# Properties

 Label 7.3.172764736.2 Degree $7$ Signature $[3, 2]$ Discriminant $172764736$ Root discriminant $15.02$ Ramified primes $2, 31, 53$ 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 - x^6 - x^5 + x^4 - 7*x^3 + 7*x^2 - 9*x - 1)

gp: K = bnfinit(x^7 - x^6 - x^5 + x^4 - 7*x^3 + 7*x^2 - 9*x - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 7, -7, 1, -1, -1, 1]);

$$x^{7} - x^{6} - x^{5} + x^{4} - 7 x^{3} + 7 x^{2} - 9 x - 1$$

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: $$172764736$$$$\medspace = 2^{6}\cdot 31^{2}\cdot 53^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $15.02$ 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: $2, 31, 53$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{61} a^{6} - \frac{5}{61} a^{5} + \frac{19}{61} a^{4} - \frac{14}{61} a^{3} - \frac{12}{61} a^{2} - \frac{6}{61} a + \frac{15}{61}$

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: $$a$$,  $$\frac{3}{61} a^{6} - \frac{15}{61} a^{5} - \frac{4}{61} a^{4} + \frac{19}{61} a^{3} - \frac{36}{61} a^{2} + \frac{43}{61} a - \frac{77}{61}$$,  $$\frac{15}{61} a^{6} - \frac{14}{61} a^{5} - \frac{20}{61} a^{4} + \frac{34}{61} a^{3} - \frac{119}{61} a^{2} + \frac{32}{61} a - \frac{19}{61}$$,  $$\frac{1}{61} a^{6} - \frac{5}{61} a^{5} + \frac{19}{61} a^{4} - \frac{14}{61} a^{3} - \frac{73}{61} a^{2} + \frac{116}{61} a - \frac{107}{61}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$50.6828270139$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{2}\cdot 50.6828270139 \cdot 1}{2\sqrt{172764736}}\approx 0.608909863124$

## 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: 8.0.466369593830464.1 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: 7.3.172764736.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ R ${\href{/LocalNumberField/59.7.0.1}{7} }$

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

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3} 31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2} 31.4.2.2x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$53$$\Q_{53}$$x + 2$$1$$1$$0Trivial[\ ] 53.2.0.1x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$