Properties

Label 6.2.15699920.1
Degree $6$
Signature $[2, 2]$
Discriminant $2^{4}\cdot 5\cdot 443^{2}$
Root discriminant $15.82$
Ramified primes $2, 5, 443$
Class number $1$
Class group Trivial
Galois Group $S_4\times C_2$ (as 6T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, -8, 22, 6, -5, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 5*x^4 + 6*x^3 + 22*x^2 - 8*x - 32)
gp: K = bnfinit(x^6 - 2*x^5 - 5*x^4 + 6*x^3 + 22*x^2 - 8*x - 32, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut 5 x^{4} \) \(\mathstrut +\mathstrut 6 x^{3} \) \(\mathstrut +\mathstrut 22 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut -\mathstrut 32 \)

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:  \(15699920=2^{4}\cdot 5\cdot 443^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $15.82$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 443$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{5} + \frac{1}{12} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{12} a + \frac{1}{3}$

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:  \( \frac{11}{24} a^{5} - \frac{1}{12} a^{4} - \frac{21}{8} a^{3} - \frac{7}{4} a^{2} + \frac{97}{12} a + \frac{29}{3} \),  \( \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{3}{2} a^{3} - 2 a^{2} + \frac{8}{3} a + \frac{25}{3} \),  \( \frac{19}{24} a^{5} - \frac{29}{12} a^{4} - \frac{5}{8} a^{3} + \frac{17}{4} a^{2} + \frac{113}{12} a - \frac{41}{3} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 75.043635062 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

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

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
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

3.3.1772.1

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

Sibling algebras

Twin sextic algebra: 4.0.44300.1 $\times$ \(\Q(\sqrt{2215}) \)
Degree 6 sibling: 6.2.27820258240.1
Degree 8 siblings: 8.0.1962490000.1, 8.0.6162187200160000.3
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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
443Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_5_443.2t1.1c1$1$ $ 2^{2} \cdot 5 \cdot 443 $ $x^{2} - 2215$ $C_2$ (as 2T1) $1$ $1$
1.2e2_443.2t1.1c1$1$ $ 2^{2} \cdot 443 $ $x^{2} - 443$ $C_2$ (as 2T1) $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
2.2e2_5e2_443.6t3.1c1$2$ $ 2^{2} \cdot 5^{2} \cdot 443 $ $x^{6} - 2 x^{5} - 97 x^{4} + 174 x^{3} + 2325 x^{2} - 3724 x - 7416$ $D_{6}$ (as 6T3) $1$ $2$
* 2.2e2_443.3t2.2c1$2$ $ 2^{2} \cdot 443 $ $x^{3} - x^{2} - 12 x + 8$ $S_3$ (as 3T2) $1$ $2$
3.2e2_5e2_443.4t5.2c1$3$ $ 2^{2} \cdot 5^{2} \cdot 443 $ $x^{4} - x^{3} + 7 x^{2} - 4 x + 6$ $S_4$ (as 4T5) $1$ $-1$
* 3.2e2_5_443.6t11.2c1$3$ $ 2^{2} \cdot 5 \cdot 443 $ $x^{6} - 2 x^{5} - 5 x^{4} + 6 x^{3} + 22 x^{2} - 8 x - 32$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.2e4_5_443e2.6t11.2c1$3$ $ 2^{4} \cdot 5 \cdot 443^{2}$ $x^{6} - 2 x^{5} - 5 x^{4} + 6 x^{3} + 22 x^{2} - 8 x - 32$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.2e4_5e2_443e2.6t8.1c1$3$ $ 2^{4} \cdot 5^{2} \cdot 443^{2}$ $x^{4} - x^{3} + 7 x^{2} - 4 x + 6$ $S_4$ (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 *.