Properties

Label 6.2.13436928.5
Degree $6$
Signature $[2, 2]$
Discriminant $2^{11}\cdot 3^{8}$
Root discriminant $15.42$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois Group $S_6$ (as 6T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 6, -4, -6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 6*x^4 - 4*x^3 + 6*x^2 - 6)
gp: K = bnfinit(x^6 - 6*x^4 - 4*x^3 + 6*x^2 - 6, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 6 x^{4} \) \(\mathstrut -\mathstrut 4 x^{3} \) \(\mathstrut +\mathstrut 6 x^{2} \) \(\mathstrut -\mathstrut 6 \)

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:  \(13436928=2^{11}\cdot 3^{8}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $15.42$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3$
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 \(a\))

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

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:  \( a + 1 \),  \( a^{5} - a^{4} - 5 a^{3} + 2 a^{2} + 4 a - 5 \),  \( 2 a^{5} - a^{4} - 14 a^{3} + 6 a^{2} + 10 a - 13 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 50.8478091524 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_6$ (as 6T16):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 720
The 11 conjugacy class representatives for $S_6$
Character table for $S_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling algebras

Twin sextic algebra: 6.2.13436928.4
Degree 6 sibling: 6.2.13436928.4
Degree 10 sibling: 10.2.22568879259648.1
Degree 12 siblings: 12.4.2888816545234944.4, 12.4.11555266180939776.6
Degree 15 siblings: Deg 15, Deg 15
Degree 20 siblings: Deg 20, Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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

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$2.6.11.3$x^{6} + 2 x^{2} + 14$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 5.2e11_3e8.6t16.3c1$5$ $ 2^{11} \cdot 3^{8}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $1$
5.2e12_3e8.12t183.1c1$5$ $ 2^{12} \cdot 3^{8}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $1$
5.2e14_3e8.12t183.5c1$5$ $ 2^{14} \cdot 3^{8}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $1$
5.2e11_3e8.6t16.4c1$5$ $ 2^{11} \cdot 3^{8}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $1$
9.2e19_3e16.10t32.1c1$9$ $ 2^{19} \cdot 3^{16}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $1$
9.2e26_3e16.20t145.1c1$9$ $ 2^{26} \cdot 3^{16}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $1$
10.2e28_3e16.30t176.1c1$10$ $ 2^{28} \cdot 3^{16}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $-2$
10.2e26_3e16.30t176.2c1$10$ $ 2^{26} \cdot 3^{16}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $-2$
16.2e42_3e32.36t1252.2c1$16$ $ 2^{42} \cdot 3^{32}$ $x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ $S_6$ (as 6T16) $1$ $0$

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 *.