# Properties

 Label 8.4.105226698752.3 Degree $8$ Signature $[4, 2]$ Discriminant $2^{31}\cdot 7^{2}$ Root discriminant $23.87$ Ramified primes $2, 7$ Class number $1$ Class group Trivial Galois Group $C_8:C_2$ (as 8T7)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![98, 0, -112, 0, -92, 0, -8, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 8*x^6 - 92*x^4 - 112*x^2 + 98)
gp: K = bnfinit(x^8 - 8*x^6 - 92*x^4 - 112*x^2 + 98, 1)

## Normalizeddefining polynomial

$$x^{8}$$ $$\mathstrut -\mathstrut 8 x^{6}$$ $$\mathstrut -\mathstrut 92 x^{4}$$ $$\mathstrut -\mathstrut 112 x^{2}$$ $$\mathstrut +\mathstrut 98$$

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

## Invariants

 Degree: $8$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[4, 2]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$105226698752=2^{31}\cdot 7^{2}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $23.87$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 7$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4711} a^{6} + \frac{76}{4711} a^{4} + \frac{1581}{4711} a^{2} + \frac{112}{673}$, $\frac{1}{4711} a^{7} + \frac{76}{4711} a^{5} + \frac{1581}{4711} a^{3} + \frac{112}{673} a$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $5$ magma: UnitRank(K); sage: UK.rank() gp: K.fu Torsion generator: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$\frac{2}{4711} a^{6} + \frac{152}{4711} a^{4} - \frac{1549}{4711} a^{2} - \frac{1795}{673}$$,  $$\frac{2}{4711} a^{6} + \frac{152}{4711} a^{4} - \frac{1549}{4711} a^{2} - \frac{449}{673}$$,  $$\frac{51}{4711} a^{6} - \frac{835}{4711} a^{4} + \frac{544}{4711} a^{2} + \frac{1001}{673}$$,  $$\frac{171}{4711} a^{7} + \frac{166}{4711} a^{6} - \frac{1137}{4711} a^{5} - \frac{1517}{4711} a^{4} - \frac{17020}{4711} a^{3} - \frac{15503}{4711} a^{2} - \frac{5076}{673} a - \frac{3617}{673}$$,  $$\frac{283}{4711} a^{7} - \frac{170}{4711} a^{6} - \frac{2047}{4711} a^{5} + \frac{1213}{4711} a^{4} - \frac{28388}{4711} a^{3} + \frac{18601}{4711} a^{2} - \frac{7338}{673} a + \frac{5861}{673}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$437.074292589$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$OD_{16}$ (as 8T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 16 The 10 conjugacy class representatives for $C_8:C_2$ Character table for $C_8:C_2$

## Intermediate fields

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

## Sibling fields

 Galois closure: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ R ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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

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];
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]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.31.2$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5] 77.4.2.2x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_7.2t1.2c1$1$ $2^{3} \cdot 7$ $x^{2} + 14$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.1c1$1$ $7$ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e3.2t1.1c1$1$ $2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e4_7.4t1.2c1$1$ $2^{4} \cdot 7$ $x^{4} + 28 x^{2} + 98$ $C_4$ (as 4T1) $0$ $-1$
* 1.2e4.4t1.1c1$1$ $2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
* 1.2e4.4t1.1c2$1$ $2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
1.2e4_7.4t1.2c2$1$ $2^{4} \cdot 7$ $x^{4} + 28 x^{2} + 98$ $C_4$ (as 4T1) $0$ $-1$
* 2.2e10_7.8t7.2c1$2$ $2^{10} \cdot 7$ $x^{8} - 8 x^{6} - 92 x^{4} - 112 x^{2} + 98$ $C_8:C_2$ (as 8T7) $0$ $0$
* 2.2e10_7.8t7.2c2$2$ $2^{10} \cdot 7$ $x^{8} - 8 x^{6} - 92 x^{4} - 112 x^{2} + 98$ $C_8:C_2$ (as 8T7) $0$ $0$

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 *.