# Properties

 Label 8.0.1358954496.9 Degree $8$ Signature $[0, 4]$ Discriminant $2^{24}\cdot 3^{4}$ Root discriminant $13.86$ Ramified primes $2, 3$ Class number $2$ Class group $[2]$ Galois Group $D_4$ (as 8T4)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 0, -108, 0, 72, 0, -12, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 12*x^6 + 72*x^4 - 108*x^2 + 81)
gp: K = bnfinit(x^8 - 12*x^6 + 72*x^4 - 108*x^2 + 81, 1)

## Normalizeddefining polynomial

$$x^{8}$$ $$\mathstrut -\mathstrut 12 x^{6}$$ $$\mathstrut +\mathstrut 72 x^{4}$$ $$\mathstrut -\mathstrut 108 x^{2}$$ $$\mathstrut +\mathstrut 81$$

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: $$1358954496=2^{24}\cdot 3^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $13.86$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{81} a^{6} + \frac{1}{27} a^{4} + \frac{1}{9} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{7} + \frac{1}{27} a^{5} + \frac{1}{9} a^{3} + \frac{1}{3} a$

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

## Class group and class number

Multiplicative Abelian group isomorphic to C2, 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: $$\frac{2}{81} a^{6} - \frac{7}{27} a^{4} + \frac{11}{9} a^{2} - \frac{1}{3}$$ (order $8$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$\frac{1}{81} a^{6} - \frac{2}{27} a^{4} + \frac{4}{9} a^{2} - \frac{2}{3}$$,  $$\frac{4}{81} a^{7} - \frac{5}{81} a^{6} - \frac{14}{27} a^{5} + \frac{19}{27} a^{4} + \frac{25}{9} a^{3} - \frac{35}{9} a^{2} - \frac{2}{3} a + \frac{7}{3}$$,  $$\frac{1}{81} a^{6} - \frac{5}{27} a^{4} + \frac{13}{9} a^{2} - a - \frac{5}{3}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$62.7365525512$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$D_4$ (as 8T4):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 8 The 5 conjugacy class representatives for $D_4$ Character table for $D_4$

## Intermediate fields

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

## Sibling fields

 Degree 4 siblings: 4.2.18432.3, 4.0.18432.1

## 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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]])

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3.2t1.2c1$1$ $2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
* 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) $1$ $-1$
*2 2.2e8_3e2.4t3.9c1$2$ $2^{8} \cdot 3^{2}$ $x^{8} - 12 x^{6} + 72 x^{4} - 108 x^{2} + 81$ $D_4$ (as 8T4) $1$ $0$

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 *.