# Properties

 Label 5.5.180769.1 Degree $5$ Signature $[5, 0]$ Discriminant $41\cdot 4409$ Root discriminant $11.26$ Ramified primes $41, 4409$ 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, 0, 7, -4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 2*x^4 - 4*x^3 + 7*x^2 - 1)
gp: K = bnfinit(x^5 - 2*x^4 - 4*x^3 + 7*x^2 - 1, 1)

## Normalizeddefining polynomial

$$x^{5}$$ $$\mathstrut -\mathstrut 2 x^{4}$$ $$\mathstrut -\mathstrut 4 x^{3}$$ $$\mathstrut +\mathstrut 7 x^{2}$$ $$\mathstrut -\mathstrut 1$$

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

## Invariants

 Degree: $5$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[5, 0]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$180769=41\cdot 4409$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.26$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $41, 4409$ 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: $$a^{4} - 2 a^{3} - 4 a^{2} + 7 a$$,  $$a - 1$$,  $$a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 2$$,  $$a^{3} - a^{2} - 4 a + 2$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$9.64320519613$$ 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 ${\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ R ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2} 41.3.0.1x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
4409Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.41_4409.2t1.1c1$1$ $41 \cdot 4409$ $x^{2} - x - 45192$ $C_2$ (as 2T1) $1$ $1$
4.41e3_4409e3.10t12.1c1$4$ $41^{3} \cdot 4409^{3}$ $x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} - 1$ $S_5$ (as 5T5) $1$ $4$
* 4.41_4409.5t5.1c1$4$ $41 \cdot 4409$ $x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} - 1$ $S_5$ (as 5T5) $1$ $4$
5.41e2_4409e2.10t13.1c1$5$ $41^{2} \cdot 4409^{2}$ $x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} - 1$ $S_5$ (as 5T5) $1$ $5$
5.41e3_4409e3.6t14.1c1$5$ $41^{3} \cdot 4409^{3}$ $x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} - 1$ $S_5$ (as 5T5) $1$ $5$
6.41e3_4409e3.20t35.1c1$6$ $41^{3} \cdot 4409^{3}$ $x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} - 1$ $S_5$ (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 *.