# Properties

 Label 6.4.582464.1 Degree $6$ Signature $[4, 1]$ Discriminant $-582464$ Root discriminant $9.14$ Ramified primes $2, 19, 479$ Class number $1$ Class group trivial Galois group $S_6$ (as 6T16)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 5, -2, 0, -2, 1]);

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[4, 1]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-582464$$$$\medspace = -\,2^{6}\cdot 19\cdot 479$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $9.14$ 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: $2, 19, 479$ 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: $4$ 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: $$a^{5} - 2 a^{4} - 2 a^{2} + 5 a$$,  $$a - 1$$,  $$a^{5} - a^{4} - a^{3} - 3 a^{2} + 3 a + 2$$,  $$a^{5} - a^{3} - 4 a^{2} - a + 1$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$5.0360819301$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{1}\cdot 5.0360819301 \cdot 1}{2\sqrt{582464}}\approx 0.33168673097$

## 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: 6.0.192977781069056.1 Degree 6 sibling: 6.0.192977781069056.1 Degree 10 sibling: 10.4.3087644497104896.1 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: Deg 45

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.6.0.1}{6} }$ R ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }$

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

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$2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3} 19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2} 19.3.0.1x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
$479$Deg $2$$1$$2$$0$$C_2$$[\ ]^{2} Deg 2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.36404.2t1.a.a$1$ $2^{2} \cdot 19 \cdot 479$ $$\Q(\sqrt{-9101})$$ $C_2$ (as 2T1) $1$ $-1$
* 5.582464.6t16.a.a$5$ $2^{6} \cdot 19 \cdot 479$ 6.4.582464.1 $S_6$ (as 6T16) $1$ $3$
5.21204019456.12t183.a.a$5$ $2^{8} \cdot 19^{2} \cdot 479^{2}$ 6.4.582464.1 $S_6$ (as 6T16) $1$ $1$
5.702...624.12t183.a.a$5$ $2^{10} \cdot 19^{4} \cdot 479^{4}$ 6.4.582464.1 $S_6$ (as 6T16) $1$ $-3$
5.192...056.6t16.a.a$5$ $2^{8} \cdot 19^{3} \cdot 479^{3}$ 6.4.582464.1 $S_6$ (as 6T16) $1$ $-1$
9.308...896.10t32.a.a$9$ $2^{12} \cdot 19^{3} \cdot 479^{3}$ 6.4.582464.1 $S_6$ (as 6T16) $1$ $3$
9.148...544.20t145.a.a$9$ $2^{18} \cdot 19^{6} \cdot 479^{6}$ 6.4.582464.1 $S_6$ (as 6T16) $1$ $-3$
10.595...176.30t164.a.a$10$ $2^{20} \cdot 19^{6} \cdot 479^{6}$ 6.4.582464.1 $S_6$ (as 6T16) $1$ $-2$
10.179...744.30t164.a.a$10$ $2^{18} \cdot 19^{4} \cdot 479^{4}$ 6.4.582464.1 $S_6$ (as 6T16) $1$ $2$
16.126...256.36t1252.a.a$16$ $2^{28} \cdot 19^{8} \cdot 479^{8}$ 6.4.582464.1 $S_6$ (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 *.