# Properties

 Label 5.1.10673289.1 Degree $5$ Signature $[1, 2]$ Discriminant $3^{6}\cdot 11^{4}$ Root discriminant $25.45$ Ramified primes $3, 11$ Class number $5$ Class group $[5]$ Galois group $A_5$ (as 5T4)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 4, 13, -5, -2, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 2*x^4 - 5*x^3 + 13*x^2 + 4*x - 32)

gp: K = bnfinit(x^5 - 2*x^4 - 5*x^3 + 13*x^2 + 4*x - 32, 1)

## Normalizeddefining polynomial

$$x^{5} - 2 x^{4} - 5 x^{3} + 13 x^{2} + 4 x - 32$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $5$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[1, 2]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$10673289=3^{6}\cdot 11^{4}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $25.45$ 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: $3, 11$ 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 $$a$$)

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

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

$C_{5}$, which has order $5$

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $2$ 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: $$\frac{1}{2} a^{4} - a^{3} - \frac{1}{2} a^{2} + \frac{9}{2} a - 3$$,  $$2 a^{4} - 9 a^{3} + 11 a^{2} + 6 a - 21$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$45.812239359$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$A_5$ (as 5T4):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A non-solvable group of order 60 The 5 conjugacy class representatives for $A_5$ Character table for $A_5$

## Intermediate fields

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

## Sibling fields

 Degree 6 sibling: 6.2.864536409.2 Degree 10 sibling: 10.2.9227446944279201.2 Degree 12 sibling: Deg 12 Degree 15 sibling: Deg 15 Degree 20 sibling: Deg 20 Degree 30 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 ${\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2} 3.3.5.3x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.3e6_11e2.12t33.1c1$3$ $3^{6} \cdot 11^{2}$ $x^{5} - 2 x^{4} - 5 x^{3} + 13 x^{2} + 4 x - 32$ $A_5$ (as 5T4) $1$ $-1$
3.3e6_11e2.12t33.1c2$3$ $3^{6} \cdot 11^{2}$ $x^{5} - 2 x^{4} - 5 x^{3} + 13 x^{2} + 4 x - 32$ $A_5$ (as 5T4) $1$ $-1$
* 4.3e6_11e4.5t4.1c1$4$ $3^{6} \cdot 11^{4}$ $x^{5} - 2 x^{4} - 5 x^{3} + 13 x^{2} + 4 x - 32$ $A_5$ (as 5T4) $1$ $0$
5.3e10_11e4.6t12.1c1$5$ $3^{10} \cdot 11^{4}$ $x^{5} - 2 x^{4} - 5 x^{3} + 13 x^{2} + 4 x - 32$ $A_5$ (as 5T4) $1$ $1$

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 *.