# Properties

 Label 5.1.129600.1 Degree $5$ Signature $[1, 2]$ Discriminant $2^{6}\cdot 3^{4}\cdot 5^{2}$ Root discriminant $10.53$ Ramified primes $2, 3, 5$ Class number $1$ Class group Trivial Galois group $A_5$ (as 5T4)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -6, -4, -2, 0, 1]);

## Normalizeddefining polynomial

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $5$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[1, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$129600=2^{6}\cdot 3^{4}\cdot 5^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $10.53$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 3, 5$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $2$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a + 1$$,  $$8 a^{4} - 7 a^{3} - 8 a^{2} - 19 a - 27$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$13.4888953709$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$A_5$ (as 5T4):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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: 6.2.262440000.2 Degree 10 sibling: 10.2.34012224000000.1 Degree 12 sibling: Deg 12 Degree 15 sibling: Deg 15 Degree 20 sibling: Deg 20 Degree 30 sibling: Deg 30

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R R ${\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.4.6.7x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 3.3.4.4x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 5.3.2.1x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
3.129600.12t33.a.a$3$ $2^{6} \cdot 3^{4} \cdot 5^{2}$ $x^{5} - 2 x^{3} - 4 x^{2} - 6 x - 4$ $A_5$ (as 5T4) $1$ $-1$
3.129600.12t33.a.b$3$ $2^{6} \cdot 3^{4} \cdot 5^{2}$ $x^{5} - 2 x^{3} - 4 x^{2} - 6 x - 4$ $A_5$ (as 5T4) $1$ $-1$
* 4.129600.5t4.a.a$4$ $2^{6} \cdot 3^{4} \cdot 5^{2}$ $x^{5} - 2 x^{3} - 4 x^{2} - 6 x - 4$ $A_5$ (as 5T4) $1$ $0$
5.262440000.6t12.a.a$5$ $2^{6} \cdot 3^{8} \cdot 5^{4}$ $x^{5} - 2 x^{3} - 4 x^{2} - 6 x - 4$ $A_5$ (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 *.