# 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)

# Related objects

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)

## Normalizeddefining polynomial

$$x^{6} - 3 x^{3} - 3 x + 4$$

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(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $9.22$ magma: Abs(Discriminant(Integers(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(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ 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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ 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$$[\ ]^{2} 29.3.2.1x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
5.515607849.6t15.b.a$5$ $3^{6} \cdot 29^{4}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $1$
* 5.613089.6t15.b.a$5$ $3^{6} \cdot 29^{2}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $1$
8.316113500535561.36t555.b.a$8$ $3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $0$
8.316113500535561.36t555.b.b$8$ $3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $0$
9.316113500535561.10t26.b.a$9$ $3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (as 6T15) $1$ $1$
10.2845021504820049.30t88.b.a$10$ $3^{14} \cdot 29^{6}$ $x^{6} - 3 x^{3} - 3 x + 4$ $A_6$ (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 *.