# Properties

 Label 8.0.324000000.5 Degree $8$ Signature $[0, 4]$ Discriminant $2^{8}\cdot 3^{4}\cdot 5^{6}$ Root discriminant $11.58$ Ramified primes $2, 3, 5$ Class number $2$ Class group $[2]$ Galois Group $C_2^2:C_4$ (as 8T10)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 0, -81, 0, 36, 0, -6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 6*x^6 + 36*x^4 - 81*x^2 + 81)
gp: K = bnfinit(x^8 - 6*x^6 + 36*x^4 - 81*x^2 + 81, 1)

## Normalizeddefining polynomial

$$x^{8}$$ $$\mathstrut -\mathstrut 6 x^{6}$$ $$\mathstrut +\mathstrut 36 x^{4}$$ $$\mathstrut -\mathstrut 81 x^{2}$$ $$\mathstrut +\mathstrut 81$$

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: $$324000000=2^{8}\cdot 3^{4}\cdot 5^{6}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.58$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3, 5$ 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 $$a$$)

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$

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

## Class group and class number

$C_{2}$, which has order $2$

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: $$-\frac{1}{27} a^{6} + \frac{1}{9} a^{4} - a^{2} + 1$$ (order $10$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$\frac{1}{27} a^{6} - \frac{2}{9} a^{4} + a^{2} - 1$$,  $$\frac{1}{27} a^{7} - \frac{2}{27} a^{6} - \frac{1}{9} a^{5} + \frac{2}{9} a^{4} + a^{3} - \frac{5}{3} a^{2} - a + 2$$,  $$\frac{2}{27} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{4} + 2 a^{3} - 3 a^{2} - 2 a + 4$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$24.6709829244$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$C_2^2:C_4$ (as 8T10):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 16 The 10 conjugacy class representatives for $C_2^2:C_4$ Character table for $C_2^2:C_4$

## 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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]])