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Properties

Label 12.8.70405533749...192.13
Degree $12$
Signature $[8, 2]$
Discriminant $2^{33}\cdot 31^{10}$
Root discriminant $117.66$
Ramified primes $2, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times A_4$ (as 12T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![492032, 0, -861056, 0, 188356, 0, 9920, 0, -2604, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2604*x^8 + 9920*x^6 + 188356*x^4 - 861056*x^2 + 492032)
 
gp: K = bnfinit(x^12 - 2604*x^8 + 9920*x^6 + 188356*x^4 - 861056*x^2 + 492032, 1)
 

Normalized defining polynomial

\( x^{12} - 2604 x^{8} + 9920 x^{6} + 188356 x^{4} - 861056 x^{2} + 492032 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7040553374918085749768192=2^{33}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.66$
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:  $2, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{3472} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{2} + \frac{3}{7}$, $\frac{1}{3472} a^{7} + \frac{1}{8} a^{3} - \frac{1}{14} a$, $\frac{1}{350672} a^{8} - \frac{11}{175336} a^{6} - \frac{71}{808} a^{4} - \frac{723}{2828} a^{2} - \frac{325}{707}$, $\frac{1}{350672} a^{9} - \frac{11}{175336} a^{7} - \frac{71}{808} a^{5} - \frac{723}{2828} a^{3} - \frac{325}{707} a$, $\frac{1}{16130912} a^{10} + \frac{1}{4032728} a^{8} + \frac{9}{1152208} a^{6} - \frac{3019}{16261} a^{4} + \frac{5505}{65044} a^{2} - \frac{113}{2323}$, $\frac{1}{64523648} a^{11} - \frac{11}{8065456} a^{9} + \frac{1699}{16130912} a^{7} - \frac{8453}{130088} a^{5} + \frac{223875}{520352} a^{3} - \frac{5697}{32522} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 526744665.998 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times A_4$ (as 12T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 16 conjugacy class representatives for $C_4\times A_4$
Character table for $C_4\times A_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.961.1, 6.6.472842752.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
$31$31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$