# Properties

 Label 8.4.258405625.1 Degree $8$ Signature $[4, 2]$ Discriminant $5^{4}\cdot 643^{2}$ Root discriminant $11.26$ Ramified primes $5, 643$ Class number $1$ Class group Trivial Galois group $V_4^2:(S_3\times C_2)$ (as 8T41)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 4*x^6 - 5*x^5 + x^4 + x^3 + 8*x^2 - 7*x + 1)

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 8, 1, 1, -5, 4, -3, 1]);

## Normalizeddefining polynomial

$$x^{8} - 3 x^{7} + 4 x^{6} - 5 x^{5} + x^{4} + x^{3} + 8 x^{2} - 7 x + 1$$

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: $$258405625=5^{4}\cdot 643^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $11.26$ 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: $5, 643$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{37} a^{7} + \frac{11}{37} a^{6} + \frac{10}{37} a^{5} - \frac{13}{37} a^{4} + \frac{4}{37} a^{3} - \frac{17}{37} a^{2} - \frac{8}{37} a - \frac{8}{37}$

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: $$12.6167062779$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_2^2:S_4:C_2$ (as 8T41):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 192 The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$ Character table for $V_4^2:(S_3\times C_2)$

## Intermediate fields

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

## Sibling fields

 Degree 8 sibling: data not computed Degree 12 siblings: data not computed Degree 16 siblings: data not computed Degree 24 siblings: data not computed Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
643Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.3215.2t1.a.a$1$ $5 \cdot 643$ $x^{2} - x + 804$ $C_2$ (as 2T1) $1$ $-1$
1.643.2t1.a.a$1$ $643$ $x^{2} - x + 161$ $C_2$ (as 2T1) $1$ $-1$
2.643.3t2.a.a$2$ $643$ $x^{3} - 2 x - 5$ $S_3$ (as 3T2) $1$ $0$
2.16075.6t3.a.a$2$ $5^{2} \cdot 643$ $x^{6} - 3 x^{5} - 4 x^{4} + 3 x^{3} + 13 x^{2} - 30 x + 36$ $D_{6}$ (as 6T3) $1$ $0$
3.413449.6t8.a.a$3$ $643^{2}$ $x^{4} - x^{3} - 2 x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.80375.6t11.b.a$3$ $5^{3} \cdot 643$ $x^{6} - 3 x^{5} + 44 x^{4} - 38 x^{3} + 650 x^{2} - 139 x + 5489$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.643.4t5.a.a$3$ $643$ $x^{4} - x^{3} - 2 x + 1$ $S_4$ (as 4T5) $1$ $1$
3.51681125.6t11.b.a$3$ $5^{3} \cdot 643^{2}$ $x^{6} - 3 x^{5} + 44 x^{4} - 38 x^{3} + 650 x^{2} - 139 x + 5489$ $S_4\times C_2$ (as 6T11) $1$ $-1$
6.33230963375.8t41.b.a$6$ $5^{3} \cdot 643^{3}$ $x^{8} - 3 x^{7} + 4 x^{6} - 5 x^{5} + x^{4} + x^{3} + 8 x^{2} - 7 x + 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $0$
* 6.51681125.8t41.b.a$6$ $5^{3} \cdot 643^{2}$ $x^{8} - 3 x^{7} + 4 x^{6} - 5 x^{5} + x^{4} + x^{3} + 8 x^{2} - 7 x + 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.21367509450125.12t108.b.a$6$ $5^{3} \cdot 643^{4}$ $x^{8} - 3 x^{7} + 4 x^{6} - 5 x^{5} + x^{4} + x^{3} + 8 x^{2} - 7 x + 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
6.33230963375.12t108.b.a$6$ $5^{3} \cdot 643^{3}$ $x^{8} - 3 x^{7} + 4 x^{6} - 5 x^{5} + x^{4} + x^{3} + 8 x^{2} - 7 x + 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $0$

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 *.