# Properties

 Label 5.3.12447.1 Degree $5$ Signature $[3, 1]$ Discriminant $-\,3^{3}\cdot 461$ Root discriminant $6.59$ Ramified primes $3, 461$ Class number $1$ Class group Trivial Galois Group $S_5$ (as 5T5)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

## Normalizeddefining polynomial

$$x^{5}$$ $$\mathstrut +\mathstrut x^{3}$$ $$\mathstrut -\mathstrut x^{2}$$ $$\mathstrut -\mathstrut 3 x$$ $$\mathstrut -\mathstrut 1$$

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

## Invariants

 Degree: $5$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[3, 1]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$-12447=-\,3^{3}\cdot 461$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $6.59$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 461$ 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: $3$ 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: $$a$$,  $$a^{4} + a^{2} - a - 2$$,  $$a^{4} + a^{2} - 2 a - 2$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$1.47353341675$$ 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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.5.0.1}{5} }$ R ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 3.3.3.2x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
461Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_461.2t1.1c1$1$ $3 \cdot 461$ $x^{2} - x + 346$ $C_2$ (as 2T1) $1$ $-1$
4.3e5_461e3.10t12.1c1$4$ $3^{5} \cdot 461^{3}$ $x^{5} + x^{3} - x^{2} - 3 x - 1$ $S_5$ (as 5T5) $1$ $-2$
* 4.3e3_461.5t5.1c1$4$ $3^{3} \cdot 461$ $x^{5} + x^{3} - x^{2} - 3 x - 1$ $S_5$ (as 5T5) $1$ $2$
5.3e6_461e2.10t13.1c1$5$ $3^{6} \cdot 461^{2}$ $x^{5} + x^{3} - x^{2} - 3 x - 1$ $S_5$ (as 5T5) $1$ $1$
5.3e7_461e3.6t14.1c1$5$ $3^{7} \cdot 461^{3}$ $x^{5} + x^{3} - x^{2} - 3 x - 1$ $S_5$ (as 5T5) $1$ $-1$
6.3e7_461e3.20t35.1c1$6$ $3^{7} \cdot 461^{3}$ $x^{5} + x^{3} - x^{2} - 3 x - 1$ $S_5$ (as 5T5) $1$ $0$

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 *.