# Properties

 Label 6.2.12425625.2 Degree $6$ Signature $[2, 2]$ Discriminant $3^{2}\cdot 5^{4}\cdot 47^{2}$ Root discriminant $15.22$ Ramified primes $3, 5, 47$ Class number $3$ Class group $[3]$ 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![1, 12, 9, -9, 9, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 9*x^4 - 9*x^3 + 9*x^2 + 12*x + 1)
gp: K = bnfinit(x^6 - 3*x^5 + 9*x^4 - 9*x^3 + 9*x^2 + 12*x + 1, 1)

## Normalizeddefining polynomial

$$x^{6}$$ $$\mathstrut -\mathstrut 3 x^{5}$$ $$\mathstrut +\mathstrut 9 x^{4}$$ $$\mathstrut -\mathstrut 9 x^{3}$$ $$\mathstrut +\mathstrut 9 x^{2}$$ $$\mathstrut +\mathstrut 12 x$$ $$\mathstrut +\mathstrut 1$$

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: $$12425625=3^{2}\cdot 5^{4}\cdot 47^{2}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $15.22$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 5, 47$ 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 $$a$$)

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

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

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

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: $$\frac{1}{6} a^{5} - \frac{2}{3} a^{4} + \frac{11}{6} a^{3} - \frac{7}{3} a^{2} + \frac{13}{6} a + \frac{11}{6}$$,  $$\frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{5}{6} a^{3} - \frac{2}{3} a^{2} + \frac{1}{6} a + \frac{1}{6}$$,  $$\frac{2}{3} a^{5} - \frac{7}{3} a^{4} + \frac{22}{3} a^{3} - \frac{29}{3} a^{2} + \frac{35}{3} a + \frac{8}{3}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$8.1176351334$$ 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: 6.2.12425625.1 Degree 6 sibling: 6.2.12425625.1 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ R ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 3.4.2.2x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{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}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2} 47.4.2.2x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.3_5_47.4t1.1c1$1$ $3 \cdot 5 \cdot 47$ $x^{4} - x^{3} + 176 x^{2} - 176 x + 6301$ $C_4$ (as 4T1) $0$ $-1$
1.3_5_47.4t1.1c2$1$ $3 \cdot 5 \cdot 47$ $x^{4} - x^{3} + 176 x^{2} - 176 x + 6301$ $C_4$ (as 4T1) $0$ $-1$
* 4.3e2_5e3_47e2.6t10.3c1$4$ $3^{2} \cdot 5^{3} \cdot 47^{2}$ $x^{6} - 3 x^{5} + 9 x^{4} - 9 x^{3} + 9 x^{2} + 12 x + 1$ $C_3^2:C_4$ (as 6T10) $1$ $0$
4.3e2_5e3_47e2.6t10.4c1$4$ $3^{2} \cdot 5^{3} \cdot 47^{2}$ $x^{6} - 3 x^{5} + 9 x^{4} - 9 x^{3} + 9 x^{2} + 12 x + 1$ $C_3^2:C_4$ (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 *.