# Properties

 Label 6.0.27848.1 Degree $6$ Signature $[0, 3]$ Discriminant $-27848$ Root discriminant $5.51$ Ramified primes $2, 59$ Class number $1$ Class group trivial Galois group $S_4\times C_2$ (as 6T11)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

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

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-27848$$$$\medspace = -\,2^{3}\cdot 59^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $5.51$ 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: $2, 59$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

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

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: $2$ 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: $$a$$,  $$\frac{1}{2} a^{5} - a^{4} + a^{3} - a^{2} - \frac{3}{2}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.599228830431$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.599228830431 \cdot 1}{2\sqrt{27848}}\approx 0.4453539458014$

## Galois group

$C_2\times S_4$ (as 6T11):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 48 The 10 conjugacy class representatives for $S_4\times C_2$ Character table for $S_4\times C_2$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling algebras

 Twin sextic algebra: 4.2.3776.2 $\times$ $$\Q(\sqrt{118})$$ Degree 6 sibling: 6.2.1643032.1 Degree 8 siblings: 8.0.14258176.2, 8.4.49632710656.3 Degree 12 siblings: 12.2.2928329928704.1, 12.0.2699554153024.1, 12.0.3176493481984.1, 12.4.11057373810786304.1, 12.0.11057373810786304.6, 12.0.11057373810786304.5 Degree 16 sibling: Deg 16 Degree 24 siblings: data not computed

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ R

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3] 2.4.0.1x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$59$$\Q_{59}$$x + 3$$1$$1$$0Trivial[\ ] \Q_{59}$$x + 3$$1$$1$$0Trivial[\ ] 59.4.2.1x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.472.2t1.a.a$1$ $2^{3} \cdot 59$ $x^{2} - 118$ $C_2$ (as 2T1) $1$ $1$
1.59.2t1.a.a$1$ $59$ $x^{2} - x + 15$ $C_2$ (as 2T1) $1$ $-1$
1.8.2t1.b.a$1$ $2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
2.3776.6t3.d.a$2$ $2^{6} \cdot 59$ $x^{6} + 16 x^{4} + 64 x^{2} - 472$ $D_{6}$ (as 6T3) $1$ $0$
* 2.59.3t2.a.a$2$ $59$ $x^{3} + 2 x - 1$ $S_3$ (as 3T2) $1$ $0$
3.3776.4t5.d.a$3$ $2^{6} \cdot 59$ $x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 3$ $S_4$ (as 4T5) $1$ $1$
* 3.472.6t11.a.a$3$ $2^{3} \cdot 59$ $x^{6} - x^{5} + 2 x^{4} + 2 x^{2} - x + 1$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.27848.6t11.a.a$3$ $2^{3} \cdot 59^{2}$ $x^{6} - x^{5} + 2 x^{4} + 2 x^{2} - x + 1$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.222784.6t8.g.a$3$ $2^{6} \cdot 59^{2}$ $x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 3$ $S_4$ (as 4T5) $1$ $-1$

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 *.