Properties

Label 7.7.18424261696.1
Degree $7$
Signature $[7, 0]$
Discriminant $2^{6}\cdot 19^{4}\cdot 47^{2}$
Root discriminant $29.27$
Ramified primes $2, 19, 47$
Class number $1$
Class group Trivial
Galois Group $\GL(3,2)$ (as 7T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -8, -14, 31, 0, -12, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 12*x^5 + 31*x^3 - 14*x^2 - 8*x + 4)
gp: K = bnfinit(x^7 - 12*x^5 + 31*x^3 - 14*x^2 - 8*x + 4, 1)

Normalized defining polynomial

\(x^{7} \) \(\mathstrut -\mathstrut 12 x^{5} \) \(\mathstrut +\mathstrut 31 x^{3} \) \(\mathstrut -\mathstrut 14 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 4 \)

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

Invariants

Degree:  $7$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[7, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(18424261696=2^{6}\cdot 19^{4}\cdot 47^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $29.27$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 19, 47$
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 \(a\))

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

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:  $6$
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:  \( \frac{7}{2} a^{6} + 2 a^{5} - 41 a^{4} - 23 a^{3} + \frac{193}{2} a^{2} + 3 a - 27 \),  \( 5 a^{6} + 3 a^{5} - 58 a^{4} - 35 a^{3} + 132 a^{2} + 11 a - 31 \),  \( a^{2} + a - 1 \),  \( a^{2} + a - 3 \),  \( \frac{1}{2} a^{6} + a^{5} - 7 a^{4} - 9 a^{3} + \frac{41}{2} a^{2} + 3 a - 5 \),  \( \frac{7}{2} a^{6} + 2 a^{5} - 41 a^{4} - 24 a^{3} + \frac{191}{2} a^{2} + 9 a - 23 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 2113.84591487 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$\PSL(2,7)$ (as 7T5):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 168
The 6 conjugacy class representatives for $\GL(3,2)$
Character table for $\GL(3,2)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 8 sibling: 8.8.58769636260854016.1
Degree 14 siblings: Deg 14, Deg 14
Degree 21 sibling: Deg 21
Degree 24 sibling: data not computed
Degree 28 sibling: data not computed
Degree 42 siblings: data not computed
Arithmetically equvalently sibling: 7.7.18424261696.2

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.2e4_19e2_47e2.42t37.1c1$3$ $ 2^{4} \cdot 19^{2} \cdot 47^{2}$ $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ $\GL(3,2)$ (as 7T5) $0$ $3$
3.2e4_19e2_47e2.42t37.1c2$3$ $ 2^{4} \cdot 19^{2} \cdot 47^{2}$ $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ $\GL(3,2)$ (as 7T5) $0$ $3$
* 6.2e6_19e4_47e2.7t5.1c1$6$ $ 2^{6} \cdot 19^{4} \cdot 47^{2}$ $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ $\GL(3,2)$ (as 7T5) $1$ $6$
7.2e8_19e6_47e4.8t37.1c1$7$ $ 2^{8} \cdot 19^{6} \cdot 47^{4}$ $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ $\GL(3,2)$ (as 7T5) $1$ $7$
8.2e10_19e6_47e4.21t14.1c1$8$ $ 2^{10} \cdot 19^{6} \cdot 47^{4}$ $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ $\GL(3,2)$ (as 7T5) $1$ $8$

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 *.