# Properties

 Label 5.5.161121.1 Degree $5$ Signature $[5, 0]$ Discriminant $3\cdot 43\cdot 1249$ Root discriminant $11.00$ Ramified primes $3, 43, 1249$ 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, 5, 3, -6, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 6*x^3 + 3*x^2 + 5*x + 1)

gp: K = bnfinit(x^5 - x^4 - 6*x^3 + 3*x^2 + 5*x + 1, 1)

## Normalizeddefining polynomial

$$x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1$$

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: $$161121=3\cdot 43\cdot 1249$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $11.00$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 43, 1249$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ 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 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: $$a$$,  $$a^{4} - 2 a^{3} - 5 a^{2} + 8 a + 2$$,  $$a^{4} - a^{3} - 5 a^{2} + 4 a + 2$$,  $$4 a^{4} - 6 a^{3} - 21 a^{2} + 23 a + 8$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$11.2054995686$$ 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 ${\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }$

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$$0Trivial[\ ] 3.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2} 43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2} 43.2.0.1x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
1249Data 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_43_1249.2t1.1c1$1$ $3 \cdot 43 \cdot 1249$ $x^{2} - x - 40280$ $C_2$ (as 2T1) $1$ $1$
4.3e3_43e3_1249e3.10t12.1c1$4$ $3^{3} \cdot 43^{3} \cdot 1249^{3}$ $x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1$ $S_5$ (as 5T5) $1$ $4$
* 4.3_43_1249.5t5.1c1$4$ $3 \cdot 43 \cdot 1249$ $x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1$ $S_5$ (as 5T5) $1$ $4$
5.3e2_43e2_1249e2.10t13.1c1$5$ $3^{2} \cdot 43^{2} \cdot 1249^{2}$ $x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1$ $S_5$ (as 5T5) $1$ $5$
5.3e3_43e3_1249e3.6t14.1c1$5$ $3^{3} \cdot 43^{3} \cdot 1249^{3}$ $x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1$ $S_5$ (as 5T5) $1$ $5$
6.3e3_43e3_1249e3.20t35.1c1$6$ $3^{3} \cdot 43^{3} \cdot 1249^{3}$ $x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1$ $S_5$ (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 *.