# Properties

 Label 6.2.837949.1 Degree $6$ Signature $[2, 2]$ Discriminant $7^{4}\cdot 349$ Root discriminant $9.71$ Ramified primes $7, 349$ Class number $1$ Class group Trivial Galois group $A_4\times C_2$ (as 6T6)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 3*x^4 - 9*x^3 + 4*x^2 - 9*x - 1)

gp: K = bnfinit(x^6 - 2*x^5 + 3*x^4 - 9*x^3 + 4*x^2 - 9*x - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 4, -9, 3, -2, 1]);

## Normalizeddefining polynomial

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$837949=7^{4}\cdot 349$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $9.71$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $7, 349$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{2}{9} a - \frac{4}{9}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$\frac{1}{9} a^{5} - \frac{1}{3} a^{4} + \frac{2}{3} a^{3} - \frac{5}{3} a^{2} + \frac{10}{9} a - \frac{10}{9}$$,  $$\frac{2}{9} a^{5} - \frac{1}{3} a^{4} - a^{2} - \frac{1}{9} a - \frac{5}{9}$$,  $$\frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{4}{3} a - \frac{1}{3}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$3.26724753988$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_2\times A_4$ (as 6T6):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 24 The 8 conjugacy class representatives for $A_4\times C_2$ Character table for $A_4\times C_2$

## 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: 4.0.5968249.1 $\times$ $$\Q(\sqrt{349})$$ Degree 8 sibling: 8.0.35619996126001.3 Degree 12 siblings: Deg 12, Deg 12

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
349Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.349.2t1.a.a$1$ $349$ $x^{2} - x - 87$ $C_2$ (as 2T1) $1$ $1$
* 1.7.3t1.a.a$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.2443.6t1.a.a$1$ $7 \cdot 349$ $x^{6} - x^{5} - 266 x^{4} + 177 x^{3} + 22799 x^{2} - 7744 x - 628489$ $C_6$ (as 6T1) $0$ $1$
1.2443.6t1.a.b$1$ $7 \cdot 349$ $x^{6} - x^{5} - 266 x^{4} + 177 x^{3} + 22799 x^{2} - 7744 x - 628489$ $C_6$ (as 6T1) $0$ $1$
* 1.7.3t1.a.b$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
3.5968249.4t4.a.a$3$ $7^{2} \cdot 349^{2}$ $x^{4} - x^{3} - 31 x^{2} - 28 x + 435$ $A_4$ (as 4T4) $1$ $-1$
* 3.17101.6t6.a.a$3$ $7^{2} \cdot 349$ $x^{6} - 2 x^{5} + 3 x^{4} - 9 x^{3} + 4 x^{2} - 9 x - 1$ $A_4\times C_2$ (as 6T6) $1$ $-1$

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 *.