# Properties

 Label 7.3.35331136.2 Degree $7$ Signature $[3, 2]$ Discriminant $2^{6}\cdot 743^{2}$ Root discriminant $11.98$ Ramified primes $2, 743$ Class number $1$ Class group Trivial Galois group $\GL(3,2)$ (as 7T5)

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 8, 4, -6, 5, 1, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 + x^5 + 5*x^4 - 6*x^3 + 4*x^2 + 8*x - 8)

gp: K = bnfinit(x^7 - x^6 + x^5 + 5*x^4 - 6*x^3 + 4*x^2 + 8*x - 8, 1)

## Normalizeddefining polynomial

$$x^{7} - x^{6} + x^{5} + 5 x^{4} - 6 x^{3} + 4 x^{2} + 8 x - 8$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $7$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[3, 2]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$35331136=2^{6}\cdot 743^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $11.98$ 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: $2, 743$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

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

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: $$\frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{7}{4} a^{3} - a + 3$$,  $$\frac{1}{2} a^{4} + \frac{1}{2} a^{2} + a - 1$$,  $$\frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{2} a^{3} + \frac{3}{2} a^{2} - 2 a + 1$$,  $$\frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{3}{4} a^{3} + \frac{1}{2} a^{2} + a + 1$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$30.7244086365$$ 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: 8.0.78018073190656.1 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: 7.3.35331136.1

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

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$$[\ ]_{3}^{2} 2.4.4.5x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
743Data not computed

## 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_743e2.42t37.1c1$3$ $2^{4} \cdot 743^{2}$ $x^{7} - x^{6} + x^{5} + 5 x^{4} - 6 x^{3} + 4 x^{2} + 8 x - 8$ $\GL(3,2)$ (as 7T5) $0$ $-1$
3.2e4_743e2.42t37.1c2$3$ $2^{4} \cdot 743^{2}$ $x^{7} - x^{6} + x^{5} + 5 x^{4} - 6 x^{3} + 4 x^{2} + 8 x - 8$ $\GL(3,2)$ (as 7T5) $0$ $-1$
* 6.2e6_743e2.7t5.1c1$6$ $2^{6} \cdot 743^{2}$ $x^{7} - x^{6} + x^{5} + 5 x^{4} - 6 x^{3} + 4 x^{2} + 8 x - 8$ $\GL(3,2)$ (as 7T5) $1$ $2$
7.2e8_743e4.8t37.1c1$7$ $2^{8} \cdot 743^{4}$ $x^{7} - x^{6} + x^{5} + 5 x^{4} - 6 x^{3} + 4 x^{2} + 8 x - 8$ $\GL(3,2)$ (as 7T5) $1$ $-1$
8.2e10_743e4.21t14.1c1$8$ $2^{10} \cdot 743^{4}$ $x^{7} - x^{6} + x^{5} + 5 x^{4} - 6 x^{3} + 4 x^{2} + 8 x - 8$ $\GL(3,2)$ (as 7T5) $1$ $0$

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 *.