# Properties

 Label 8.2.5754585088.2 Degree $8$ Signature $[2, 3]$ Discriminant $-5754585088$ Root discriminant $16.60$ Ramified primes $2, 7$ Class number $1$ Class group trivial Galois group $QD_{16}$ (as 8T8)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 4*x^6 + 4*x^4 - 4*x^2 - 7)

gp: K = bnfinit(x^8 + 4*x^6 + 4*x^4 - 4*x^2 - 7, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, -4, 0, 4, 0, 4, 0, 1]);

$$x^{8} + 4 x^{6} + 4 x^{4} - 4 x^{2} - 7$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-5754585088$$$$\medspace = -\,2^{24}\cdot 7^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $16.60$ 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: $2, 7$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $4$ 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: $$a^{2} + 2$$,  $$a^{6} + 2 a^{4} - a^{2} - 5$$,  $$a^{7} + 2 a^{5} - a^{4} - a^{3} - 3 a + 3$$,  $$a^{7} + a^{6} + 5 a^{5} + 5 a^{4} + 9 a^{3} + 9 a^{2} + 5 a + 6$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$117.703504989$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 117.703504989 \cdot 1}{2\sqrt{5754585088}}\approx 0.769754060051$

## Galois group

$SD_{16}$ (as 8T8):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 16 The 7 conjugacy class representatives for $QD_{16}$ Character table for $QD_{16}$

## Intermediate fields

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

## Sibling fields

 Galois closure: 16.0.1622647227216566419456.5

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.8.0.1}{8} }$ ${\href{/padicField/5.8.0.1}{8} }$ R ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }$

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$$[2, 3, 4] 7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2} 7.2.1.2x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.8.2t1.a.a$1$ $2^{3}$ $$\Q(\sqrt{2})$$ $C_2$ (as 2T1) $1$ $1$
1.7.2t1.a.a$1$ $7$ $$\Q(\sqrt{-7})$$ $C_2$ (as 2T1) $1$ $-1$
1.56.2t1.b.a$1$ $2^{3} \cdot 7$ $$\Q(\sqrt{-14})$$ $C_2$ (as 2T1) $1$ $-1$
* 2.224.4t3.c.a$2$ $2^{5} \cdot 7$ 4.2.1792.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.1792.8t8.a.a$2$ $2^{8} \cdot 7$ 8.2.5754585088.2 $QD_{16}$ (as 8T8) $0$ $0$
* 2.1792.8t8.a.b$2$ $2^{8} \cdot 7$ 8.2.5754585088.2 $QD_{16}$ (as 8T8) $0$ $0$

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 *.