# Properties

 Label 8.0.2067245.1 Degree $8$ Signature $[0, 4]$ Discriminant $2067245$ Root discriminant $6.16$ Ramified primes $5, 643$ Class number $1$ Class group trivial Galois group $C_2 \wr S_4$ (as 8T44)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

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

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$2067245$$$$\medspace = 5\cdot 643^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $6.16$ 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))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

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

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: $3$ 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: $$a^{7} - a^{6} + a^{5} - a^{3} + a - 1$$,  $$a^{6} - a^{5} - a^{2} + 1$$,  $$a^{4} + a - 1$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.453534943932$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 0.453534943932 \cdot 1}{2\sqrt{2067245}}\approx 0.2458126658969$

## Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
643Data not computed

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.643.2t1.a.a$1$ $643$ $$\Q(\sqrt{-643})$$ $C_2$ (as 2T1) $1$ $-1$
1.3215.2t1.a.a$1$ $5 \cdot 643$ $$\Q(\sqrt{-3215})$$ $C_2$ (as 2T1) $1$ $-1$
1.5.2t1.a.a$1$ $5$ $$\Q(\sqrt{5})$$ $C_2$ (as 2T1) $1$ $1$
2.643.3t2.a.a$2$ $643$ 3.1.643.1 $S_3$ (as 3T2) $1$ $0$
2.16075.6t3.b.a$2$ $5^{2} \cdot 643$ 6.0.33230963375.1 $D_{6}$ (as 6T3) $1$ $0$
* 3.643.4t5.a.a$3$ $643$ 4.2.643.1 $S_4$ (as 4T5) $1$ $1$
3.413449.6t8.a.a$3$ $643^{2}$ 4.2.643.1 $S_4$ (as 4T5) $1$ $-1$
3.80375.6t11.a.a$3$ $5^{3} \cdot 643$ 6.2.51681125.1 $S_4\times C_2$ (as 6T11) $1$ $1$
3.51681125.6t11.a.a$3$ $5^{3} \cdot 643^{2}$ 6.2.51681125.1 $S_4\times C_2$ (as 6T11) $1$ $-1$
* 4.3215.8t44.b.a$4$ $5 \cdot 643$ 8.0.2067245.1 $C_2 \wr S_4$ (as 8T44) $1$ $-2$
4.80375.8t44.b.a$4$ $5^{3} \cdot 643$ 8.0.2067245.1 $C_2 \wr S_4$ (as 8T44) $1$ $-2$
4.1329238535.8t44.b.a$4$ $5 \cdot 643^{3}$ 8.0.2067245.1 $C_2 \wr S_4$ (as 8T44) $1$ $2$
4.33230963375.8t44.b.a$4$ $5^{3} \cdot 643^{3}$ 8.0.2067245.1 $C_2 \wr S_4$ (as 8T44) $1$ $2$
6.51681125.8t41.b.a$6$ $5^{3} \cdot 643^{2}$ 8.4.258405625.1 $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.213...125.12t108.b.a$6$ $5^{3} \cdot 643^{4}$ 8.4.258405625.1 $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
6.33230963375.8t41.b.a$6$ $5^{3} \cdot 643^{3}$ 8.4.258405625.1 $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $0$
6.33230963375.12t108.b.a$6$ $5^{3} \cdot 643^{3}$ 8.4.258405625.1 $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $0$
8.267...625.24t708.b.a$8$ $5^{6} \cdot 643^{4}$ 8.0.2067245.1 $C_2 \wr S_4$ (as 8T44) $1$ $0$
8.427...025.24t708.b.a$8$ $5^{2} \cdot 643^{4}$ 8.0.2067245.1 $C_2 \wr S_4$ (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 *.