Properties

Label 6.2.613089.1
Degree $6$
Signature $[2, 2]$
Discriminant $3^{6}\cdot 29^{2}$
Root discriminant $9.22$
Ramified primes $3, 29$
Class number $1$
Class group Trivial
Galois Group $A_6$ (as 6T15)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(613089=3^{6}\cdot 29^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.22$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$

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:  \( \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{8}{7} a^{2} - \frac{5}{7} a - \frac{3}{7} \),  \( \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{4}{7} a^{2} + \frac{1}{7} a - \frac{5}{7} \),  \( \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{5}{7} a^{2} + \frac{3}{7} a - \frac{1}{7} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 5.16689103695 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$A_6$ (as 6T15):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

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

Sibling algebras

Twin sextic algebra: 6.2.515607849.1
Degree 6 sibling: 6.2.515607849.1
Degree 10 sibling: 10.2.316113500535561.1
Degree 15 siblings: Deg 15, Deg 15
Degree 20 sibling: Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.3e6_29e4.6t15.1c1$5$ $ 3^{6} \cdot 29^{4}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $1$
* 5.3e6_29e2.6t15.1c1$5$ $ 3^{6} \cdot 29^{2}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $1$
8.3e12_29e6.36t555.1c1$8$ $ 3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $0$
8.3e12_29e6.36t555.1c2$8$ $ 3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $0$
9.3e12_29e6.10t26.1c1$9$ $ 3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $1$
10.3e14_29e6.30t88.1c1$10$ $ 3^{14} \cdot 29^{6}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $-2$

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 *.