Properties

Label 6.4.124659.1
Degree $6$
Signature $[4, 1]$
Discriminant $-\,3^{8}\cdot 19$
Root discriminant $7.07$
Ramified primes $3, 19$
Class number $1$
Class group Trivial
Galois Group $A_4\times C_2$ (as 6T6)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 3 x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut +\mathstrut 3 x \) \(\mathstrut -\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $6$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[4, 1]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-124659=-\,3^{8}\cdot 19\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $7.07$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 19$
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 \(a\))

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

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial Abelian group, 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:  $4$
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:  \( a^{2} - 1 \),  \( a^{4} - 2 a^{2} - a + 1 \),  \( a^{3} - a - 1 \),  \( a \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1.54949123381 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times A_4$ (as 6T6):

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

\(\Q(\zeta_{9})^+\)

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: 4.0.29241.1 $\times$ \(\Q(\sqrt{-19}) \)
Degree 8 sibling: 8.0.855036081.1
Degree 12 siblings: 12.4.5609891727441.1, 12.0.2025170913606201.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.19.2t1.1c1$1$ $ 19 $ $x^{2} - x + 5$ $C_2$ (as 2T1) $1$ $-1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2_19.6t1.3c1$1$ $ 3^{2} \cdot 19 $ $x^{6} - 3 x^{5} + 12 x^{4} - 17 x^{3} + 87 x^{2} - 114 x + 323$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_19.6t1.3c2$1$ $ 3^{2} \cdot 19 $ $x^{6} - 3 x^{5} + 12 x^{4} - 17 x^{3} + 87 x^{2} - 114 x + 323$ $C_6$ (as 6T1) $0$ $-1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
3.3e4_19e2.4t4.1c1$3$ $ 3^{4} \cdot 19^{2}$ $x^{4} - x^{3} + 3 x^{2} + x + 20$ $A_4$ (as 4T4) $1$ $-1$
3.3e4_19.6t6.1c1$3$ $ 3^{4} \cdot 19 $ $x^{6} - 3 x^{4} - 2 x^{3} + 3 x^{2} + 3 x - 1$ $A_4\times C_2$ (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 *.