# Properties

 Label 7.3.624200256.2 Degree $7$ Signature $[3, 2]$ Discriminant $2^{6}\cdot 3^{4}\cdot 347^{2}$ Root discriminant $18.05$ Ramified primes $2, 3, 347$ 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![3, -3, 9, -9, 1, 3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 3*x^6 + 3*x^5 + x^4 - 9*x^3 + 9*x^2 - 3*x + 3)
gp: K = bnfinit(x^7 - 3*x^6 + 3*x^5 + x^4 - 9*x^3 + 9*x^2 - 3*x + 3, 1)

## Normalizeddefining polynomial

$$x^{7}$$ $$\mathstrut -\mathstrut 3 x^{6}$$ $$\mathstrut +\mathstrut 3 x^{5}$$ $$\mathstrut +\mathstrut x^{4}$$ $$\mathstrut -\mathstrut 9 x^{3}$$ $$\mathstrut +\mathstrut 9 x^{2}$$ $$\mathstrut -\mathstrut 3 x$$ $$\mathstrut +\mathstrut 3$$

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: $$624200256=2^{6}\cdot 3^{4}\cdot 347^{2}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $18.05$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3, 347$ 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 $$a$$)

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

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: $$\frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{2}{3} a^{4} + 2 a^{3} - 2 a^{2} - 2 a + 3$$,  $$a^{2} - 2$$,  $$\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$$,  $$a^{3} - 2 a^{2} + a - 1$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$210.641544965$$ 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: 8.0.676433957622336.2 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: 7.3.624200256.1

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3} 3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2} 3.4.3.1x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
347Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.2e3_3e2_347e2.42t37.1c1$3$ $2^{3} \cdot 3^{2} \cdot 347^{2}$ $x^{7} - 3 x^{6} + 3 x^{5} + x^{4} - 9 x^{3} + 9 x^{2} - 3 x + 3$ $\GL(3,2)$ (as 7T5) $0$ $-1$
3.2e3_3e2_347e2.42t37.1c2$3$ $2^{3} \cdot 3^{2} \cdot 347^{2}$ $x^{7} - 3 x^{6} + 3 x^{5} + x^{4} - 9 x^{3} + 9 x^{2} - 3 x + 3$ $\GL(3,2)$ (as 7T5) $0$ $-1$
* 6.2e6_3e4_347e2.7t5.1c1$6$ $2^{6} \cdot 3^{4} \cdot 347^{2}$ $x^{7} - 3 x^{6} + 3 x^{5} + x^{4} - 9 x^{3} + 9 x^{2} - 3 x + 3$ $\GL(3,2)$ (as 7T5) $1$ $2$
7.2e6_3e6_347e4.8t37.1c1$7$ $2^{6} \cdot 3^{6} \cdot 347^{4}$ $x^{7} - 3 x^{6} + 3 x^{5} + x^{4} - 9 x^{3} + 9 x^{2} - 3 x + 3$ $\GL(3,2)$ (as 7T5) $1$ $-1$
8.2e6_3e6_347e4.21t14.1c1$8$ $2^{6} \cdot 3^{6} \cdot 347^{4}$ $x^{7} - 3 x^{6} + 3 x^{5} + x^{4} - 9 x^{3} + 9 x^{2} - 3 x + 3$ $\GL(3,2)$ (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 *.