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Properties

Label 7.3.2834497600.2
Degree $7$
Signature $[3, 2]$
Discriminant $2^{6}\cdot 5^{2}\cdot 11^{6}$
Root discriminant $22.41$
Ramified primes $2, 5, 11$
Class number $1$
Class group Trivial
Galois group $\GL(3,2)$ (as 7T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -12, -12, 8, -1, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 9*x^5 - x^4 + 8*x^3 - 12*x^2 - 12*x - 2)
 
gp: K = bnfinit(x^7 - x^6 - 9*x^5 - x^4 + 8*x^3 - 12*x^2 - 12*x - 2, 1)
 

Normalized defining polynomial

\( x^{7} - x^{6} - 9 x^{5} - x^{4} + 8 x^{3} - 12 x^{2} - 12 x - 2 \)

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:  \(2834497600=2^{6}\cdot 5^{2}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.41$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{10} a^{6} - \frac{2}{5} a^{5} + \frac{3}{10} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$

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:  \( \frac{3}{5} a^{6} - \frac{7}{5} a^{5} - \frac{16}{5} a^{4} + 3 a^{3} - \frac{6}{5} a^{2} - \frac{23}{5} a - \frac{7}{5} \),  \( \frac{37}{10} a^{6} - \frac{24}{5} a^{5} - \frac{319}{10} a^{4} + 6 a^{3} + \frac{138}{5} a^{2} - \frac{266}{5} a - \frac{144}{5} \),  \( \frac{11}{10} a^{6} - \frac{7}{5} a^{5} - \frac{97}{10} a^{4} + a^{3} + \frac{49}{5} a^{2} - \frac{43}{5} a - \frac{12}{5} \),  \( 4 a^{6} - 7 a^{5} - 30 a^{4} + 16 a^{3} + 18 a^{2} - 55 a - 13 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 559.719690981 \)
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: 8.0.283449760000.1
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: 7.3.2834497600.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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$$[\ ]_{3}^{2}$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$

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_5e2_11e3.42t37.1c1$3$ $ 2^{4} \cdot 5^{2} \cdot 11^{3}$ $x^{7} - x^{6} - 9 x^{5} - x^{4} + 8 x^{3} - 12 x^{2} - 12 x - 2$ $\GL(3,2)$ (as 7T5) $0$ $-1$
3.2e4_5e2_11e3.42t37.1c2$3$ $ 2^{4} \cdot 5^{2} \cdot 11^{3}$ $x^{7} - x^{6} - 9 x^{5} - x^{4} + 8 x^{3} - 12 x^{2} - 12 x - 2$ $\GL(3,2)$ (as 7T5) $0$ $-1$
* 6.2e6_5e2_11e6.7t5.1c1$6$ $ 2^{6} \cdot 5^{2} \cdot 11^{6}$ $x^{7} - x^{6} - 9 x^{5} - x^{4} + 8 x^{3} - 12 x^{2} - 12 x - 2$ $\GL(3,2)$ (as 7T5) $1$ $2$
7.2e8_5e4_11e6.8t37.1c1$7$ $ 2^{8} \cdot 5^{4} \cdot 11^{6}$ $x^{7} - x^{6} - 9 x^{5} - x^{4} + 8 x^{3} - 12 x^{2} - 12 x - 2$ $\GL(3,2)$ (as 7T5) $1$ $-1$
8.2e10_5e4_11e6.21t14.1c1$8$ $ 2^{10} \cdot 5^{4} \cdot 11^{6}$ $x^{7} - x^{6} - 9 x^{5} - x^{4} + 8 x^{3} - 12 x^{2} - 12 x - 2$ $\GL(3,2)$ (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 *.