# Properties

 Label 6.2.82625.1 Degree $6$ Signature $[2, 2]$ Discriminant $5^{3}\cdot 661$ Root discriminant $6.60$ Ramified primes $5, 661$ 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, 2, -3, 1, -1, 1]);

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

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

## Normalizeddefining polynomial

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

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: $$82625=5^{3}\cdot 661$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $6.60$ magma: Abs(Discriminant(K))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $5, 661$ 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 $$a$$)

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

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: $$a$$,  $$a^{5} - 2 a^{2}$$,  $$a^{5} - a^{4} + a^{3} - 3 a^{2} + 2 a - 1$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$1.0798068604$$ 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: 6.2.1444023905.1 Degree 6 sibling: 6.2.1444023905.1 Degree 9 sibling: Deg 9 Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
661Data not computed

## 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_661.2t1.1c1$1$ $5 \cdot 661$ $x^{2} - x - 826$ $C_2$ (as 2T1) $1$ $1$
* 1.5.2t1.1c1$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.661.2t1.1c1$1$ $661$ $x^{2} - x - 165$ $C_2$ (as 2T1) $1$ $1$
2.5_661.4t3.1c1$2$ $5 \cdot 661$ $x^{4} - 2 x^{3} + 30 x^{2} - 29 x + 45$ $D_{4}$ (as 4T3) $1$ $-2$
4.5e2_661e3.12t36.1c1$4$ $5^{2} \cdot 661^{3}$ $x^{6} - x^{5} + x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
* 4.5e2_661.6t13.2c1$4$ $5^{2} \cdot 661$ $x^{6} - x^{5} + x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.5_661e2.6t13.2c1$4$ $5 \cdot 661^{2}$ $x^{6} - x^{5} + x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.5e3_661e2.12t34.1c1$4$ $5^{3} \cdot 661^{2}$ $x^{6} - x^{5} + x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1$ $C_3^2:D_4$ (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 *.