# Properties

 Label 6.2.21160000.4 Degree $6$ Signature $[2, 2]$ Discriminant $2^{6}\cdot 5^{4}\cdot 23^{2}$ Root discriminant $16.63$ Ramified primes $2, 5, 23$ Class number $3$ Class group $[3]$ Galois group $C_3^2:C_4$ (as 6T10)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 5*x^4 + 10*x^3 - 10*x^2 + 44*x + 76)

gp: K = bnfinit(x^6 - x^5 + 5*x^4 + 10*x^3 - 10*x^2 + 44*x + 76, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![76, 44, -10, 10, 5, -1, 1]);

## Normalizeddefining polynomial

$$x^{6} - x^{5} + 5 x^{4} + 10 x^{3} - 10 x^{2} + 44 x + 76$$

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: $$21160000=2^{6}\cdot 5^{4}\cdot 23^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $16.63$ 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: $2, 5, 23$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{50} a^{5} - \frac{2}{5} a^{3} + \frac{3}{10} a^{2} - \frac{2}{5} a + \frac{12}{25}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{3}$, which has order $3$

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: $$\frac{3}{50} a^{5} - \frac{1}{5} a^{4} + \frac{3}{5} a^{3} - \frac{3}{10} a^{2} - \frac{2}{5} a + \frac{106}{25}$$,  $$\frac{1}{10} a^{5} - \frac{3}{10} a^{4} + \frac{7}{10} a^{3} - \frac{4}{5} a^{2} - \frac{9}{5} a + \frac{13}{5}$$,  $$\frac{1}{25} a^{5} + \frac{1}{5} a^{3} + \frac{3}{5} a^{2} + \frac{1}{5} a - \frac{1}{25}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$13.2486785184$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_3:S_3.C_2$ (as 6T10):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 36 The 6 conjugacy class representatives for $C_3^2:C_4$ Character table for $C_3^2:C_4$

## Intermediate fields

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

## Sibling algebras

 Galois closure: data not computed Twin sextic algebra: 6.2.21160000.3 Degree 6 sibling: 6.2.21160000.3 Degree 9 sibling: data not computed Degree 12 siblings: data not computed Degree 18 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2} 2.4.6.4x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.4.3.2x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 23.4.2.2x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

## 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) $1$ $1$
1.920.4t1.a.a$1$ $2^{3} \cdot 5 \cdot 23$ $x^{4} + 230 x^{2} + 10580$ $C_4$ (as 4T1) $0$ $-1$
1.920.4t1.a.b$1$ $2^{3} \cdot 5 \cdot 23$ $x^{4} + 230 x^{2} + 10580$ $C_4$ (as 4T1) $0$ $-1$
* 4.4232000.6t10.c.a$4$ $2^{6} \cdot 5^{3} \cdot 23^{2}$ $x^{6} - x^{5} + 5 x^{4} + 10 x^{3} - 10 x^{2} + 44 x + 76$ $C_3^2:C_4$ (as 6T10) $1$ $0$
4.4232000.6t10.d.a$4$ $2^{6} \cdot 5^{3} \cdot 23^{2}$ $x^{6} - x^{5} + 5 x^{4} + 10 x^{3} - 10 x^{2} + 44 x + 76$ $C_3^2:C_4$ (as 6T10) $1$ $0$

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 *.