# Properties

 Label 6.6.1229312.1 Degree $6$ Signature $[6, 0]$ Discriminant $2^{9}\cdot 7^{4}$ Root discriminant $10.35$ Ramified primes $2, 7$ Class number $1$ Class group Trivial Galois group $C_6$ (as 6T1)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 24, 0, -10, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 10*x^4 + 24*x^2 - 8)

gp: K = bnfinit(x^6 - 10*x^4 + 24*x^2 - 8, 1)

## Normalizeddefining polynomial

$$x^{6} - 10 x^{4} + 24 x^{2} - 8$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $6$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[6, 0]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$1229312=2^{9}\cdot 7^{4}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $10.35$ 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: $2, 7$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Gal(K/\Q)|$: $6$ This field is Galois and abelian over $\Q$. Conductor: $$56=2^{3}\cdot 7$$ Dirichlet character group: $\lbrace$$\chi_{56}(1,·), \chi_{56}(37,·), \chi_{56}(25,·), \chi_{56}(9,·), \chi_{56}(29,·), \chi_{56}(53,·)$$\rbrace$ This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$

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: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental units: $$\frac{1}{2} a^{2} - 2$$,  $$\frac{1}{4} a^{4} - \frac{3}{2} a^{2} + 1$$,  $$\frac{1}{4} a^{5} - 2 a^{3} + 3 a + 1$$,  $$\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - 2 a^{3} + \frac{3}{2} a^{2} + 3 a - 1$$,  $$\frac{1}{4} a^{5} - 2 a^{3} + \frac{1}{2} a^{2} + 3 a - 2$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$8.7234383498$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$C_6$ (as 6T1):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A cyclic group of order 6 The 6 conjugacy class representatives for $C_6$ Character table for $C_6$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: $$\Q(\zeta_{7})^+$$ $\times$ $$\Q(\sqrt{2})$$ $\times$ $$\Q$$

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3} 77.3.2.2x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3.2t1.1c1$1$ $2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.7.3t1.1c1$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e3_7.6t1.2c1$1$ $2^{3} \cdot 7$ $x^{6} - 10 x^{4} + 24 x^{2} - 8$ $C_6$ (as 6T1) $0$ $1$
* 1.7.3t1.1c2$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e3_7.6t1.2c2$1$ $2^{3} \cdot 7$ $x^{6} - 10 x^{4} + 24 x^{2} - 8$ $C_6$ (as 6T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.