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Properties

Label 12.0.1518874382041088.21
Degree $12$
Signature $[0, 6]$
Discriminant $2^{31}\cdot 29^{4}$
Root discriminant $18.41$
Ramified primes $2, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:D_4$ (as 12T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 0, -72, 0, 81, 0, -52, 0, 18, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^10 + 18*x^8 - 52*x^6 + 81*x^4 - 72*x^2 + 32)
 
gp: K = bnfinit(x^12 - 4*x^10 + 18*x^8 - 52*x^6 + 81*x^4 - 72*x^2 + 32, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{10} + 18 x^{8} - 52 x^{6} + 81 x^{4} - 72 x^{2} + 32 \)

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:  \(1518874382041088=2^{31}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.41$
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 \(a\))

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

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

Class group and class number

Trivial group, which has order $1$ (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:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1}{8} a^{10} + \frac{1}{4} a^{8} - \frac{13}{8} a^{6} + 3 a^{4} - \frac{9}{4} a^{2} + 1 \) (order $4$)
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:  \( 3390.07895528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:D_4$ (as 12T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 15 conjugacy class representatives for $C_3^2:D_4$
Character table for $C_3^2:D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.512.1, 6.0.861184.1

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

Sibling fields

Degree 24 sibling: data not computed
Degree 36 siblings: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.4.9.2$x^{4} - 2 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.8.22.87$x^{8} + 4 x^{7} + 6 x^{4} + 12 x^{2} + 2$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$