# Properties

 Label 7.3.15429184.1 Degree $7$ Signature $[3, 2]$ Discriminant $15429184$ Root discriminant $10.64$ Ramified primes $2, 491$ Class number $1$ Class group trivial Galois group $\GL(3,2)$ (as 7T5)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 7, 1, -5, 7, -3, 1]);

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $7$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[3, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$15429184$$$$\medspace = 2^{6}\cdot 491^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $10.64$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 491$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $4$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{3} - a^{2} + 2 a + 1$$,  $$a$$,  $$a^{4} - a^{3} + 2 a^{2} + 3 a$$,  $$a^{5} - 2 a^{4} + 4 a^{3} - a^{2} + a$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$18.7523072477$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{2}\cdot 18.7523072477 \cdot 1}{2\sqrt{15429184}}\approx 0.753881279603$

## Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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: 8.0.14878732431616.5 Degree 14 siblings: Deg 14, Deg 14 Degree 21 sibling: Deg 21 Degree 24 sibling: Deg 24 Degree 28 sibling: Deg 28 Degree 42 siblings: Deg 42, Deg 42, Deg 42 Arithmetically equvalently sibling: 7.3.15429184.2

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.6.6.7x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
491Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
3.3857296.42t37.a.a$3$ $2^{4} \cdot 491^{2}$ $x^{7} - 3 x^{6} + 7 x^{5} - 5 x^{4} + x^{3} + 7 x^{2} - 3 x - 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
3.3857296.42t37.a.b$3$ $2^{4} \cdot 491^{2}$ $x^{7} - 3 x^{6} + 7 x^{5} - 5 x^{4} + x^{3} + 7 x^{2} - 3 x - 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
* 6.15429184.7t5.a.a$6$ $2^{6} \cdot 491^{2}$ $x^{7} - 3 x^{6} + 7 x^{5} - 5 x^{4} + x^{3} + 7 x^{2} - 3 x - 1$ $\GL(3,2)$ (as 7T5) $1$ $2$
7.148...616.8t37.a.a$7$ $2^{8} \cdot 491^{4}$ $x^{7} - 3 x^{6} + 7 x^{5} - 5 x^{4} + x^{3} + 7 x^{2} - 3 x - 1$ $\GL(3,2)$ (as 7T5) $1$ $-1$
8.595...464.21t14.a.a$8$ $2^{10} \cdot 491^{4}$ $x^{7} - 3 x^{6} + 7 x^{5} - 5 x^{4} + x^{3} + 7 x^{2} - 3 x - 1$ $\GL(3,2)$ (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 *.