Properties

Label 7.3.28291761.2
Degree $7$
Signature $[3, 2]$
Discriminant $3^{6}\cdot 197^{2}$
Root discriminant $11.60$
Ramified primes $3, 197$
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, 5, -6, 3, -6, 3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 2*x^6 + 3*x^5 - 6*x^4 + 3*x^3 - 6*x^2 + 5*x - 1)
gp: K = bnfinit(x^7 - 2*x^6 + 3*x^5 - 6*x^4 + 3*x^3 - 6*x^2 + 5*x - 1, 1)

Normalized defining polynomial

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

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:  \(28291761=3^{6}\cdot 197^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.60$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 197$
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 \(a\))

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

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:  \( a \),  \( \frac{2}{3} a^{6} - a^{5} + a^{4} - 2 a^{3} - a^{2} - 2 a + \frac{4}{3} \),  \( 3 a^{6} - 5 a^{5} + 7 a^{4} - 15 a^{3} + 4 a^{2} - 16 a + 9 \),  \( \frac{8}{3} a^{6} - 4 a^{5} + 6 a^{4} - 13 a^{3} + 2 a^{2} - 16 a + \frac{16}{3} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 20.6177124051 \)
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: 8.0.9881774573841.1
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: 7.3.28291761.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }$ R ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
$197$$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.2.2$x^{4} - 197 x^{2} + 116427$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.3e4_197e2.42t37.2c1$3$ $ 3^{4} \cdot 197^{2}$ $x^{7} - 2 x^{6} + 3 x^{5} - 6 x^{4} + 3 x^{3} - 6 x^{2} + 5 x - 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
3.3e4_197e2.42t37.2c2$3$ $ 3^{4} \cdot 197^{2}$ $x^{7} - 2 x^{6} + 3 x^{5} - 6 x^{4} + 3 x^{3} - 6 x^{2} + 5 x - 1$ $\GL(3,2)$ (as 7T5) $0$ $-1$
* 6.3e6_197e2.7t5.2c1$6$ $ 3^{6} \cdot 197^{2}$ $x^{7} - 2 x^{6} + 3 x^{5} - 6 x^{4} + 3 x^{3} - 6 x^{2} + 5 x - 1$ $\GL(3,2)$ (as 7T5) $1$ $2$
7.3e8_197e4.8t37.2c1$7$ $ 3^{8} \cdot 197^{4}$ $x^{7} - 2 x^{6} + 3 x^{5} - 6 x^{4} + 3 x^{3} - 6 x^{2} + 5 x - 1$ $\GL(3,2)$ (as 7T5) $1$ $-1$
8.3e10_197e4.21t14.2c1$8$ $ 3^{10} \cdot 197^{4}$ $x^{7} - 2 x^{6} + 3 x^{5} - 6 x^{4} + 3 x^{3} - 6 x^{2} + 5 x - 1$ $\GL(3,2)$ (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 *.