Properties

Label 8.2.19954863104.2
Degree $8$
Signature $[2, 3]$
Discriminant $-\,2^{10}\cdot 11^{7}$
Root discriminant $19.39$
Ramified primes $2, 11$
Class number $1$
Class group Trivial
Galois Group $\textrm{GL(2,3)}$ (as 8T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-99, -44, 0, -44, 22, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 22*x^4 - 44*x^3 - 44*x - 99)
gp: K = bnfinit(x^8 + 22*x^4 - 44*x^3 - 44*x - 99, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut +\mathstrut 22 x^{4} \) \(\mathstrut -\mathstrut 44 x^{3} \) \(\mathstrut -\mathstrut 44 x \) \(\mathstrut -\mathstrut 99 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 3]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-19954863104=-\,2^{10}\cdot 11^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $19.39$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{400828} a^{7} + \frac{35207}{400828} a^{6} - \frac{27741}{400828} a^{5} - \frac{59965}{400828} a^{4} + \frac{173757}{400828} a^{3} + \frac{25719}{400828} a^{2} - \frac{182033}{400828} a - \frac{197397}{400828}$

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:  $4$
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:  \( 324.186778143 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$\GL(2,3)$ (as 8T23):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 48
The 8 conjugacy class representatives for $\textrm{GL(2,3)}$
Character table for $\textrm{GL(2,3)}$

Intermediate fields

4.2.21296.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 24 sibling: data not computed
Arithmetically equvalently sibling: 8.2.19954863104.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.8.10.3$x^{8} + 4 x^{2} + 20$$8$$1$$10$$\textrm{GL(2,3)}$$[4/3, 4/3, 3/2]_{3}^{2}$
$11$11.8.7.2$x^{8} - 11$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_11.3t2.1c1$2$ $ 2^{2} \cdot 11 $ $x^{3} - x^{2} + x + 1$ $S_3$ (as 3T2) $1$ $0$
2.2e3_11e2.24t22.1c1$2$ $ 2^{3} \cdot 11^{2}$ $x^{8} + 22 x^{4} - 44 x^{3} - 44 x - 99$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.2e3_11e2.24t22.1c2$2$ $ 2^{3} \cdot 11^{2}$ $x^{8} + 22 x^{4} - 44 x^{3} - 44 x - 99$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.2e4_11e2.6t8.1c1$3$ $ 2^{4} \cdot 11^{2}$ $x^{4} - 2 x^{3} - 4 x^{2} - 6 x - 2$ $S_4$ (as 4T5) $1$ $-1$
* 3.2e4_11e3.4t5.1c1$3$ $ 2^{4} \cdot 11^{3}$ $x^{4} - 2 x^{3} - 4 x^{2} - 6 x - 2$ $S_4$ (as 4T5) $1$ $1$
* 4.2e6_11e4.8t23.1c1$4$ $ 2^{6} \cdot 11^{4}$ $x^{8} + 22 x^{4} - 44 x^{3} - 44 x - 99$ $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$

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 *.