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Properties

Label 8.2.17331306928.4
Degree $8$
Signature $[2, 3]$
Discriminant $-\,2^{4}\cdot 13^{3}\cdot 79^{3}$
Root discriminant $19.05$
Ramified primes $2, 13, 79$
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![-2, -7, 4, 1, 13, -19, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 11*x^6 - 19*x^5 + 13*x^4 + x^3 + 4*x^2 - 7*x - 2)
 
gp: K = bnfinit(x^8 - 4*x^7 + 11*x^6 - 19*x^5 + 13*x^4 + x^3 + 4*x^2 - 7*x - 2, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 11 x^{6} - 19 x^{5} + 13 x^{4} + x^{3} + 4 x^{2} - 7 x - 2 \)

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:  \(-17331306928=-\,2^{4}\cdot 13^{3}\cdot 79^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.05$
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:  $2, 13, 79$
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 \(a\))

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

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:  \( \frac{1}{2} a^{6} - \frac{3}{2} a^{5} + 4 a^{4} - \frac{11}{2} a^{3} + \frac{1}{2} a^{2} + 2 a + 1 \),  \( \frac{1}{4} a^{7} - \frac{3}{4} a^{6} + \frac{9}{4} a^{5} - \frac{13}{4} a^{4} + \frac{9}{4} a^{3} - \frac{3}{4} a^{2} + \frac{1}{2} a - 1 \),  \( \frac{1}{4} a^{7} - a^{6} + 3 a^{5} - \frac{11}{2} a^{4} + \frac{11}{2} a^{3} - 3 a^{2} + \frac{5}{4} a + \frac{1}{2} \),  \( 3 a^{6} - 14 a^{5} + 41 a^{4} - 78 a^{3} + 72 a^{2} - 8 a - 19 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 359.86487448 \)
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

4.2.4108.1

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: 8.2.17331306928.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ R ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.13_79.2t1.1c1$1$ $ 13 \cdot 79 $ $x^{2} - x + 257$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_13_79.3t2.1c1$2$ $ 2^{2} \cdot 13 \cdot 79 $ $x^{3} + 10 x - 2$ $S_3$ (as 3T2) $1$ $0$
2.2e2_13_79.24t22.4c1$2$ $ 2^{2} \cdot 13 \cdot 79 $ $x^{8} - 4 x^{7} + 11 x^{6} - 19 x^{5} + 13 x^{4} + x^{3} + 4 x^{2} - 7 x - 2$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.2e2_13_79.24t22.4c2$2$ $ 2^{2} \cdot 13 \cdot 79 $ $x^{8} - 4 x^{7} + 11 x^{6} - 19 x^{5} + 13 x^{4} + x^{3} + 4 x^{2} - 7 x - 2$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.2e2_13e2_79e2.6t8.1c1$3$ $ 2^{2} \cdot 13^{2} \cdot 79^{2}$ $x^{4} - x^{3} - 2 x - 2$ $S_4$ (as 4T5) $1$ $-1$
* 3.2e2_13_79.4t5.1c1$3$ $ 2^{2} \cdot 13 \cdot 79 $ $x^{4} - x^{3} - 2 x - 2$ $S_4$ (as 4T5) $1$ $1$
* 4.2e2_13e2_79e2.8t23.4c1$4$ $ 2^{2} \cdot 13^{2} \cdot 79^{2}$ $x^{8} - 4 x^{7} + 11 x^{6} - 19 x^{5} + 13 x^{4} + x^{3} + 4 x^{2} - 7 x - 2$ $\textrm{GL(2,3)}$ (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 *.