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Properties

Label 27.27.4873283973...8969.1
Degree $27$
Signature $[27, 0]$
Discriminant $13^{18}\cdot 37^{24}$
Root discriminant $136.96$
Ramified primes $13, 37$
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![70153, -488126, -3332540, 2604404, 26431312, -4261315, -92239753, 2045189, 171925849, -486454, -185148056, 3352006, 120468998, -5298545, -48977088, 3524558, 12673590, -1244428, -2090049, 252089, 215668, -29833, -13298, 2014, 442, -71, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 71*x^25 + 442*x^24 + 2014*x^23 - 13298*x^22 - 29833*x^21 + 215668*x^20 + 252089*x^19 - 2090049*x^18 - 1244428*x^17 + 12673590*x^16 + 3524558*x^15 - 48977088*x^14 - 5298545*x^13 + 120468998*x^12 + 3352006*x^11 - 185148056*x^10 - 486454*x^9 + 171925849*x^8 + 2045189*x^7 - 92239753*x^6 - 4261315*x^5 + 26431312*x^4 + 2604404*x^3 - 3332540*x^2 - 488126*x + 70153)
 
gp: K = bnfinit(x^27 - 6*x^26 - 71*x^25 + 442*x^24 + 2014*x^23 - 13298*x^22 - 29833*x^21 + 215668*x^20 + 252089*x^19 - 2090049*x^18 - 1244428*x^17 + 12673590*x^16 + 3524558*x^15 - 48977088*x^14 - 5298545*x^13 + 120468998*x^12 + 3352006*x^11 - 185148056*x^10 - 486454*x^9 + 171925849*x^8 + 2045189*x^7 - 92239753*x^6 - 4261315*x^5 + 26431312*x^4 + 2604404*x^3 - 3332540*x^2 - 488126*x + 70153, 1)
 

Normalized defining polynomial

\( x^{27} - 6 x^{26} - 71 x^{25} + 442 x^{24} + 2014 x^{23} - 13298 x^{22} - 29833 x^{21} + 215668 x^{20} + 252089 x^{19} - 2090049 x^{18} - 1244428 x^{17} + 12673590 x^{16} + 3524558 x^{15} - 48977088 x^{14} - 5298545 x^{13} + 120468998 x^{12} + 3352006 x^{11} - 185148056 x^{10} - 486454 x^{9} + 171925849 x^{8} + 2045189 x^{7} - 92239753 x^{6} - 4261315 x^{5} + 26431312 x^{4} + 2604404 x^{3} - 3332540 x^{2} - 488126 x + 70153 \)

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:  \(4873283973806953022959748626642827335637181614041911288969=13^{18}\cdot 37^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.96$
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:  $13, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(481=13\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{481}(256,·)$, $\chi_{481}(1,·)$, $\chi_{481}(386,·)$, $\chi_{481}(451,·)$, $\chi_{481}(456,·)$, $\chi_{481}(9,·)$, $\chi_{481}(269,·)$, $\chi_{481}(334,·)$, $\chi_{481}(16,·)$, $\chi_{481}(81,·)$, $\chi_{481}(248,·)$, $\chi_{481}(211,·)$, $\chi_{481}(404,·)$, $\chi_{481}(367,·)$, $\chi_{481}(157,·)$, $\chi_{481}(417,·)$, $\chi_{481}(419,·)$, $\chi_{481}(100,·)$, $\chi_{481}(144,·)$, $\chi_{481}(107,·)$, $\chi_{481}(477,·)$, $\chi_{481}(308,·)$, $\chi_{481}(53,·)$, $\chi_{481}(118,·)$, $\chi_{481}(120,·)$, $\chi_{481}(380,·)$, $\chi_{481}(445,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{31} a^{18} - \frac{14}{31} a^{17} + \frac{1}{31} a^{16} - \frac{3}{31} a^{15} - \frac{8}{31} a^{14} + \frac{7}{31} a^{13} + \frac{12}{31} a^{12} + \frac{13}{31} a^{10} - \frac{4}{31} a^{9} + \frac{7}{31} a^{8} + \frac{11}{31} a^{7} - \frac{14}{31} a^{6} - \frac{12}{31} a^{5} + \frac{1}{31} a^{4} - \frac{5}{31} a^{3} + \frac{14}{31} a^{2} - \frac{15}{31} a$, $\frac{1}{31} a^{19} - \frac{9}{31} a^{17} + \frac{11}{31} a^{16} + \frac{12}{31} a^{15} - \frac{12}{31} a^{14} - \frac{14}{31} a^{13} + \frac{13}{31} a^{12} + \frac{13}{31} a^{11} - \frac{8}{31} a^{10} + \frac{13}{31} a^{9} - \frac{15}{31} a^{8} - \frac{15}{31} a^{7} + \frac{9}{31} a^{6} - \frac{12}{31} a^{5} + \frac{9}{31} a^{4} + \frac{6}{31} a^{3} - \frac{5}{31} a^{2} + \frac{7}{31} a$, $\frac{1}{31} a^{20} + \frac{9}{31} a^{17} - \frac{10}{31} a^{16} - \frac{8}{31} a^{15} + \frac{7}{31} a^{14} + \frac{14}{31} a^{13} - \frac{3}{31} a^{12} - \frac{8}{31} a^{11} + \frac{6}{31} a^{10} + \frac{11}{31} a^{9} - \frac{14}{31} a^{8} + \frac{15}{31} a^{7} - \frac{14}{31} a^{6} - \frac{6}{31} a^{5} + \frac{15}{31} a^{4} + \frac{12}{31} a^{3} + \frac{9}{31} a^{2} - \frac{11}{31} a$, $\frac{1}{31} a^{21} - \frac{8}{31} a^{17} + \frac{14}{31} a^{16} + \frac{3}{31} a^{15} - \frac{7}{31} a^{14} - \frac{4}{31} a^{13} + \frac{8}{31} a^{12} + \frac{6}{31} a^{11} - \frac{13}{31} a^{10} - \frac{9}{31} a^{9} + \frac{14}{31} a^{8} + \frac{11}{31} a^{7} - \frac{4}{31} a^{6} - \frac{1}{31} a^{5} + \frac{3}{31} a^{4} - \frac{8}{31} a^{3} - \frac{13}{31} a^{2} + \frac{11}{31} a$, $\frac{1}{31} a^{22} - \frac{5}{31} a^{17} + \frac{11}{31} a^{16} - \frac{6}{31} a^{14} + \frac{2}{31} a^{13} + \frac{9}{31} a^{12} - \frac{13}{31} a^{11} + \frac{2}{31} a^{10} + \frac{13}{31} a^{9} + \frac{5}{31} a^{8} - \frac{9}{31} a^{7} + \frac{11}{31} a^{6} + \frac{9}{31} a^{3} - \frac{1}{31} a^{2} + \frac{4}{31} a$, $\frac{1}{2263} a^{23} - \frac{1}{2263} a^{22} - \frac{1}{2263} a^{21} - \frac{7}{2263} a^{20} - \frac{17}{2263} a^{19} - \frac{7}{2263} a^{18} + \frac{886}{2263} a^{17} + \frac{538}{2263} a^{16} + \frac{1027}{2263} a^{15} + \frac{29}{73} a^{14} + \frac{261}{2263} a^{13} - \frac{68}{2263} a^{12} + \frac{14}{31} a^{11} - \frac{342}{2263} a^{10} + \frac{362}{2263} a^{9} - \frac{743}{2263} a^{8} - \frac{390}{2263} a^{7} + \frac{865}{2263} a^{6} + \frac{922}{2263} a^{5} - \frac{99}{2263} a^{4} - \frac{612}{2263} a^{3} + \frac{632}{2263} a^{2} + \frac{128}{2263} a$, $\frac{1}{97309} a^{24} - \frac{1}{97309} a^{23} + \frac{1167}{97309} a^{22} - \frac{80}{97309} a^{21} - \frac{90}{97309} a^{20} - \frac{80}{97309} a^{19} + \frac{1105}{97309} a^{18} - \frac{39466}{97309} a^{17} + \frac{44681}{97309} a^{16} + \frac{22361}{97309} a^{15} + \frac{8218}{97309} a^{14} - \frac{9485}{97309} a^{13} + \frac{145}{1333} a^{12} + \frac{31194}{97309} a^{11} + \frac{40585}{97309} a^{10} + \frac{21522}{97309} a^{9} - \frac{48497}{97309} a^{8} + \frac{19115}{97309} a^{7} - \frac{15795}{97309} a^{6} - \frac{46600}{97309} a^{5} + \frac{47422}{97309} a^{4} - \frac{20100}{97309} a^{3} + \frac{29839}{97309} a^{2} - \frac{174}{1333} a + \frac{4}{43}$, $\frac{1}{43107887} a^{25} + \frac{11}{43107887} a^{24} + \frac{4810}{43107887} a^{23} + \frac{594123}{43107887} a^{22} + \frac{541481}{43107887} a^{21} + \frac{240070}{43107887} a^{20} + \frac{35319}{43107887} a^{19} - \frac{503807}{43107887} a^{18} - \frac{4118354}{43107887} a^{17} + \frac{20480003}{43107887} a^{16} + \frac{20459761}{43107887} a^{15} + \frac{14279862}{43107887} a^{14} - \frac{8158210}{43107887} a^{13} + \frac{19462505}{43107887} a^{12} - \frac{17866623}{43107887} a^{11} + \frac{19605530}{43107887} a^{10} - \frac{8273359}{43107887} a^{9} - \frac{12165023}{43107887} a^{8} + \frac{2269286}{43107887} a^{7} + \frac{3041578}{43107887} a^{6} - \frac{7877248}{43107887} a^{5} - \frac{4257705}{43107887} a^{4} + \frac{12851265}{43107887} a^{3} - \frac{17032482}{43107887} a^{2} - \frac{19598765}{43107887} a - \frac{7563}{19049}$, $\frac{1}{15560111481162628312742141612974719254081955595935637719659773811} a^{26} - \frac{159880898860149690730683460159893504773401060183256290635}{15560111481162628312742141612974719254081955595935637719659773811} a^{25} + \frac{13119602753777496883485430753410193930443707957340118885268}{15560111481162628312742141612974719254081955595935637719659773811} a^{24} + \frac{381017496630026682402585155766589354062021494032643447996641}{15560111481162628312742141612974719254081955595935637719659773811} a^{23} + \frac{208778187191037938803843131614701372176387644147381283492914141}{15560111481162628312742141612974719254081955595935637719659773811} a^{22} - \frac{129534887488685653323821316752720600858865779422905326976676686}{15560111481162628312742141612974719254081955595935637719659773811} a^{21} - \frac{208782439124192336276302329056532557080213171182433844478050224}{15560111481162628312742141612974719254081955595935637719659773811} a^{20} - \frac{87777054089231246335464927590757374487505417608189995877842059}{15560111481162628312742141612974719254081955595935637719659773811} a^{19} - \frac{25797415965838812527253049078248472933106215635629480830688484}{15560111481162628312742141612974719254081955595935637719659773811} a^{18} - \frac{419599349468425960091266522976576150460636296626111207504675188}{15560111481162628312742141612974719254081955595935637719659773811} a^{17} - \frac{2552460747253948217005819905818053109821100601341661179283810999}{15560111481162628312742141612974719254081955595935637719659773811} a^{16} + \frac{1991986497189392320706076783247314238291558810585849304834482402}{15560111481162628312742141612974719254081955595935637719659773811} a^{15} - \frac{2298727156825478386931012093778348203614289361318585044381039699}{15560111481162628312742141612974719254081955595935637719659773811} a^{14} + \frac{5154282919825455030660479343238775287440446815533085029569147442}{15560111481162628312742141612974719254081955595935637719659773811} a^{13} - \frac{7383937273788646238177322460044057171971944007636388859763283412}{15560111481162628312742141612974719254081955595935637719659773811} a^{12} - \frac{4695211418245950189687181256877634392681997887553386335295743036}{15560111481162628312742141612974719254081955595935637719659773811} a^{11} + \frac{5127048851949765139809383795842796612593234591206307737041081020}{15560111481162628312742141612974719254081955595935637719659773811} a^{10} + \frac{652104230204412377732232108370416444831640654115186697498590775}{15560111481162628312742141612974719254081955595935637719659773811} a^{9} - \frac{5303777872557193239563076489137264568235735062876766619906779617}{15560111481162628312742141612974719254081955595935637719659773811} a^{8} - \frac{7351243073544350030163245221835147535058843698267833169761427547}{15560111481162628312742141612974719254081955595935637719659773811} a^{7} + \frac{523068457537499906816844426189916613153597686951640510441166739}{15560111481162628312742141612974719254081955595935637719659773811} a^{6} + \frac{5256963898649261645922860533462132477411710224808977194303924703}{15560111481162628312742141612974719254081955595935637719659773811} a^{5} + \frac{6921523101473334840090198536805313662582222911997832392657455}{213152212070720935790988241273626291151807610903227913967942107} a^{4} - \frac{4881584437398854397438563017503822891588866690385287552238537680}{15560111481162628312742141612974719254081955595935637719659773811} a^{3} + \frac{5412349858734268785654774275020396768623739512326199398332543741}{15560111481162628312742141612974719254081955595935637719659773811} a^{2} + \frac{1859971267049786465646663645031162742837566300956377232796497548}{15560111481162628312742141612974719254081955595935637719659773811} a + \frac{3292424939490543484203914775308103653991096335395355468134870}{6875877808732933412612523912052461004897019706555739160256197}$

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:  \( 326889469217849300000 \) (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

3.3.169.1, 3.3.1369.1, 3.3.231361.2, 3.3.231361.1, 9.9.12384271322498881.1, 9.9.16954067440500968089.2, 9.9.3512479453921.1, 9.9.16954067440500968089.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{27}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
13Data not computed
37Data not computed