Properties

Label 6.2.14440000.2
Degree $6$
Signature $[2, 2]$
Discriminant $2^{6}\cdot 5^{4}\cdot 19^{2}$
Root discriminant $15.60$
Ramified primes $2, 5, 19$
Class number $1$
Class group Trivial
Galois Group $C_3^2:C_4$ (as 6T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -20, -24, -12, -1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - x^4 - 12*x^3 - 24*x^2 - 20*x - 4)
gp: K = bnfinit(x^6 - x^5 - x^4 - 12*x^3 - 24*x^2 - 20*x - 4, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut -\mathstrut 12 x^{3} \) \(\mathstrut -\mathstrut 24 x^{2} \) \(\mathstrut -\mathstrut 20 x \) \(\mathstrut -\mathstrut 4 \)

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:  \(14440000=2^{6}\cdot 5^{4}\cdot 19^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $15.60$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 19$
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 \(a\))

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

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:  \( \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{21}{8} a^{2} - \frac{19}{4} a - \frac{11}{4} \),  \( \frac{1}{8} a^{5} - \frac{9}{8} a^{2} - 2 a - \frac{5}{4} \),  \( \frac{3}{8} a^{5} - 2 a^{3} - \frac{43}{8} a^{2} - 4 a - \frac{3}{4} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 32.9681500507 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3:S_3.C_2$ (as 6T10):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 36
The 6 conjugacy class representatives for $C_3^2:C_4$
Character table for $C_3^2:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \)

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: 6.2.5212840000.4
Degree 6 sibling: 6.2.5212840000.4
Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
Degree 18 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 ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.2e3_5.4t1.2c1$1$ $ 2^{3} \cdot 5 $ $x^{4} + 10 x^{2} + 20$ $C_4$ (as 4T1) $0$ $-1$
1.2e3_5.4t1.2c2$1$ $ 2^{3} \cdot 5 $ $x^{4} + 10 x^{2} + 20$ $C_4$ (as 4T1) $0$ $-1$
4.2e6_5e3_19e2.6t10.1c1$4$ $ 2^{6} \cdot 5^{3} \cdot 19^{2}$ $x^{6} - x^{5} - x^{4} - 12 x^{3} - 24 x^{2} - 20 x - 4$ $C_3^2:C_4$ (as 6T10) $1$ $0$
4.2e6_5e3_19e4.6t10.1c1$4$ $ 2^{6} \cdot 5^{3} \cdot 19^{4}$ $x^{6} - x^{5} - x^{4} - 12 x^{3} - 24 x^{2} - 20 x - 4$ $C_3^2:C_4$ (as 6T10) $1$ $0$

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 *.