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Properties

Label 27.27.7361439730...5329.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 19^{18}$
Root discriminant $326.28$
Ramified primes $3, 19$
Class number Not computed
Class group Not computed
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![-319768342177193387, -1135430553480940593, 0, 1812704918715185859, 0, -858649698338772249, 0, 189376374570957288, 0, -23533613799022470, 0, 1824136571981646, 0, -93545465229828, 0, 3282297025608, 0, -80025043581, 0, 1354686795, 0, -15617943, 0, 116964, 0, -513, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 513*x^25 + 116964*x^23 - 15617943*x^21 + 1354686795*x^19 - 80025043581*x^17 + 3282297025608*x^15 - 93545465229828*x^13 + 1824136571981646*x^11 - 23533613799022470*x^9 + 189376374570957288*x^7 - 858649698338772249*x^5 + 1812704918715185859*x^3 - 1135430553480940593*x - 319768342177193387)
 
gp: K = bnfinit(x^27 - 513*x^25 + 116964*x^23 - 15617943*x^21 + 1354686795*x^19 - 80025043581*x^17 + 3282297025608*x^15 - 93545465229828*x^13 + 1824136571981646*x^11 - 23533613799022470*x^9 + 189376374570957288*x^7 - 858649698338772249*x^5 + 1812704918715185859*x^3 - 1135430553480940593*x - 319768342177193387, 1)
 

Normalized defining polynomial

\( x^{27} - 513 x^{25} + 116964 x^{23} - 15617943 x^{21} + 1354686795 x^{19} - 80025043581 x^{17} + 3282297025608 x^{15} - 93545465229828 x^{13} + 1824136571981646 x^{11} - 23533613799022470 x^{9} + 189376374570957288 x^{7} - 858649698338772249 x^{5} + 1812704918715185859 x^{3} - 1135430553480940593 x - 319768342177193387 \)

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:  \(73614397307175798532497185733881845387702404681973260785482012855329=3^{94}\cdot 19^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $326.28$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1539=3^{4}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1539}(448,·)$, $\chi_{1539}(1,·)$, $\chi_{1539}(514,·)$, $\chi_{1539}(1027,·)$, $\chi_{1539}(961,·)$, $\chi_{1539}(1132,·)$, $\chi_{1539}(343,·)$, $\chi_{1539}(1474,·)$, $\chi_{1539}(463,·)$, $\chi_{1539}(976,·)$, $\chi_{1539}(1489,·)$, $\chi_{1539}(277,·)$, $\chi_{1539}(790,·)$, $\chi_{1539}(1303,·)$, $\chi_{1539}(856,·)$, $\chi_{1539}(1369,·)$, $\chi_{1539}(292,·)$, $\chi_{1539}(805,·)$, $\chi_{1539}(1318,·)$, $\chi_{1539}(106,·)$, $\chi_{1539}(619,·)$, $\chi_{1539}(172,·)$, $\chi_{1539}(685,·)$, $\chi_{1539}(1198,·)$, $\chi_{1539}(121,·)$, $\chi_{1539}(634,·)$, $\chi_{1539}(1147,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{19} a^{3}$, $\frac{1}{19} a^{4}$, $\frac{1}{19} a^{5}$, $\frac{1}{361} a^{6}$, $\frac{1}{361} a^{7}$, $\frac{1}{361} a^{8}$, $\frac{1}{6859} a^{9}$, $\frac{1}{6859} a^{10}$, $\frac{1}{6859} a^{11}$, $\frac{1}{130321} a^{12}$, $\frac{1}{130321} a^{13}$, $\frac{1}{41808670973} a^{14} + \frac{6937}{41808670973} a^{13} - \frac{14}{2200456367} a^{12} - \frac{90181}{2200456367} a^{11} + \frac{77}{115813493} a^{10} - \frac{94756}{2200456367} a^{9} - \frac{210}{6095447} a^{8} - \frac{29236}{115813493} a^{7} + \frac{294}{320813} a^{6} - \frac{72829}{6095447} a^{5} - \frac{3724}{320813} a^{4} - \frac{110157}{6095447} a^{3} + \frac{17689}{320813} a^{2} + \frac{153228}{320813} a - \frac{13718}{320813}$, $\frac{1}{794364748487} a^{15} - \frac{15}{41808670973} a^{13} + \frac{6937}{41808670973} a^{12} + \frac{90}{2200456367} a^{11} - \frac{83244}{2200456367} a^{10} - \frac{275}{115813493} a^{9} + \frac{53785}{115813493} a^{8} + \frac{450}{6095447} a^{7} - \frac{4538}{115813493} a^{6} - \frac{378}{320813} a^{5} + \frac{44356}{6095447} a^{4} + \frac{2660}{320813} a^{3} + \frac{67287}{320813} a^{2} - \frac{5415}{320813} a - \frac{124494}{320813}$, $\frac{1}{794364748487} a^{16} + \frac{110992}{41808670973} a^{13} - \frac{120}{2200456367} a^{12} - \frac{152707}{2200456367} a^{11} + \frac{880}{115813493} a^{10} - \frac{78612}{2200456367} a^{9} - \frac{2700}{6095447} a^{8} - \frac{6435}{6095447} a^{7} - \frac{148513}{115813493} a^{6} - \frac{85640}{6095447} a^{5} - \frac{48361}{6095447} a^{4} - \frac{53089}{6095447} a^{3} - \frac{60893}{320813} a^{2} - \frac{71765}{320813} a + \frac{115043}{320813}$, $\frac{1}{794364748487} a^{17} - \frac{136}{2200456367} a^{13} - \frac{5040}{41808670973} a^{12} + \frac{1088}{115813493} a^{11} - \frac{128530}{2200456367} a^{10} - \frac{66888}{2200456367} a^{9} + \frac{13875}{115813493} a^{8} + \frac{110917}{115813493} a^{7} - \frac{115476}{115813493} a^{6} - \frac{137154}{6095447} a^{5} + \frac{115436}{6095447} a^{4} - \frac{153027}{6095447} a^{3} - \frac{33693}{320813} a^{2} - \frac{28377}{320813} a + \frac{9758}{320813}$, $\frac{1}{15092930221253} a^{18} + \frac{98922}{41808670973} a^{13} - \frac{816}{2200456367} a^{12} + \frac{6428}{115813493} a^{11} + \frac{6732}{115813493} a^{10} - \frac{40421}{2200456367} a^{9} - \frac{97795}{115813493} a^{8} - \frac{81763}{115813493} a^{7} - \frac{139515}{115813493} a^{6} - \frac{21005}{6095447} a^{5} + \frac{145484}{6095447} a^{4} + \frac{63166}{6095447} a^{3} + \frac{74095}{320813} a^{2} - \frac{80863}{320813} a + \frac{59230}{320813}$, $\frac{1}{15092930221253} a^{19} - \frac{51}{115813493} a^{13} + \frac{81403}{41808670973} a^{12} - \frac{155114}{2200456367} a^{11} - \frac{76644}{2200456367} a^{10} + \frac{5571}{2200456367} a^{9} + \frac{17027}{115813493} a^{8} + \frac{135695}{115813493} a^{7} - \frac{139592}{115813493} a^{6} + \frac{38281}{6095447} a^{5} - \frac{109836}{6095447} a^{4} + \frac{20136}{6095447} a^{3} + \frac{122794}{320813} a^{2} - \frac{109175}{320813} a - \frac{26994}{320813}$, $\frac{1}{15092930221253} a^{20} + \frac{114936}{41808670973} a^{13} - \frac{144980}{41808670973} a^{12} + \frac{129053}{2200456367} a^{11} - \frac{7428}{2200456367} a^{10} + \frac{31078}{2200456367} a^{9} + \frac{141982}{115813493} a^{8} - \frac{79374}{115813493} a^{7} + \frac{46804}{115813493} a^{6} + \frac{33785}{6095447} a^{5} + \frac{153013}{6095447} a^{4} - \frac{154159}{6095447} a^{3} - \frac{62191}{320813} a^{2} + \frac{145005}{320813} a - \frac{82267}{320813}$, $\frac{1}{286765674203807} a^{21} - \frac{315}{115813493} a^{13} + \frac{134092}{41808670973} a^{12} + \frac{129225}{2200456367} a^{11} + \frac{83673}{2200456367} a^{10} + \frac{1019}{115813493} a^{9} + \frac{7313}{6095447} a^{8} - \frac{61206}{115813493} a^{7} - \frac{51898}{115813493} a^{6} - \frac{92143}{6095447} a^{5} - \frac{52301}{6095447} a^{4} - \frac{72765}{6095447} a^{3} - \frac{31976}{320813} a^{2} - \frac{92068}{320813} a - \frac{106775}{320813}$, $\frac{1}{286765674203807} a^{22} + \frac{95880}{41808670973} a^{13} + \frac{117816}{41808670973} a^{12} - \frac{61197}{2200456367} a^{11} - \frac{117541}{2200456367} a^{10} + \frac{41180}{2200456367} a^{9} - \frac{154474}{115813493} a^{8} - \frac{38519}{115813493} a^{7} - \frac{103902}{115813493} a^{6} - \frac{14441}{6095447} a^{5} - \frac{101265}{6095447} a^{4} - \frac{4775}{6095447} a^{3} - \frac{84943}{320813} a^{2} - \frac{101224}{320813} a - \frac{149564}{320813}$, $\frac{1}{286765674203807} a^{23} + \frac{2295}{2200456367} a^{13} - \frac{2129}{2200456367} a^{12} - \frac{115237}{2200456367} a^{11} - \frac{35979}{2200456367} a^{10} + \frac{54244}{2200456367} a^{9} + \frac{113585}{115813493} a^{8} + \frac{100597}{115813493} a^{7} + \frac{106874}{115813493} a^{6} - \frac{72503}{6095447} a^{5} - \frac{152006}{6095447} a^{4} + \frac{37722}{6095447} a^{3} + \frac{15787}{320813} a^{2} - \frac{18869}{320813} a - \frac{51460}{320813}$, $\frac{1}{5448547809872333} a^{24} + \frac{118106}{41808670973} a^{13} - \frac{146393}{41808670973} a^{12} - \frac{62193}{2200456367} a^{11} + \frac{5364}{2200456367} a^{10} + \frac{67391}{2200456367} a^{9} + \frac{112041}{115813493} a^{8} - \frac{116521}{115813493} a^{7} - \frac{152966}{115813493} a^{6} - \frac{59673}{6095447} a^{5} - \frac{45682}{6095447} a^{4} + \frac{25458}{6095447} a^{3} + \frac{129118}{320813} a^{2} - \frac{117460}{320813} a + \frac{43136}{320813}$, $\frac{1}{5448547809872333} a^{25} - \frac{91313}{41808670973} a^{13} + \frac{78107}{41808670973} a^{12} - \frac{69050}{2200456367} a^{11} - \frac{124293}{2200456367} a^{10} - \frac{105468}{2200456367} a^{9} - \frac{147878}{115813493} a^{8} - \frac{116269}{115813493} a^{7} - \frac{107203}{115813493} a^{6} - \frac{141964}{6095447} a^{5} - \frac{124243}{6095447} a^{4} + \frac{159791}{6095447} a^{3} - \frac{160238}{320813} a^{2} - \frac{41702}{320813} a + \frac{72458}{320813}$, $\frac{1}{5448547809872333} a^{26} - \frac{89287}{41808670973} a^{13} + \frac{63832}{41808670973} a^{12} + \frac{126951}{2200456367} a^{11} + \frac{27243}{2200456367} a^{10} - \frac{50383}{2200456367} a^{9} - \frac{609}{6095447} a^{8} + \frac{71715}{115813493} a^{7} - \frac{156787}{115813493} a^{6} + \frac{94770}{6095447} a^{5} + \frac{70170}{6095447} a^{4} - \frac{152544}{6095447} a^{3} - \frac{99500}{320813} a^{2} - \frac{157360}{320813} a + \frac{143031}{320813}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{27})^+\)

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 $27$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ $27$

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
$19$19.9.6.1$x^{9} - 38 x^{6} + 361 x^{3} - 109744$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
19.9.6.1$x^{9} - 38 x^{6} + 361 x^{3} - 109744$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
19.9.6.1$x^{9} - 38 x^{6} + 361 x^{3} - 109744$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$