# Properties

 Label 8.0.558140625.1 Degree $8$ Signature $[0, 4]$ Discriminant $558140625$ Root discriminant $$12.40$$ Ramified primes see page Class number $2$ Class group $[2]$ Galois group $Q_8:C_2$ (as 8T11)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 10*x^6 + 11*x^5 + 34*x^4 - 29*x^3 - 55*x^2 + 4*x + 76)

gp: K = bnfinit(x^8 - x^7 - 10*x^6 + 11*x^5 + 34*x^4 - 29*x^3 - 55*x^2 + 4*x + 76, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![76, 4, -55, -29, 34, 11, -10, -1, 1]);

$$x^{8} - x^{7} - 10x^{6} + 11x^{5} + 34x^{4} - 29x^{3} - 55x^{2} + 4x + 76$$

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: $$558140625$$ 558140625 $$\medspace = 3^{6}\cdot 5^{6}\cdot 7^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$12.40$$ 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: $$3$$, $$5$$, $$7$$ 3, 5, 7 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $4$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{6}+\frac{1}{8}a^{4}-\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: No Index: Not computed Inessential primes: $2$

## Class group and class number

$C_{2}$, which has order $2$

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: $$\frac{1}{4} a^{5} - \frac{5}{4} a^{3} + \frac{5}{4} a + \frac{5}{2}$$ (1)/(4)*a^(5) - (5)/(4)*a^(3) + (5)/(4)*a + (5)/(2)  (order $6$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{1}{8}a^{7}-\frac{7}{8}a^{5}+\frac{17}{8}a^{3}+a^{2}-2a-2$, $\frac{1}{8}a^{7}-\frac{3}{8}a^{6}-\frac{1}{8}a^{5}+\frac{17}{8}a^{4}-\frac{17}{8}a^{3}-\frac{19}{8}a^{2}+\frac{1}{4}a+7$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{3}{4}a^{3}+\frac{5}{2}a^{2}-\frac{7}{4}a+\frac{1}{2}$ 1/8*a^7 - 7/8*a^5 + 17/8*a^3 + a^2 - 2*a - 2, 1/8*a^7 - 3/8*a^6 - 1/8*a^5 + 17/8*a^4 - 17/8*a^3 - 19/8*a^2 + 1/4*a + 7, 1/4*a^5 - 1/2*a^4 - 3/4*a^3 + 5/2*a^2 - 7/4*a + 1/2 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$31.3606243426$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 31.3606243426 \cdot 2}{6\sqrt{558140625}}\approx 0.689621990597$

## Galois group

$D_4:C_2$ (as 8T11):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 16 The 10 conjugacy class representatives for $Q_8:C_2$ Character table for $Q_8:C_2$

## Intermediate fields

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

## Sibling fields

 Galois closure: 16.0.747961818418212890625.3 Degree 8 siblings: 8.4.27348890625.1, 8.0.27348890625.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.2.0.1}{2} }^{4}$ R R R ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
$$3$$ 3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2} $$5$$ 5.8.6.1x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$$7$$ 7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2} 7.4.0.1x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.105.2t1.a.a$1$ $3 \cdot 5 \cdot 7$ $$\Q(\sqrt{105})$$ $C_2$ (as 2T1) $1$ $1$
* 1.3.2t1.a.a$1$ $3$ $$\Q(\sqrt{-3})$$ $C_2$ (as 2T1) $1$ $-1$
1.35.2t1.a.a$1$ $5 \cdot 7$ $$\Q(\sqrt{-35})$$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.a.a$1$ $7$ $$\Q(\sqrt{-7})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $5$ $$\Q(\sqrt{5})$$ $C_2$ (as 2T1) $1$ $1$
* 1.15.2t1.a.a$1$ $3 \cdot 5$ $$\Q(\sqrt{-15})$$ $C_2$ (as 2T1) $1$ $-1$
1.21.2t1.a.a$1$ $3 \cdot 7$ $$\Q(\sqrt{21})$$ $C_2$ (as 2T1) $1$ $1$
* 2.1575.8t11.a.a$2$ $3^{2} \cdot 5^{2} \cdot 7$ 8.0.558140625.1 $Q_8:C_2$ (as 8T11) $0$ $0$
* 2.1575.8t11.a.b$2$ $3^{2} \cdot 5^{2} \cdot 7$ 8.0.558140625.1 $Q_8:C_2$ (as 8T11) $0$ $0$

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 *.