# Properties

 Label 6.2.7884864.1 Degree $6$ Signature $[2, 2]$ Discriminant $2^{6}\cdot 3^{6}\cdot 13^{2}$ Root discriminant $14.11$ Ramified primes $2, 3, 13$ Class number $1$ Class group Trivial Galois group $A_6$ (as 6T15)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -6, -6, -6, 0, 0, 1]);

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

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

## Normalizeddefining polynomial

$$x^{6} - 6 x^{3} - 6 x^{2} - 6 x - 2$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $6$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[2, 2]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$7884864=2^{6}\cdot 3^{6}\cdot 13^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $14.11$ 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: $2, 3, 13$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

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

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: $3$ 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: $$a^{4} - a^{3} - 5 a - 1$$,  $$a^{5} - 6 a^{2} - 5 a - 5$$,  $$2 a^{5} - 2 a^{4} + a^{3} - 10 a^{2} - 4 a - 7$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$26.7489008607$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$A_6$ (as 6T15):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A non-solvable group of order 360 The 7 conjugacy class representatives for $A_6$ Character table for $A_6$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Sibling algebras

 Twin sextic algebra: 6.2.1971216.1 Degree 6 sibling: 6.2.1971216.1 Degree 10 sibling: 10.2.15542770074624.1 Degree 15 siblings: Deg 15, 15.3.30638157055420022784.1 Degree 20 sibling: 20.4.241577701592627342528741376.1 Degree 30 siblings: data not computed Degree 36 sibling: data not computed Degree 40 sibling: data not computed Degree 45 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 R ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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

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$2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2} 33.6.6.2x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 13.4.2.2x^{4} - 13 x^{2} + 338$$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$1$ $1$ $x$ $C_1$ $1$ $1$
5.1971216.6t15.a.a$5$ $2^{4} \cdot 3^{6} \cdot 13^{2}$ $x^{6} - 6 x^{3} - 6 x^{2} - 6 x - 2$ $A_6$ (as 6T15) $1$ $1$
* 5.7884864.6t15.a.a$5$ $2^{6} \cdot 3^{6} \cdot 13^{2}$ $x^{6} - 6 x^{3} - 6 x^{2} - 6 x - 2$ $A_6$ (as 6T15) $1$ $1$
8.15542770074624.36t555.a.a$8$ $2^{10} \cdot 3^{12} \cdot 13^{4}$ $x^{6} - 6 x^{3} - 6 x^{2} - 6 x - 2$ $A_6$ (as 6T15) $1$ $0$
8.15542770074624.36t555.a.b$8$ $2^{10} \cdot 3^{12} \cdot 13^{4}$ $x^{6} - 6 x^{3} - 6 x^{2} - 6 x - 2$ $A_6$ (as 6T15) $1$ $0$
9.15542770074624.10t26.a.a$9$ $2^{10} \cdot 3^{12} \cdot 13^{4}$ $x^{6} - 6 x^{3} - 6 x^{2} - 6 x - 2$ $A_6$ (as 6T15) $1$ $1$
10.94562213134012416.30t88.a.a$10$ $2^{12} \cdot 3^{14} \cdot 13^{6}$ $x^{6} - 6 x^{3} - 6 x^{2} - 6 x - 2$ $A_6$ (as 6T15) $1$ $-2$

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 *.