# Properties

 Label 6.2.722701.1 Degree $6$ Signature $[2, 2]$ Discriminant $7^{5}\cdot 43$ Root discriminant $9.47$ Ramified primes $7, 43$ 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 + 4*x^4 - 8*x^3 + 2*x^2 - 11*x + 1)

gp: K = bnfinit(x^6 - 2*x^5 + 4*x^4 - 8*x^3 + 2*x^2 - 11*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -11, 2, -8, 4, -2, 1]);

## Normalizeddefining polynomial

$$x^{6} - 2 x^{5} + 4 x^{4} - 8 x^{3} + 2 x^{2} - 11 x + 1$$

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: $$722701=7^{5}\cdot 43$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $9.47$ 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: $7, 43$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{43} a^{5} + \frac{6}{43} a^{4} + \frac{9}{43} a^{3} + \frac{21}{43} a^{2} - \frac{2}{43} a + \frac{16}{43}$

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: $$\frac{6}{43} a^{5} - \frac{7}{43} a^{4} + \frac{11}{43} a^{3} - \frac{46}{43} a^{2} - \frac{12}{43} a - \frac{33}{43}$$,  $$\frac{6}{43} a^{5} - \frac{7}{43} a^{4} + \frac{11}{43} a^{3} - \frac{46}{43} a^{2} - \frac{12}{43} a - \frac{76}{43}$$,  $$\frac{6}{43} a^{5} - \frac{7}{43} a^{4} + \frac{11}{43} a^{3} - \frac{3}{43} a^{2} - \frac{12}{43} a + \frac{10}{43}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$3.31543456345$$ 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: Deg 24 Twin sextic algebra: $$\Q(\sqrt{301})$$ $\times$ 4.0.90601.1 Degree 8 sibling: 8.0.402218518849.3 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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6} 43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.301.2t1.a.a$1$ $7 \cdot 43$ $x^{2} - x - 75$ $C_2$ (as 2T1) $1$ $1$
* 1.7.3t1.a.a$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.301.6t1.c.a$1$ $7 \cdot 43$ $x^{6} - x^{5} - 76 x^{4} + 76 x^{3} + 1618 x^{2} - 1618 x - 7699$ $C_6$ (as 6T1) $0$ $1$
1.301.6t1.c.b$1$ $7 \cdot 43$ $x^{6} - x^{5} - 76 x^{4} + 76 x^{3} + 1618 x^{2} - 1618 x - 7699$ $C_6$ (as 6T1) $0$ $1$
* 1.7.3t1.a.b$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
3.90601.4t4.a.a$3$ $7^{2} \cdot 43^{2}$ $x^{4} - x^{3} - 7 x^{2} + 9 x + 38$ $A_4$ (as 4T4) $1$ $-1$
* 3.14749.6t6.a.a$3$ $7^{3} \cdot 43$ $x^{6} - 2 x^{5} + 4 x^{4} - 8 x^{3} + 2 x^{2} - 11 x + 1$ $A_4\times C_2$ (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 *.