# Properties

 Label 6.2.1036800.1 Degree $6$ Signature $[2, 2]$ Discriminant $2^{9}\cdot 3^{4}\cdot 5^{2}$ Root discriminant $10.06$ Ramified primes $2, 3, 5$ Class number $1$ Class group Trivial Galois group $S_6$ (as 6T16)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - x^4 + 6*x^3 - 2*x^2 - 4*x - 1)

gp: K = bnfinit(x^6 - 2*x^5 - x^4 + 6*x^3 - 2*x^2 - 4*x - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, -2, 6, -1, -2, 1]);

## Normalizeddefining polynomial

$$x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$1036800=2^{9}\cdot 3^{4}\cdot 5^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $10.06$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 3, 5$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a$$,  $$a^{5} - 2 a^{4} - a^{3} + 6 a^{2} - 3 a - 3$$,  $$a^{3} - 2 a^{2} + a + 2$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$7.87392875368$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$S_6$ (as 6T16):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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: 6.2.25920000.1 Degree 6 sibling: 6.2.25920000.1 Degree 10 sibling: 10.2.3359232000000.1 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: Deg 30, Deg 30, Deg 30, Deg 30, Deg 30, Deg 30 Degree 36 sibling: Deg 36 Degree 40 siblings: Deg 40, Deg 40, Deg 40 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 ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3} 3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4} 55.3.0.1x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.2.1$x^{3} - 5$$3$$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$1$ $1$ $x$ $C_1$ $1$ $1$
1.8.2t1.a.a$1$ $2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 5.1036800.6t16.b.a$5$ $2^{9} \cdot 3^{4} \cdot 5^{2}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (as 6T16) $1$ $1$
5.207360000.12t183.b.a$5$ $2^{12} \cdot 3^{4} \cdot 5^{4}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (as 6T16) $1$ $1$
5.8294400.12t183.b.a$5$ $2^{12} \cdot 3^{4} \cdot 5^{2}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (as 6T16) $1$ $1$
5.25920000.6t16.b.a$5$ $2^{9} \cdot 3^{4} \cdot 5^{4}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (as 6T16) $1$ $1$
9.3359232000000.10t32.b.a$9$ $2^{15} \cdot 3^{8} \cdot 5^{6}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (as 6T16) $1$ $1$
9.1719926784000000.20t145.b.a$9$ $2^{24} \cdot 3^{8} \cdot 5^{6}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (as 6T16) $1$ $1$
10.1719926784000000.30t164.c.a$10$ $2^{24} \cdot 3^{8} \cdot 5^{6}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (as 6T16) $1$ $-2$
10.1719926784000000.30t164.d.a$10$ $2^{24} \cdot 3^{8} \cdot 5^{6}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (as 6T16) $1$ $-2$
16.8916100448256000000000000.36t1252.b.a$16$ $2^{36} \cdot 3^{12} \cdot 5^{12}$ $x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ $S_6$ (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 *.