# Properties

 Label 6.0.31211.1 Degree $6$ Signature $[0, 3]$ Discriminant $-31211$ Root discriminant $5.61$ Ramified primes $23, 59$ Class number $1$ Class group trivial Galois group $S_4\times C_2$ (as 6T11)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 2, -1, -1, 0, 1]);

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-31211$$$$\medspace = -\,23^{2}\cdot 59$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $5.61$ 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: $23, 59$ 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 $$a$$)

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

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: $2$ 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: $$a^{5} - a^{4} - a^{2} + 2 a + 1$$,  $$a^{5} - a^{3} - a^{2} + 2 a + 2$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.482550424804$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.482550424804 \cdot 1}{2\sqrt{31211}}\approx 0.3387649497998$

## Galois group

$C_2\times S_4$ (as 6T11):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 48 The 10 conjugacy class representatives for $S_4\times C_2$ Character table for $S_4\times C_2$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: 4.2.80063.1 $\times$ $$\Q(\sqrt{1357})$$ Degree 6 sibling: 6.2.717853.1 Degree 8 siblings: 8.0.6410083969.1, 8.4.3390934419601.1 Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12 Degree 16 sibling: Deg 16 Degree 24 siblings: data not computed

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ R

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
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 23.4.2.1x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$59$$\Q_{59}$$x + 3$$1$$1$$0Trivial[\ ] \Q_{59}$$x + 3$$1$$1$$0Trivial[\ ] 59.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.59.2t1.a.a$1$ $59$ $x^{2} - x + 15$ $C_2$ (as 2T1) $1$ $-1$
1.23.2t1.a.a$1$ $23$ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
1.1357.2t1.a.a$1$ $23 \cdot 59$ $x^{2} - x - 339$ $C_2$ (as 2T1) $1$ $1$
2.80063.6t3.a.a$2$ $23 \cdot 59^{2}$ $x^{6} - 2 x^{5} + 31 x^{4} - 163 x^{3} + 358 x^{2} - 1995 x - 71909$ $D_{6}$ (as 6T3) $1$ $0$
* 2.23.3t2.b.a$2$ $23$ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
3.80063.4t5.a.a$3$ $23 \cdot 59^{2}$ $x^{4} - x^{3} - x^{2} + 8 x + 5$ $S_4$ (as 4T5) $1$ $1$
3.31211.6t11.a.a$3$ $23^{2} \cdot 59$ $x^{6} - x^{4} - x^{3} + 2 x^{2} + 3 x + 1$ $S_4\times C_2$ (as 6T11) $1$ $1$
* 3.1357.6t11.a.a$3$ $23 \cdot 59$ $x^{6} - x^{4} - x^{3} + 2 x^{2} + 3 x + 1$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.1841449.6t8.c.a$3$ $23^{2} \cdot 59^{2}$ $x^{4} - x^{3} - x^{2} + 8 x + 5$ $S_4$ (as 4T5) $1$ $-1$

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 *.