# Properties

 Label 6.0.12446784.1 Degree $6$ Signature $[0, 3]$ Discriminant $-\,2^{6}\cdot 3^{4}\cdot 7^{4}$ Root discriminant $15.22$ Ramified primes $2, 3, 7$ Class number $2$ Class group $[2]$ Galois group $S_6$ (as 6T16)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -8, 9, -2, 5, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 5*x^4 - 2*x^3 + 9*x^2 - 8*x + 16)

gp: K = bnfinit(x^6 + 5*x^4 - 2*x^3 + 9*x^2 - 8*x + 16, 1)

## Normalizeddefining polynomial

$$x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $6$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[0, 3]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$-12446784=-\,2^{6}\cdot 3^{4}\cdot 7^{4}$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $15.22$ magma: Abs(Discriminant(K))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3, 7$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{32} a^{5} - \frac{3}{8} a^{4} - \frac{11}{32} a^{3} + \frac{1}{16} a^{2} - \frac{15}{32} a + \frac{3}{8}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $2$ 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: $$\frac{1}{4} a^{5} + \frac{5}{4} a^{3} - \frac{1}{2} a^{2} + \frac{5}{4} a - 2$$,  $$\frac{1}{32} a^{5} - \frac{3}{8} a^{4} - \frac{11}{32} a^{3} - \frac{31}{16} a^{2} - \frac{47}{32} a - \frac{29}{8}$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$17.0352826889$$ 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: 6.4.63011844.2 Degree 6 sibling: 6.4.63011844.2 Degree 10 sibling: 10.4.196073702927424.1 Degree 12 siblings: Deg 12, Deg 12 Degree 15 siblings: Deg 15, Deg 15 Degree 20 siblings: Deg 20, Deg 20, 20.0.9841893626795960030057286598656.1 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 ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2] 2.4.4.2x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3} 3.3.4.4x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] 7.5.4.1x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2.2t1.1c1$1$ $2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 5.2e6_3e4_7e4.6t16.2c1$5$ $2^{6} \cdot 3^{4} \cdot 7^{4}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (as 6T16) $1$ $-1$
5.2e8_3e8_7e4.12t183.2c1$5$ $2^{8} \cdot 3^{8} \cdot 7^{4}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (as 6T16) $1$ $-3$
5.2e4_3e4_7e4.12t183.2c1$5$ $2^{4} \cdot 3^{4} \cdot 7^{4}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (as 6T16) $1$ $1$
5.2e2_3e8_7e4.6t16.2c1$5$ $2^{2} \cdot 3^{8} \cdot 7^{4}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (as 6T16) $1$ $3$
9.2e6_3e12_7e8.10t32.2c1$9$ $2^{6} \cdot 3^{12} \cdot 7^{8}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (as 6T16) $1$ $3$
9.2e12_3e12_7e8.20t145.2c1$9$ $2^{12} \cdot 3^{12} \cdot 7^{8}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (as 6T16) $1$ $-3$
10.2e8_3e12_7e8.30t176.2c1$10$ $2^{8} \cdot 3^{12} \cdot 7^{8}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (as 6T16) $1$ $2$
10.2e12_3e12_7e8.30t176.2c1$10$ $2^{12} \cdot 3^{12} \cdot 7^{8}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (as 6T16) $1$ $-2$
16.2e16_3e24_7e12.36t1252.2c1$16$ $2^{16} \cdot 3^{24} \cdot 7^{12}$ $x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ $S_6$ (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 *.