Properties

Label 8.2.10403062487.3
Degree $8$
Signature $[2, 3]$
Discriminant $-\,37^{3}\cdot 59^{3}$
Root discriminant $17.87$
Ramified primes $37, 59$
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![4, -27, -15, 18, 12, -1, -5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 5*x^6 - x^5 + 12*x^4 + 18*x^3 - 15*x^2 - 27*x + 4)
gp: K = bnfinit(x^8 - x^7 - 5*x^6 - x^5 + 12*x^4 + 18*x^3 - 15*x^2 - 27*x + 4, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut -\mathstrut 5 x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut +\mathstrut 12 x^{4} \) \(\mathstrut +\mathstrut 18 x^{3} \) \(\mathstrut -\mathstrut 15 x^{2} \) \(\mathstrut -\mathstrut 27 x \) \(\mathstrut +\mathstrut 4 \)

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:  \(-10403062487=-\,37^{3}\cdot 59^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $17.87$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $37, 59$
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 \(a\))

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

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:  $4$
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:  \( \frac{1}{4} a^{5} - \frac{3}{4} a^{3} - \frac{3}{4} a^{2} + \frac{3}{4} a + 2 \),  \( \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{3}{4} a^{4} - \frac{1}{4} a^{3} + \frac{9}{4} a^{2} + \frac{5}{2} a - 3 \),  \( \frac{1}{2} a^{7} + \frac{3}{4} a^{6} - a^{5} - \frac{15}{4} a^{4} - \frac{13}{4} a^{3} + \frac{11}{4} a^{2} + \frac{7}{2} a - 1 \),  \( a^{7} - \frac{3}{4} a^{6} - 5 a^{5} - \frac{7}{4} a^{4} + \frac{45}{4} a^{3} + \frac{75}{4} a^{2} - 13 a - 29 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 243.223791751 \)
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

4.2.2183.1

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: 8.2.10403062487.4

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ R

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
37Data not computed
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.1$x^{2} - 59$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.37_59.2t1.1c1$1$ $ 37 \cdot 59 $ $x^{2} - x + 546$ $C_2$ (as 2T1) $1$ $-1$
2.37_59.3t2.1c1$2$ $ 37 \cdot 59 $ $x^{3} - x - 9$ $S_3$ (as 3T2) $1$ $0$
2.37_59.24t22.3c1$2$ $ 37 \cdot 59 $ $x^{8} - x^{7} - 5 x^{6} - x^{5} + 12 x^{4} + 18 x^{3} - 15 x^{2} - 27 x + 4$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.37_59.24t22.3c2$2$ $ 37 \cdot 59 $ $x^{8} - x^{7} - 5 x^{6} - x^{5} + 12 x^{4} + 18 x^{3} - 15 x^{2} - 27 x + 4$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.37e2_59e2.6t8.1c1$3$ $ 37^{2} \cdot 59^{2}$ $x^{4} - 2 x^{3} + 3 x^{2} + x - 1$ $S_4$ (as 4T5) $1$ $-1$
* 3.37_59.4t5.1c1$3$ $ 37 \cdot 59 $ $x^{4} - 2 x^{3} + 3 x^{2} + x - 1$ $S_4$ (as 4T5) $1$ $1$
* 4.37e2_59e2.8t23.3c1$4$ $ 37^{2} \cdot 59^{2}$ $x^{8} - x^{7} - 5 x^{6} - x^{5} + 12 x^{4} + 18 x^{3} - 15 x^{2} - 27 x + 4$ $\textrm{GL(2,3)}$ (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 *.