Properties

Label 5.5.179024.1
Degree $5$
Signature $[5, 0]$
Discriminant $2^{4}\cdot 67\cdot 167$
Root discriminant $11.24$
Ramified primes $2, 67, 167$
Class number $1$
Class group Trivial
Galois Group $S_5$ (as 5T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 6, 0, -8, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 8*x^3 + 6*x - 2)
gp: K = bnfinit(x^5 - 8*x^3 + 6*x - 2, 1)

Normalized defining polynomial

\(x^{5} \) \(\mathstrut -\mathstrut 8 x^{3} \) \(\mathstrut +\mathstrut 6 x \) \(\mathstrut -\mathstrut 2 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $5$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[5, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(179024=2^{4}\cdot 67\cdot 167\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.24$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 67, 167$
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 \(a\))

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

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial Abelian group, 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:  \( a^{4} + a^{3} - 8 a^{2} - 7 a + 5 \),  \( 2 a^{4} + a^{3} - 15 a^{2} - 8 a + 7 \),  \( 3 a^{4} + 2 a^{3} - 23 a^{2} - 15 a + 11 \),  \( 5 a^{4} + 2 a^{3} - 39 a^{2} - 16 a + 23 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 15.1926139168 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_5$ (as 5T5):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.1.1$x^{2} - 167$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.67_167.2t1.1c1$1$ $ 67 \cdot 167 $ $x^{2} - x - 2797$ $C_2$ (as 2T1) $1$ $1$
4.2e4_67e3_167e3.10t12.1c1$4$ $ 2^{4} \cdot 67^{3} \cdot 167^{3}$ $x^{5} - 8 x^{3} + 6 x - 2$ $S_5$ (as 5T5) $1$ $4$
* 4.2e4_67_167.5t5.1c1$4$ $ 2^{4} \cdot 67 \cdot 167 $ $x^{5} - 8 x^{3} + 6 x - 2$ $S_5$ (as 5T5) $1$ $4$
5.2e4_67e2_167e2.10t13.1c1$5$ $ 2^{4} \cdot 67^{2} \cdot 167^{2}$ $x^{5} - 8 x^{3} + 6 x - 2$ $S_5$ (as 5T5) $1$ $5$
5.2e4_67e3_167e3.6t14.1c1$5$ $ 2^{4} \cdot 67^{3} \cdot 167^{3}$ $x^{5} - 8 x^{3} + 6 x - 2$ $S_5$ (as 5T5) $1$ $5$
6.2e4_67e3_167e3.20t35.1c1$6$ $ 2^{4} \cdot 67^{3} \cdot 167^{3}$ $x^{5} - 8 x^{3} + 6 x - 2$ $S_5$ (as 5T5) $1$ $6$

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 *.