850805916667920729.1

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
31Data not computed