# Properties

 Label 8.0.5416809268248576.2 Degree $8$ Signature $[0, 4]$ Discriminant $5.417\times 10^{15}$ Root discriminant $92.62$ Ramified primes $2, 3, 11$ Class number $200$ Class group $[2, 10, 10]$ Galois group $Q_8$ (as 8T5)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 132*x^6 + 5346*x^4 + 69696*x^2 + 278784)

gp: K = bnfinit(x^8 + 132*x^6 + 5346*x^4 + 69696*x^2 + 278784, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![278784, 0, 69696, 0, 5346, 0, 132, 0, 1]);

$$x^{8} + 132 x^{6} + 5346 x^{4} + 69696 x^{2} + 278784$$

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: $$5416809268248576$$$$\medspace = 2^{22}\cdot 3^{6}\cdot 11^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $92.62$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{66} a^{4}$, $\frac{1}{132} a^{5} - \frac{1}{2} a$, $\frac{1}{16368} a^{6} + \frac{25}{4092} a^{4} + \frac{25}{248} a^{2} + \frac{1}{31}$, $\frac{1}{65472} a^{7} + \frac{25}{16368} a^{5} + \frac{273}{992} a^{3} + \frac{8}{31} a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}\times C_{10}\times C_{10}$, which has order $200$

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: $$\frac{1}{16368} a^{6} + \frac{25}{4092} a^{4} + \frac{25}{248} a^{2} + \frac{1}{31}$$,  $$\frac{3}{1364} a^{6} + \frac{256}{1023} a^{4} + \frac{473}{62} a^{2} + \frac{1431}{31}$$,  $$\frac{8}{1023} a^{6} + \frac{28}{31} a^{4} + \frac{896}{31} a^{2} + \frac{5491}{31}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$262.769412592$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 262.769412592 \cdot 200}{2\sqrt{5416809268248576}}\approx 0.556445342547$

## 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4] 2.4.11.2x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2} 1111.8.6.1x^{8} + 143 x^{4} + 5929$$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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.264.2t1.a.a$1$ $2^{3} \cdot 3 \cdot 11$ $x^{2} - 66$ $C_2$ (as 2T1) $1$ $1$
* 1.8.2t1.a.a$1$ $2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.33.2t1.a.a$1$ $3 \cdot 11$ $x^{2} - x - 8$ $C_2$ (as 2T1) $1$ $1$
*2 2.278784.8t5.d.a$2$ $2^{8} \cdot 3^{2} \cdot 11^{2}$ $x^{8} + 132 x^{6} + 5346 x^{4} + 69696 x^{2} + 278784$ $Q_8$ (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 *.