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Properties

Label 12.4.41920632350...072.16
Degree $12$
Signature $[4, 4]$
Discriminant $2^{35}\cdot 29^{11}$
Root discriminant $165.40$
Ramified primes $2, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2.S_3^2$ (as 12T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![133632, 0, -781376, 0, 124033, 0, -696, 0, -580, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 580*x^8 - 696*x^6 + 124033*x^4 - 781376*x^2 + 133632)
 
gp: K = bnfinit(x^12 - 580*x^8 - 696*x^6 + 124033*x^4 - 781376*x^2 + 133632, 1)
 

Normalized defining polynomial

\( x^{12} - 580 x^{8} - 696 x^{6} + 124033 x^{4} - 781376 x^{2} + 133632 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(419206323505881263292547072=2^{35}\cdot 29^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $165.40$
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:  $2, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} - \frac{3}{7} a^{6} - \frac{1}{7} a^{2} + \frac{3}{7}$, $\frac{1}{84} a^{9} + \frac{8}{21} a^{7} - \frac{3}{7} a^{3} - \frac{11}{84} a$, $\frac{1}{6756485028048} a^{10} - \frac{26947719718}{422280314253} a^{8} + \frac{230496321985}{563040419004} a^{6} - \frac{49554768373}{281520209502} a^{4} + \frac{2935120191121}{6756485028048} a^{2} + \frac{11227879897}{46920034917}$, $\frac{1}{27025940112192} a^{11} - \frac{6839133325}{1689121257012} a^{9} + \frac{525422255749}{2252161676016} a^{7} - \frac{49554768373}{1126080838008} a^{5} + \frac{11622029512897}{27025940112192} a^{3} + \frac{33570753667}{187680139668} a$

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:  $7$
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:  \( 3134372714.58 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.S_3^2$ (as 12T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2.S_3^2$
Character table for $C_2.S_3^2$

Intermediate fields

\(\Q(\sqrt{58}) \), 4.4.49948672.1, 6.2.42006833152.5

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 24 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$2$2.4.11.9$x^{4} + 12 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.8.24.2$x^{8} + 14 x^{4} + 4 x^{2} + 8 x + 6$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$29$29.12.11.4$x^{12} + 232$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$