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Properties

Label 12.0.9159316554578733.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{19}\cdot 199^{3}$
Root discriminant $21.39$
Ramified primes $3, 199$
Class number $4$
Class group $[2, 2]$
Galois group $D_4\times A_4$ (as 12T51)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50329, -23661, 42426, -12427, 14679, -2718, 2919, -306, 360, -16, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 27*x^10 - 16*x^9 + 360*x^8 - 306*x^7 + 2919*x^6 - 2718*x^5 + 14679*x^4 - 12427*x^3 + 42426*x^2 - 23661*x + 50329)
 
gp: K = bnfinit(x^12 + 27*x^10 - 16*x^9 + 360*x^8 - 306*x^7 + 2919*x^6 - 2718*x^5 + 14679*x^4 - 12427*x^3 + 42426*x^2 - 23661*x + 50329, 1)
 

Normalized defining polynomial

\( x^{12} + 27 x^{10} - 16 x^{9} + 360 x^{8} - 306 x^{7} + 2919 x^{6} - 2718 x^{5} + 14679 x^{4} - 12427 x^{3} + 42426 x^{2} - 23661 x + 50329 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9159316554578733=3^{19}\cdot 199^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.39$
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:  $3, 199$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{2611398710212731} a^{11} + \frac{432142014121333}{2611398710212731} a^{10} + \frac{44605706463502}{2611398710212731} a^{9} - \frac{7608762519598}{870466236737577} a^{8} - \frac{106761913806307}{870466236737577} a^{7} - \frac{14094428875487}{290155412245859} a^{6} - \frac{202910451971983}{870466236737577} a^{5} - \frac{250872368590096}{870466236737577} a^{4} + \frac{17422975831177}{290155412245859} a^{3} - \frac{625257929732863}{2611398710212731} a^{2} - \frac{552593986965370}{2611398710212731} a + \frac{391017114721883}{2611398710212731}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{104964438890}{870466236737577} a^{11} - \frac{162651530749}{870466236737577} a^{10} - \frac{2342368665350}{870466236737577} a^{9} - \frac{1059812747477}{290155412245859} a^{8} - \frac{27924298871030}{870466236737577} a^{7} - \frac{10285852916047}{290155412245859} a^{6} - \frac{67460117719283}{290155412245859} a^{5} - \frac{143357220602006}{870466236737577} a^{4} - \frac{292508137749963}{290155412245859} a^{3} - \frac{225126149920966}{870466236737577} a^{2} - \frac{2086353618014743}{870466236737577} a - \frac{265316929314613}{870466236737577} \) (order $18$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2380.75581344 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times A_4$ (as 12T51):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 20 conjugacy class representatives for $D_4\times A_4$
Character table for $D_4\times A_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }$ R ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.12.19.43$x^{12} + 3 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{6} + 3 x^{3} + 3$$12$$1$$19$$D_4 \times C_3$$[2]_{4}^{2}$
199Data not computed