# Properties

 Label 8.4.3186376704.2 Degree $8$ Signature $[4, 2]$ Discriminant $2^{14}\cdot 3^{4}\cdot 7^{4}$ Root discriminant $15.41$ Ramified primes $2, 3, 7$ Class number $1$ Class group Trivial Galois group $(A_4\wr C_2):C_2$ (as 8T45)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 4*x^6 + 4*x^5 + 4*x^4 + 4*x^3 + 4*x^2 - 8*x - 6)

gp: K = bnfinit(x^8 - 2*x^7 - 4*x^6 + 4*x^5 + 4*x^4 + 4*x^3 + 4*x^2 - 8*x - 6, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, -8, 4, 4, 4, 4, -4, -2, 1]);

## Normalizeddefining polynomial

$$x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[4, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$3186376704=2^{14}\cdot 3^{4}\cdot 7^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $15.41$ 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: $2, 3, 7$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{11} a^{7} + \frac{3}{11} a^{6} + \frac{4}{11} a^{4} + \frac{2}{11} a^{3} + \frac{3}{11} a^{2} - \frac{3}{11} a - \frac{1}{11}$

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: $5$ 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: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$146.850411209$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$(A_4\wr C_2):C_2$ (as 8T45):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 576 The 16 conjugacy class representatives for $(A_4\wr C_2):C_2$ Character table for $(A_4\wr C_2):C_2$

## Intermediate fields

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

## Sibling fields

 Degree 12 siblings: data not computed Degree 16 siblings: data not computed Degree 18 siblings: data not computed Degree 24 siblings: data not computed Degree 32 siblings: data not computed Degree 36 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.8.14.2$x^{8} + 2 x^{7} + 2$$8$$1$$14$$A_4\times C_2$$[2, 2, 2]^{3} 3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2} 3.4.0.1x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.a.a$1$ $7$ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.28.2t1.a.a$1$ $2^{2} \cdot 7$ $x^{2} - 7$ $C_2$ (as 2T1) $1$ $1$
2.567.3t2.b.a$2$ $3^{4} \cdot 7$ $x^{3} - 3 x - 5$ $S_3$ (as 3T2) $1$ $0$
2.15876.6t3.c.a$2$ $2^{2} \cdot 3^{4} \cdot 7^{2}$ $x^{6} - 12 x^{4} - 14 x^{3} - 27 x^{2} - 126 x - 126$ $D_{6}$ (as 6T3) $1$ $0$
2.324.3t2.b.a$2$ $2^{2} \cdot 3^{4}$ $x^{3} - 3 x - 4$ $S_3$ (as 3T2) $1$ $0$
2.9072.6t3.b.a$2$ $2^{4} \cdot 3^{4} \cdot 7$ $x^{6} - 6 x^{4} + 9 x^{2} - 7$ $D_{6}$ (as 6T3) $1$ $0$
4.63504.6t9.a.a$4$ $2^{4} \cdot 3^{4} \cdot 7^{2}$ $x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} - x^{2} - 2 x - 3$ $S_3^2$ (as 6T9) $1$ $0$
* 6.113799168.8t45.a.a$6$ $2^{12} \cdot 3^{4} \cdot 7^{3}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $2$
6.113799168.12t161.a.a$6$ $2^{12} \cdot 3^{4} \cdot 7^{3}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-2$
9.256096265048064.18t185.a.a$9$ $2^{12} \cdot 3^{12} \cdot 7^{6}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.746636341248.12t165.a.a$9$ $2^{12} \cdot 3^{12} \cdot 7^{3}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
9.47784725839872.18t185.a.a$9$ $2^{18} \cdot 3^{12} \cdot 7^{3}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.16390160963076096.18t179.a.a$9$ $2^{18} \cdot 3^{12} \cdot 7^{6}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
12.6882294149039505014784.24t1503.a.a$12$ $2^{24} \cdot 3^{20} \cdot 7^{6}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $0$

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 *.