# Properties

 Label 6.2.37015056.2 Degree $6$ Signature $[2, 2]$ Discriminant $2^{4}\cdot 3^{4}\cdot 13^{4}$ Root discriminant $18.26$ Ramified primes $2, 3, 13$ Class number $1$ Class group Trivial Galois Group $C_3^2:C_4$ (as 6T10)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -18, 9, -6, 6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 6*x^4 - 6*x^3 + 9*x^2 - 18*x - 4)
gp: K = bnfinit(x^6 + 6*x^4 - 6*x^3 + 9*x^2 - 18*x - 4, 1)

## Normalizeddefining polynomial

$$x^{6}$$ $$\mathstrut +\mathstrut 6 x^{4}$$ $$\mathstrut -\mathstrut 6 x^{3}$$ $$\mathstrut +\mathstrut 9 x^{2}$$ $$\mathstrut -\mathstrut 18 x$$ $$\mathstrut -\mathstrut 4$$

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: $$37015056=2^{4}\cdot 3^{4}\cdot 13^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $18.26$ magma: Abs(Discriminant(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(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$

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: $$\frac{1}{2} a^{3} + \frac{3}{2} a$$,  $$\frac{5}{6} a^{5} - \frac{1}{3} a^{4} + \frac{7}{3} a^{3} - \frac{19}{3} a^{2} + \frac{11}{6} a + \frac{2}{3}$$,  $$\frac{1}{2} a^{4} - 2 a^{3} + \frac{7}{2} a^{2} - 6 a + 5$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$92.9323542043$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

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

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 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: 6.2.333135504.1 Degree 6 sibling: 6.2.333135504.1 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2} 2.4.4.2x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0Trivial[\ ] 3.2.1.1x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2} 1313.2.1.1x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.13.2t1.1c1$1$ $13$ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
1.2e2_3_13.4t1.1c1$1$ $2^{2} \cdot 3 \cdot 13$ $x^{4} + 39 x^{2} + 117$ $C_4$ (as 4T1) $0$ $-1$
1.2e2_3_13.4t1.1c2$1$ $2^{2} \cdot 3 \cdot 13$ $x^{4} + 39 x^{2} + 117$ $C_4$ (as 4T1) $0$ $-1$
4.2e4_3e4_13e3.6t10.2c1$4$ $2^{4} \cdot 3^{4} \cdot 13^{3}$ $x^{6} + 6 x^{4} - 6 x^{3} + 9 x^{2} - 18 x - 4$ $C_3^2:C_4$ (as 6T10) $1$ $0$
4.2e4_3e6_13e3.6t10.2c1$4$ $2^{4} \cdot 3^{6} \cdot 13^{3}$ $x^{6} + 6 x^{4} - 6 x^{3} + 9 x^{2} - 18 x - 4$ $C_3^2:C_4$ (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 *.