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Properties

Label 12.6.41085390865563648.2
Degree $12$
Signature $[6, 3]$
Discriminant $-\,2^{33}\cdot 3^{14}$
Root discriminant $24.24$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_3\wr C_2$ (as 12T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18, 72, -72, -48, 267, -72, -296, 24, 90, -8, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - 8*x^9 + 90*x^8 + 24*x^7 - 296*x^6 - 72*x^5 + 267*x^4 - 48*x^3 - 72*x^2 + 72*x - 18)
 
gp: K = bnfinit(x^12 - 12*x^10 - 8*x^9 + 90*x^8 + 24*x^7 - 296*x^6 - 72*x^5 + 267*x^4 - 48*x^3 - 72*x^2 + 72*x - 18, 1)
 

Normalized defining polynomial

\( x^{12} - 12 x^{10} - 8 x^{9} + 90 x^{8} + 24 x^{7} - 296 x^{6} - 72 x^{5} + 267 x^{4} - 48 x^{3} - 72 x^{2} + 72 x - 18 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-41085390865563648=-\,2^{33}\cdot 3^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.24$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{36} a^{10} - \frac{1}{18} a^{9} + \frac{1}{36} a^{8} - \frac{1}{9} a^{7} - \frac{1}{36} a^{6} + \frac{7}{18} a^{5} + \frac{5}{12} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{1297224} a^{11} + \frac{15611}{1297224} a^{10} + \frac{77435}{1297224} a^{9} - \frac{10285}{432408} a^{8} + \frac{61231}{1297224} a^{7} + \frac{111349}{1297224} a^{6} - \frac{263455}{1297224} a^{5} + \frac{41561}{432408} a^{4} - \frac{27689}{216204} a^{3} + \frac{82945}{216204} a^{2} + \frac{14887}{72068} a - \frac{18339}{72068}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 68022.0886135 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 12T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{6}) \), 4.2.18432.2, 6.4.35831808.1 x3

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

Sibling fields

Degree 6 siblings: data not computed
Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: 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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.4.11.13$x^{4} + 4 x^{2} + 14$$4$$1$$11$$D_{4}$$[3, 4]^{2}$
2.4.11.13$x^{4} + 4 x^{2} + 14$$4$$1$$11$$D_{4}$$[3, 4]^{2}$
2.4.11.13$x^{4} + 4 x^{2} + 14$$4$$1$$11$$D_{4}$$[3, 4]^{2}$
$3$3.6.7.6$x^{6} + 3 x^{3} + 3 x^{2} + 3$$6$$1$$7$$S_3\times C_3$$[3/2]_{2}^{3}$
3.6.7.6$x^{6} + 3 x^{3} + 3 x^{2} + 3$$6$$1$$7$$S_3\times C_3$$[3/2]_{2}^{3}$