# Properties

 Label 6.2.4723920.2 Degree $6$ Signature $[2, 2]$ Discriminant $2^{4}\cdot 3^{10}\cdot 5$ Root discriminant $12.95$ Ramified primes $2, 3, 5$ Class number $1$ Class group Trivial Galois Group $S_6$ (as 6T16)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 9, -4, 3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 3*x^4 - 4*x^3 + 9*x^2 - 6*x + 1)
gp: K = bnfinit(x^6 + 3*x^4 - 4*x^3 + 9*x^2 - 6*x + 1, 1)

## Normalizeddefining polynomial

$$x^{6}$$ $$\mathstrut +\mathstrut 3 x^{4}$$ $$\mathstrut -\mathstrut 4 x^{3}$$ $$\mathstrut +\mathstrut 9 x^{2}$$ $$\mathstrut -\mathstrut 6 x$$ $$\mathstrut +\mathstrut 1$$

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

## Invariants

 Degree: $6$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[2, 2]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$4723920=2^{4}\cdot 3^{10}\cdot 5$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $12.95$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3, 5$ 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 $$a$$)

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

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: $$a$$,  $$a^{5} - a^{4} + a^{3} - 7 a^{2} + 11 a - 2$$,  $$2 a^{5} + 6 a^{3} - 7 a^{2} + 19 a - 9$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$39.7763369901$$ 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: 6.2.118098000.1 Degree 6 sibling: 6.2.118098000.1 Degree 10 sibling: 10.2.12397455648000.1 Degree 12 siblings: Deg 12, Deg 12 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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.5.4.1x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.6.10.10$x^{6} + 3 x^{5} + 24$$6$$1$$10$$C_3^2:D_4$$[9/4, 9/4]_{4}^{2} 55.2.1.2x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$

## 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) $1$ $1$
* 5.2e4_3e10_5.6t16.2c1$5$ $2^{4} \cdot 3^{10} \cdot 5$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.2e4_3e10_5e2.12t183.2c1$5$ $2^{4} \cdot 3^{10} \cdot 5^{2}$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.2e4_3e10_5e4.12t183.2c1$5$ $2^{4} \cdot 3^{10} \cdot 5^{4}$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.2e4_3e10_5e3.6t16.2c1$5$ $2^{4} \cdot 3^{10} \cdot 5^{3}$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (as 6T16) $1$ $1$
9.2e8_3e18_5e3.10t32.2c1$9$ $2^{8} \cdot 3^{18} \cdot 5^{3}$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (as 6T16) $1$ $1$
9.2e8_3e18_5e6.20t145.2c1$9$ $2^{8} \cdot 3^{18} \cdot 5^{6}$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (as 6T16) $1$ $1$
10.2e8_3e20_5e6.30t176.2c1$10$ $2^{8} \cdot 3^{20} \cdot 5^{6}$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (as 6T16) $1$ $-2$
10.2e8_3e20_5e4.30t176.2c1$10$ $2^{8} \cdot 3^{20} \cdot 5^{4}$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (as 6T16) $1$ $-2$
16.2e12_3e36_5e8.36t1252.2c1$16$ $2^{12} \cdot 3^{36} \cdot 5^{8}$ $x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ $S_6$ (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 *.