# Properties

 Label 12.0.1339147769319424.1 Degree $12$ Signature $[0, 6]$ Discriminant $2^{12}\cdot 83^{6}$ Root discriminant $18.22$ Ramified primes $2, 83$ Class number $4$ Class group $[4]$ Galois group $C_2 \times S_4$ (as 12T24)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^10 + 5*x^8 + 51*x^6 - 35*x^4 + 44*x^2 + 4)

gp: K = bnfinit(x^12 - 5*x^10 + 5*x^8 + 51*x^6 - 35*x^4 + 44*x^2 + 4, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 44, 0, -35, 0, 51, 0, 5, 0, -5, 0, 1]);

## Normalizeddefining polynomial

$$x^{12} - 5 x^{10} + 5 x^{8} + 51 x^{6} - 35 x^{4} + 44 x^{2} + 4$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $12$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 6]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$1339147769319424=2^{12}\cdot 83^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $18.22$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 83$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $4$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{100924} a^{10} - \frac{1}{2} a^{9} - \frac{2699}{100924} a^{8} - \frac{1}{2} a^{7} + \frac{4583}{100924} a^{6} - \frac{1}{2} a^{5} - \frac{33823}{100924} a^{4} - \frac{1}{2} a^{3} - \frac{15245}{100924} a^{2} - \frac{1}{2} a - \frac{2997}{50462}$, $\frac{1}{201848} a^{11} + \frac{98225}{201848} a^{9} - \frac{1}{2} a^{8} - \frac{96341}{201848} a^{7} - \frac{1}{2} a^{6} + \frac{67101}{201848} a^{5} - \frac{1}{2} a^{4} - \frac{15245}{201848} a^{3} - \frac{1}{2} a^{2} - \frac{2997}{100924} a - \frac{1}{2}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{4}$, which has order $4$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $5$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$\frac{303}{4388} a^{11} - \frac{1629}{4388} a^{9} + \frac{2041}{4388} a^{7} + \frac{15163}{4388} a^{5} - \frac{16223}{4388} a^{3} + \frac{6807}{2194} a$$ (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$1141.74484523$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_2\times S_4$ (as 12T24):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 48 The 10 conjugacy class representatives for $C_2 \times S_4$ Character table for $C_2 \times S_4$

## Intermediate fields

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

## Sibling fields

 Degree 6 siblings: 6.2.2287148.1, some data not computed Degree 8 siblings: data not computed Degree 12 siblings: data not computed Degree 16 sibling: data not computed Degree 24 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2] 2.2.2.1x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4} 8383.4.2.1x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 83.4.2.1x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$