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)

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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)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut -\mathstrut 4 x^{5} \) \(\mathstrut +\mathstrut 5 x^{4} \) \(\mathstrut -\mathstrut 8 x^{3} \) \(\mathstrut +\mathstrut 4 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 16 \)

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(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.77$
magma: Abs(Discriminant(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(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
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 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:  $3$
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

4.4.26569.1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$$[\ ]^{2}$
163.6.4.1$x^{6} + 5216 x^{3} + 35363339$$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 *.