# Properties

 Label 8.4.2522550625.1 Degree $8$ Signature $[4, 2]$ Discriminant $5^{4}\cdot 7^{4}\cdot 41^{2}$ Root discriminant $14.97$ Ramified primes $5, 7, 41$ Class number $1$ Class group Trivial Galois group $C_2^4:C_6$ (as 8T33)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 21, -13, 11, -11, 9, -1, -3, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 - x^6 + 9*x^5 - 11*x^4 + 11*x^3 - 13*x^2 + 21*x - 9)

gp: K = bnfinit(x^8 - 3*x^7 - x^6 + 9*x^5 - 11*x^4 + 11*x^3 - 13*x^2 + 21*x - 9, 1)

## Normalizeddefining polynomial

$$x^{8} - 3 x^{7} - x^{6} + 9 x^{5} - 11 x^{4} + 11 x^{3} - 13 x^{2} + 21 x - 9$$

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: $$2522550625=5^{4}\cdot 7^{4}\cdot 41^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $14.97$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $5, 7, 41$ magma: PrimeDivisors(Discriminant(Integers(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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1179} a^{7} - \frac{146}{393} a^{6} - \frac{469}{1179} a^{5} + \frac{19}{393} a^{4} - \frac{47}{1179} a^{3} + \frac{413}{1179} a^{2} - \frac{460}{1179} a - \frac{103}{393}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Trivial group, which has 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: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$62.2273770026$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 96 The 10 conjugacy class representatives for $C_2^4:C_6$ Character table for $C_2^4:C_6$

## Intermediate fields

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

## Sibling fields

 Degree 8 sibling: data not computed Degree 12 siblings: data not computed Degree 16 sibling: data not computed Degree 24 siblings: data not computed Degree 32 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.6.3.1x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 7.6.4.3x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2} 41.2.0.1x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.5_7.6t1.1c1$1$ $5 \cdot 7$ $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ $C_6$ (as 6T1) $0$ $1$
1.7.3t1.1c1$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.7.3t1.1c2$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.5_7.6t1.1c2$1$ $5 \cdot 7$ $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ $C_6$ (as 6T1) $0$ $1$
3.7e2_41e2.4t4.1c1$3$ $7^{2} \cdot 41^{2}$ $x^{4} - x^{3} + 4 x^{2} - 7 x + 8$ $A_4$ (as 4T4) $1$ $-1$
3.5e3_7e2_41e2.6t6.1c1$3$ $5^{3} \cdot 7^{2} \cdot 41^{2}$ $x^{6} - 2 x^{5} + 38 x^{4} + 12 x^{3} + 32 x^{2} + 719 x - 3361$ $A_4\times C_2$ (as 6T6) $1$ $-1$
6.5e3_7e4_41e4.8t33.1c1$6$ $5^{3} \cdot 7^{4} \cdot 41^{4}$ $x^{8} - 3 x^{7} - x^{6} + 9 x^{5} - 11 x^{4} + 11 x^{3} - 13 x^{2} + 21 x - 9$ $C_2^4:C_6$ (as 8T33) $1$ $-2$
* 6.5e3_7e4_41e2.8t33.1c1$6$ $5^{3} \cdot 7^{4} \cdot 41^{2}$ $x^{8} - 3 x^{7} - x^{6} + 9 x^{5} - 11 x^{4} + 11 x^{3} - 13 x^{2} + 21 x - 9$ $C_2^4:C_6$ (as 8T33) $1$ $2$

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 *.