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Properties

Label 12.0.3656158440062976.16
Degree $12$
Signature $[0, 6]$
Discriminant $2^{20}\cdot 3^{20}$
Root discriminant $19.81$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_3\wr C_2$ (as 12T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, 504, 0, 417, 0, 164, 0, 54, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 12*x^10 + 54*x^8 + 164*x^6 + 417*x^4 + 504*x^2 + 16)
 
gp: K = bnfinit(x^12 + 12*x^10 + 54*x^8 + 164*x^6 + 417*x^4 + 504*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{12} + 12 x^{10} + 54 x^{8} + 164 x^{6} + 417 x^{4} + 504 x^{2} + 16 \)

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:  \(3656158440062976=2^{20}\cdot 3^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.81$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{6} - \frac{1}{8} a^{4} + \frac{1}{16} a^{2} - \frac{1}{4}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{5} + \frac{1}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{32} a^{8} - \frac{1}{32} a^{7} + \frac{1}{16} a^{5} - \frac{3}{32} a^{4} - \frac{1}{32} a^{3} + \frac{7}{16} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{9} + \frac{1}{64} a^{7} - \frac{1}{32} a^{6} - \frac{5}{64} a^{5} + \frac{1}{16} a^{4} - \frac{1}{64} a^{3} - \frac{1}{32} a^{2} - \frac{3}{16} a + \frac{1}{8}$, $\frac{1}{64} a^{10} - \frac{1}{64} a^{8} - \frac{1}{64} a^{6} - \frac{3}{64} a^{4} + \frac{7}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{11} - \frac{1}{128} a^{10} - \frac{1}{128} a^{9} + \frac{1}{128} a^{8} - \frac{1}{128} a^{7} - \frac{3}{128} a^{6} + \frac{13}{128} a^{5} - \frac{5}{128} a^{4} + \frac{3}{32} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8}$

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:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1}{64} a^{9} - \frac{9}{64} a^{7} + \frac{1}{32} a^{6} - \frac{27}{64} a^{5} + \frac{3}{16} a^{4} - \frac{87}{64} a^{3} + \frac{9}{32} a^{2} - \frac{45}{16} a + \frac{7}{8} \) (order $12$)
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:  \( 4079.63492337 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 12T34):

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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 6.4.60466176.4

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.16.5$x^{8} + 4 x^{6} + 40 x^{2} + 4$$4$$2$$16$$D_4$$[2, 3]^{2}$
$3$3.12.20.32$x^{12} + 27 x^{11} + 18 x^{10} + 6 x^{9} - 63 x^{8} + 9 x^{7} - 3 x^{6} + 54 x^{5} + 54 x^{4} - 117 x^{3} + 54 x^{2} + 27 x - 45$$6$$2$$20$12T34$[9/4, 9/4]_{4}^{2}$