# Properties

 Label 8.0.21667237072994304.2 Degree $8$ Signature $[0, 4]$ Discriminant $2^{24}\cdot 3^{6}\cdot 11^{6}$ Root discriminant $110.15$ Ramified primes $2, 3, 11$ Class number $1296$ Class group $[2, 18, 36]$ Galois group $Q_8$ (as 8T5)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 132*x^6 + 5940*x^4 + 100188*x^2 + 393129)

gp: K = bnfinit(x^8 + 132*x^6 + 5940*x^4 + 100188*x^2 + 393129, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![393129, 0, 100188, 0, 5940, 0, 132, 0, 1]);

## Normalizeddefining polynomial

$$x^{8} + 132 x^{6} + 5940 x^{4} + 100188 x^{2} + 393129$$

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: $$21667237072994304=2^{24}\cdot 3^{6}\cdot 11^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $110.15$ 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: $2, 3, 11$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $8$ This field is Galois over $\Q$. This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{165} a^{4} - \frac{2}{5}$, $\frac{1}{165} a^{5} - \frac{2}{5} a$, $\frac{1}{825} a^{6} + \frac{1}{825} a^{4} - \frac{2}{25} a^{2} - \frac{2}{25}$, $\frac{1}{15675} a^{7} - \frac{13}{5225} a^{5} + \frac{123}{475} a^{3} + \frac{53}{475} a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}\times C_{18}\times C_{36}$, which has order $1296$

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: $$\frac{1}{825} a^{6} + \frac{76}{825} a^{4} + \frac{48}{25} a^{2} + \frac{198}{25}$$,  $$\frac{1}{275} a^{6} + \frac{298}{825} a^{4} + \frac{244}{25} a^{2} + \frac{1104}{25}$$,  $$\frac{1}{825} a^{6} + \frac{37}{275} a^{4} + \frac{123}{25} a^{2} + \frac{1478}{25}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$267.195532151$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$Q_8$ (as 8T5):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 8 The 5 conjugacy class representatives for $Q_8$ Character table for $Q_8$

## Intermediate fields

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

## 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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.24.4$x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 14$$8$$1$$24$$Q_8$$[2, 3, 4] 33.8.6.1x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$11$11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.24.2t1.a.a$1$ $2^{3} \cdot 3$ $x^{2} - 6$ $C_2$ (as 2T1) $1$ $1$
* 1.44.2t1.a.a$1$ $2^{2} \cdot 11$ $x^{2} - 11$ $C_2$ (as 2T1) $1$ $1$
* 1.264.2t1.a.a$1$ $2^{3} \cdot 3 \cdot 11$ $x^{2} - 66$ $C_2$ (as 2T1) $1$ $1$
*2 2.278784.8t5.h.a$2$ $2^{8} \cdot 3^{2} \cdot 11^{2}$ $x^{8} + 132 x^{6} + 5940 x^{4} + 100188 x^{2} + 393129$ $Q_8$ (as 8T5) $-1$ $-2$

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 *.