# Properties

 Label 6.2.255476.1 Degree $6$ Signature $[2, 2]$ Discriminant $2^{2}\cdot 13\cdot 17^{3}$ Root discriminant $7.97$ Ramified primes $2, 13, 17$ Class number $1$ Class group Trivial Galois group $C_3^2:D_4$ (as 6T13)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

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

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

## Normalizeddefining polynomial

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

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: $$255476=2^{2}\cdot 13\cdot 17^{3}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $7.97$ 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: $2, 13, 17$ 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: $$a$$,  $$a^{4} - a^{2} - 1$$,  $$a^{3} - 2 a$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$3.10473042386$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$S_3\wr C_2$ (as 6T13):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 72 The 9 conjugacy class representatives for $C_3^2:D_4$ Character table for $C_3^2:D_4$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: 6.2.597584.1 Degree 6 sibling: 6.2.597584.1 Degree 9 sibling: 9.1.690807104.1 Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, 12.0.143393766507472.1 Degree 18 siblings: Deg 18, Deg 18, Deg 18 Degree 24 siblings: data not computed Degree 36 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R R ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3} 2.3.2.1x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{13}$$x + 2$$1$$1$$0Trivial[\ ] 13.2.1.2x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 1717.6.3.1x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

## 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_17.2t1.1c1$1$ $13 \cdot 17$ $x^{2} - x - 55$ $C_2$ (as 2T1) $1$ $1$
* 1.17.2t1.1c1$1$ $17$ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
1.13.2t1.1c1$1$ $13$ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
2.13_17.4t3.2c1$2$ $13 \cdot 17$ $x^{4} - x^{3} + x^{2} - 2 x + 4$ $D_{4}$ (as 4T3) $1$ $-2$
4.2e2_13e3_17e2.12t34.2c1$4$ $2^{2} \cdot 13^{3} \cdot 17^{2}$ $x^{6} - x^{5} - 2 x^{4} + x^{3} + 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
* 4.2e2_13_17e2.6t13.2c1$4$ $2^{2} \cdot 13 \cdot 17^{2}$ $x^{6} - x^{5} - 2 x^{4} + x^{3} + 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e4_13e2_17.6t13.2c1$4$ $2^{4} \cdot 13^{2} \cdot 17$ $x^{6} - x^{5} - 2 x^{4} + x^{3} + 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e4_13e2_17e3.12t34.3c1$4$ $2^{4} \cdot 13^{2} \cdot 17^{3}$ $x^{6} - x^{5} - 2 x^{4} + x^{3} + 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$

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 *.