# Properties

 Label 8.0.134217728.1 Degree $8$ Signature $[0, 4]$ Discriminant $2^{27}$ Root discriminant $10.37$ Ramified prime $2$ Class number $1$ (GRH) Class group Trivial (GRH) Galois group $D_{8}$ (as 8T6)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -8, 0, 10, 0, -4, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^6 + 10*x^4 - 8*x^2 + 2)

gp: K = bnfinit(x^8 - 4*x^6 + 10*x^4 - 8*x^2 + 2, 1)

## Normalizeddefining polynomial

$$x^{8} - 4 x^{6} + 10 x^{4} - 8 x^{2} + 2$$

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: $$134217728=2^{27}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $10.37$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

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

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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: $$-2 a^{6} + 7 a^{4} - 16 a^{2} + 7$$ (order $4$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental units: $$4 a^{6} - 14 a^{4} + 33 a^{2} - 15$$,  $$2 a^{7} - 2 a^{6} - 7 a^{5} + 7 a^{4} + 16 a^{3} - 16 a^{2} - 7 a + 7$$,  $$2 a^{7} - 7 a^{5} + 16 a^{3} + a^{2} - 8 a - 1$$ (assuming GRH) magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$19.6078940119$$ (assuming GRH) magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$D_8$ (as 8T6):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 16 The 7 conjugacy class representatives for $D_{8}$ Character table for $D_{8}$

## Intermediate fields

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

## Sibling fields

 Galois closure: data not computed Degree 8 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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]])

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3.2t1.1c1$1$ $2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.2e2.2t1.1c1$1$ $2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e3.2t1.2c1$1$ $2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e7.4t3.1c1$2$ $2^{7}$ $x^{4} - 2 x^{2} - 1$ $D_{4}$ (as 4T3) $1$ $0$
* 2.2e9.8t6.2c1$2$ $2^{9}$ $x^{8} - 4 x^{6} + 10 x^{4} - 8 x^{2} + 2$ $D_{8}$ (as 8T6) $1$ $0$
* 2.2e9.8t6.2c2$2$ $2^{9}$ $x^{8} - 4 x^{6} + 10 x^{4} - 8 x^{2} + 2$ $D_{8}$ (as 8T6) $1$ $0$

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 *.