# Properties

 Label 8.0.2708009.1 Degree $8$ Signature $[0, 4]$ Discriminant $41\cdot 257^{2}$ Root discriminant $6.37$ Ramified primes $41, 257$ Class number $1$ Class group Trivial Galois Group $C_2 \wr S_4$ (as 8T44)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -1, -1, 1, 1, -1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - x^6 + x^5 + x^4 - x^3 - x^2 + x + 1)
gp: K = bnfinit(x^8 - x^7 - x^6 + x^5 + x^4 - x^3 - x^2 + x + 1, 1)

## Normalizeddefining polynomial

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

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

## Invariants

 Degree: $8$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[0, 4]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$2708009=41\cdot 257^{2}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $6.37$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $41, 257$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $3$ 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: $$a^{7} - a^{6} - a^{5} + a^{4} + a^{3} - a^{2} - a + 1$$,  $$a^{5} - a^{4}$$,  $$a^{3} - a^{2} + 1$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$0.563263410972$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$C_2^3:S_4.C_2$ (as 8T44):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 384 The 20 conjugacy class representatives for $C_2 \wr S_4$ Character table for $C_2 \wr S_4$

## Intermediate fields

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

## Sibling fields

 Degree 8 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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
$41$$\Q_{41}$$x + 6$$1$$1$$0Trivial[\ ] \Q_{41}$$x + 6$$1$$1$$0Trivial[\ ] 41.2.1.2x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
257Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.257.2t1.1c1$1$ $257$ $x^{2} - x - 64$ $C_2$ (as 2T1) $1$ $1$
1.41_257.2t1.1c1$1$ $41 \cdot 257$ $x^{2} - x - 2634$ $C_2$ (as 2T1) $1$ $1$
1.41.2t1.1c1$1$ $41$ $x^{2} - x - 10$ $C_2$ (as 2T1) $1$ $1$
2.257.3t2.1c1$2$ $257$ $x^{3} - x^{2} - 4 x + 3$ $S_3$ (as 3T2) $1$ $2$
2.41e2_257.6t3.2c1$2$ $41^{2} \cdot 257$ $x^{6} - 2 x^{5} - 267 x^{4} + 504 x^{3} + 17720 x^{2} - 31624 x - 249501$ $D_{6}$ (as 6T3) $1$ $2$
* 3.257.4t5.1c1$3$ $257$ $x^{4} + x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.257e2.6t8.1c1$3$ $257^{2}$ $x^{4} + x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.41e3_257e2.6t11.2c1$3$ $41^{3} \cdot 257^{2}$ $x^{6} - 2 x^{5} + 18 x^{4} + 56 x^{3} - 257 x^{2} - 794 x - 1671$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.41e3_257.6t11.2c1$3$ $41^{3} \cdot 257$ $x^{6} - 2 x^{5} + 18 x^{4} + 56 x^{3} - 257 x^{2} - 794 x - 1671$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 4.41_257.8t44.1c1$4$ $41 \cdot 257$ $x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
4.41e3_257.8t44.1c1$4$ $41^{3} \cdot 257$ $x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
4.41_257e3.8t44.1c1$4$ $41 \cdot 257^{3}$ $x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
4.41e3_257e3.8t44.1c1$4$ $41^{3} \cdot 257^{3}$ $x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
6.41e3_257e2.8t41.1c1$6$ $41^{3} \cdot 257^{2}$ $x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.41e3_257e4.12t111.1c1$6$ $41^{3} \cdot 257^{4}$ $x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.41e3_257e3.8t41.1c1$6$ $41^{3} \cdot 257^{3}$ $x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
6.41e3_257e3.12t108.1c1$6$ $41^{3} \cdot 257^{3}$ $x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
8.41e6_257e4.24t708.1c1$8$ $41^{6} \cdot 257^{4}$ $x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
8.41e2_257e4.24t1151.1c1$8$ $41^{2} \cdot 257^{4}$ $x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$

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 *.