# Properties

 Label 8.0.705911761.1 Degree $8$ Signature $[0, 4]$ Discriminant $163^{4}$ Root discriminant $12.77$ Ramified prime $163$ Class number $1$ Class group Trivial Galois group $\SL(2,3)$ (as 8T12)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -8, 4, -8, 5, -4, 1, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + x^6 - 4*x^5 + 5*x^4 - 8*x^3 + 4*x^2 - 8*x + 16)

gp: K = bnfinit(x^8 - x^7 + x^6 - 4*x^5 + 5*x^4 - 8*x^3 + 4*x^2 - 8*x + 16, 1)

## Normalizeddefining polynomial

$$x^{8} - x^{7} + x^{6} - 4 x^{5} + 5 x^{4} - 8 x^{3} + 4 x^{2} - 8 x + 16$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $8$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[0, 4]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$705911761=163^{4}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $12.77$ 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: $163$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is a CM field.

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

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

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: $3$ 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: $$\frac{5}{8} a^{7} + \frac{3}{8} a^{6} + \frac{9}{8} a^{5} - a^{4} + \frac{13}{8} a^{3} - 3 a^{2} - 2 a - 8$$,  $$\frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{3}{4} a^{2} - \frac{3}{2} a - 2$$,  $$\frac{1}{8} a^{7} - \frac{3}{8} a^{6} + \frac{3}{8} a^{5} - \frac{3}{4} a^{4} + \frac{13}{8} a^{3} - \frac{5}{4} a^{2} + \frac{1}{2} a - 2$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$62.8368450498$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$\SL(2,3)$ (as 8T12):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 24 The 7 conjugacy class representatives for $\SL(2,3)$ Character table for $\SL(2,3)$

## 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$163$163.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2} 163.6.4.1x^{6} + 5216 x^{3} + 35363339$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.163.3t1.1c1$1$ $163$ $x^{3} - x^{2} - 54 x + 169$ $C_3$ (as 3T1) $0$ $1$
1.163.3t1.1c2$1$ $163$ $x^{3} - x^{2} - 54 x + 169$ $C_3$ (as 3T1) $0$ $1$
2.163e2.24t7.2c1$2$ $163^{2}$ $x^{8} - x^{7} + x^{6} - 4 x^{5} + 5 x^{4} - 8 x^{3} + 4 x^{2} - 8 x + 16$ $\SL(2,3)$ (as 8T12) $-1$ $-2$
* 2.163.8t12.1c1$2$ $163$ $x^{8} - x^{7} + x^{6} - 4 x^{5} + 5 x^{4} - 8 x^{3} + 4 x^{2} - 8 x + 16$ $\SL(2,3)$ (as 8T12) $0$ $-2$
* 2.163.8t12.1c2$2$ $163$ $x^{8} - x^{7} + x^{6} - 4 x^{5} + 5 x^{4} - 8 x^{3} + 4 x^{2} - 8 x + 16$ $\SL(2,3)$ (as 8T12) $0$ $-2$
* 3.163e2.4t4.1c1$3$ $163^{2}$ $x^{4} - x^{3} - 7 x^{2} + 2 x + 9$ $A_4$ (as 4T4) $1$ $3$

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 *.