# Properties

 Label 8.0.1438916737499136.2 Degree $8$ Signature $[0, 4]$ Discriminant $1.439\times 10^{15}$ Root discriminant $78.48$ Ramified primes $2, 3, 7$ Class number $400$ Class group $[2, 10, 20]$ Galois group $Q_8$ (as 8T5)

# Learn more

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 84*x^6 + 2268*x^4 + 19404*x^2 + 441)

gp: K = bnfinit(x^8 + 84*x^6 + 2268*x^4 + 19404*x^2 + 441, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![441, 0, 19404, 0, 2268, 0, 84, 0, 1]);

$$x^{8} + 84 x^{6} + 2268 x^{4} + 19404 x^{2} + 441$$

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: $$1438916737499136$$$$\medspace = 2^{24}\cdot 3^{6}\cdot 7^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $78.48$ 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: $2, 3, 7$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Gal(K/\Q) }$: $8$ This field is Galois over $\Q$. This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{21} a^{4}$, $\frac{1}{21} a^{5}$, $\frac{1}{3003} a^{6} - \frac{10}{3003} a^{4} + \frac{37}{143} a^{2} + \frac{20}{143}$, $\frac{1}{3003} a^{7} - \frac{10}{3003} a^{5} + \frac{37}{143} a^{3} + \frac{20}{143} a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}\times C_{10}\times C_{20}$, which has order $400$

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: $$\frac{2}{3003} a^{6} + \frac{41}{1001} a^{4} + \frac{74}{143} a^{2} + \frac{40}{143}$$,  $$\frac{2}{429} a^{6} + \frac{718}{3003} a^{4} + \frac{375}{143} a^{2} - \frac{6}{143}$$,  $$\frac{5}{3003} a^{6} + \frac{379}{3003} a^{4} + \frac{328}{143} a^{2} - \frac{43}{143}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$116.658894546$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 116.658894546 \cdot 400}{2\sqrt{1438916737499136}}\approx 0.958626639837$

## 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

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R R ${\href{/padicField/5.4.0.1}{4} }^{2}$ R ${\href{/padicField/11.1.0.1}{1} }^{8}$ ${\href{/padicField/13.1.0.1}{1} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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$2.8.24.8$x^{8} + 8 x^{7} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 22$$8$$1$$24$$Q_8$$[2, 3, 4] 33.8.6.1x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.56.2t1.a.a$1$ $2^{3} \cdot 7$ $$\Q(\sqrt{14})$$ $C_2$ (as 2T1) $1$ $1$
* 1.168.2t1.a.a$1$ $2^{3} \cdot 3 \cdot 7$ $$\Q(\sqrt{42})$$ $C_2$ (as 2T1) $1$ $1$
* 1.12.2t1.a.a$1$ $2^{2} \cdot 3$ $$\Q(\sqrt{3})$$ $C_2$ (as 2T1) $1$ $1$
*2 2.112896.8t5.d.a$2$ $2^{8} \cdot 3^{2} \cdot 7^{2}$ 8.0.1438916737499136.2 $Q_8$ (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 *.