# Properties

 Label 6.2.51681125.1 Degree $6$ Signature $[2, 2]$ Discriminant $5^{3}\cdot 643^{2}$ Root discriminant $19.30$ Ramified primes $5, 643$ Class number $4$ Class group $[4]$ Galois group $S_4\times C_2$ (as 6T11)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 4*x^4 - 22*x^3 + 4*x^2 - x - 1)

gp: K = bnfinit(x^6 - x^5 - 4*x^4 - 22*x^3 + 4*x^2 - x - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 4, -22, -4, -1, 1]);

## Normalizeddefining polynomial

$$x^{6} - x^{5} - 4 x^{4} - 22 x^{3} + 4 x^{2} - x - 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$51681125=5^{3}\cdot 643^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $19.30$ 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: $5, 643$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{10} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a - \frac{1}{2}$

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: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{5} - a^{4} - 4 a^{3} - 22 a^{2} + 4 a - 1$$,  $$\frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{3}{5} a^{3} - \frac{19}{5} a^{2} + \frac{26}{5} a + \frac{8}{5}$$,  $$\frac{11}{5} a^{5} - \frac{8}{5} a^{4} - \frac{48}{5} a^{3} - \frac{256}{5} a^{2} - \frac{24}{5} a + \frac{12}{5}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$51.418342098$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_2\times S_4$ (as 6T11):

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 $S_4\times C_2$ Character table for $S_4\times C_2$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: 4.2.643.1 $\times$ $$\Q(\sqrt{-3215})$$ Degree 6 sibling: 6.0.33230963375.2 Degree 8 siblings: 8.4.258405625.2, 8.0.106837547250625.3 Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12 Degree 16 sibling: Deg 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.4.2.1x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
643Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.a.a$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.643.2t1.a.a$1$ $643$ $x^{2} - x + 161$ $C_2$ (as 2T1) $1$ $-1$
1.3215.2t1.a.a$1$ $5 \cdot 643$ $x^{2} - x + 804$ $C_2$ (as 2T1) $1$ $-1$
2.16075.6t3.b.a$2$ $5^{2} \cdot 643$ $x^{6} - 16 x^{4} - 55 x^{3} + 64 x^{2} + 440 x + 1560$ $D_{6}$ (as 6T3) $1$ $0$
* 2.643.3t2.a.a$2$ $643$ $x^{3} - 2 x - 5$ $S_3$ (as 3T2) $1$ $0$
3.643.4t5.a.a$3$ $643$ $x^{4} - x^{3} - 2 x + 1$ $S_4$ (as 4T5) $1$ $1$
3.51681125.6t11.a.a$3$ $5^{3} \cdot 643^{2}$ $x^{6} - x^{5} - 4 x^{4} - 22 x^{3} + 4 x^{2} - x - 1$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 3.80375.6t11.a.a$3$ $5^{3} \cdot 643$ $x^{6} - x^{5} - 4 x^{4} - 22 x^{3} + 4 x^{2} - x - 1$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.413449.6t8.a.a$3$ $643^{2}$ $x^{4} - x^{3} - 2 x + 1$ $S_4$ (as 4T5) $1$ $-1$

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 *.