Properties

Label 7.3.30558784.1
Degree $7$
Signature $[3, 2]$
Discriminant $2^{6}\cdot 691^{2}$
Root discriminant $11.73$
Ramified primes $2, 691$
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![1, -2, -1, 3, -5, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 3*x^6 + 4*x^5 - 5*x^4 + 3*x^3 - x^2 - 2*x + 1)
 
gp: K = bnfinit(x^7 - 3*x^6 + 4*x^5 - 5*x^4 + 3*x^3 - x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{7} - 3 x^{6} + 4 x^{5} - 5 x^{4} + 3 x^{3} - x^{2} - 2 x + 1 \)

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:  \(30558784=2^{6}\cdot 691^{2}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.73$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 691$
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 \(a\))

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

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:  \( a^{6} - 3 a^{5} + 4 a^{4} - 5 a^{3} + 3 a^{2} - a - 2 \),  \( a^{6} - 3 a^{5} + 3 a^{4} - 3 a^{3} + 2 a^{2} + a - 2 \),  \( 2 a^{6} - 5 a^{5} + 5 a^{4} - 6 a^{3} + 2 a^{2} - a - 4 \),  \( 2 a^{6} - 5 a^{5} + 5 a^{4} - 7 a^{3} + 2 a^{2} - 4 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18.3119129313 \)
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: 8.0.933839279558656.2
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: 7.3.30558784.2

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
691Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.2e6_691e2.42t37.1c1$3$ $ 2^{6} \cdot 691^{2}$ $x^{7} - 3 x^{6} + 4 x^{5} - 5 x^{4} + 3 x^{3} - x^{2} - 2 x + 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
3.2e6_691e2.42t37.1c2$3$ $ 2^{6} \cdot 691^{2}$ $x^{7} - 3 x^{6} + 4 x^{5} - 5 x^{4} + 3 x^{3} - x^{2} - 2 x + 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
* 6.2e6_691e2.7t5.1c1$6$ $ 2^{6} \cdot 691^{2}$ $x^{7} - 3 x^{6} + 4 x^{5} - 5 x^{4} + 3 x^{3} - x^{2} - 2 x + 1$ $\GL(3,2)$ (as 7T5) $1$ $2$
7.2e12_691e4.8t37.1c1$7$ $ 2^{12} \cdot 691^{4}$ $x^{7} - 3 x^{6} + 4 x^{5} - 5 x^{4} + 3 x^{3} - x^{2} - 2 x + 1$ $\GL(3,2)$ (as 7T5) $1$ $-1$
8.2e12_691e4.21t14.1c1$8$ $ 2^{12} \cdot 691^{4}$ $x^{7} - 3 x^{6} + 4 x^{5} - 5 x^{4} + 3 x^{3} - x^{2} - 2 x + 1$ $\GL(3,2)$ (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 *.