# Properties

 Label 7.3.15864289.1 Degree $7$ Signature $[3, 2]$ Discriminant $7^{2}\cdot 569^{2}$ Root discriminant $10.68$ Ramified primes $7, 569$ Class number $1$ Class group Trivial Galois group $\GL(3,2)$ (as 7T5)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -5, 4, -4, 1, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 + x^5 - 4*x^4 + 4*x^3 - 5*x^2 + 2*x + 1)

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

## Normalizeddefining polynomial

$$x^{7} - x^{6} + x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$$

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: $$15864289=7^{2}\cdot 569^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $10.68$ 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: $7, 569$ 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 $$a$$)

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

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: $$\frac{4}{7} a^{6} - \frac{3}{7} a^{5} + \frac{5}{7} a^{4} - \frac{13}{7} a^{3} + \frac{11}{7} a^{2} - \frac{19}{7} a + \frac{5}{7}$$,  $$a$$,  $$\frac{4}{7} a^{6} - \frac{3}{7} a^{5} + \frac{5}{7} a^{4} - \frac{13}{7} a^{3} + \frac{11}{7} a^{2} - \frac{19}{7} a - \frac{2}{7}$$,  $$a - 1$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$8.28712673096$$ 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: 8.0.251675665475521.3 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: 7.3.15864289.2

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] 7.4.2.1x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
569Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.7e2_569e2.42t37.1c1$3$ $7^{2} \cdot 569^{2}$ $x^{7} - x^{6} + x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
3.7e2_569e2.42t37.1c2$3$ $7^{2} \cdot 569^{2}$ $x^{7} - x^{6} + x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
* 6.7e2_569e2.7t5.1c1$6$ $7^{2} \cdot 569^{2}$ $x^{7} - x^{6} + x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$ $\GL(3,2)$ (as 7T5) $1$ $2$
7.7e4_569e4.8t37.1c1$7$ $7^{4} \cdot 569^{4}$ $x^{7} - x^{6} + x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$ $\GL(3,2)$ (as 7T5) $1$ $-1$
8.7e4_569e4.21t14.1c1$8$ $7^{4} \cdot 569^{4}$ $x^{7} - x^{6} + x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$ $\GL(3,2)$ (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 *.