# Properties

 Label 5.1.25000000.4 Degree $5$ Signature $[1, 2]$ Discriminant $2^{6}\cdot 5^{8}$ Root discriminant $30.17$ Ramified primes $2, 5$ Class number $3$ Class group $[3]$ Galois group $A_5$ (as 5T4)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 10*x^3 - 20*x^2 + 110*x + 116)

gp: K = bnfinit(x^5 - 10*x^3 - 20*x^2 + 110*x + 116, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![116, 110, -20, -10, 0, 1]);

## Normalizeddefining polynomial

$$x^{5} - 10 x^{3} - 20 x^{2} + 110 x + 116$$

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: $$25000000=2^{6}\cdot 5^{8}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $30.17$ 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: $2, 5$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{382} a^{4} + \frac{50}{191} a^{3} + \frac{29}{191} a^{2} + \frac{25}{191} a + \frac{72}{191}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{3}$, which has order $3$

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: $$\frac{17}{191} a^{4} - \frac{19}{191} a^{3} - \frac{160}{191} a^{2} - \frac{105}{191} a + \frac{1875}{191}$$,  $$\frac{789}{191} a^{4} - \frac{1129}{191} a^{3} - \frac{8291}{191} a^{2} - \frac{9064}{191} a + \frac{103875}{191}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$47.9274830176$$ 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: 6.2.25000000.1 Degree 10 sibling: 10.2.625000000000000.5 Degree 12 sibling: 12.0.40000000000000000.5 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 R ${\href{/LocalNumberField/3.5.0.1}{5} }$ R ${\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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}$
$5$5.5.8.6$x^{5} + 5 x^{4} + 5$$5$$1$$8$$D_{5}$$[2]^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
3.40000.12t33.e.a$3$ $2^{6} \cdot 5^{4}$ $x^{5} - 10 x^{3} - 20 x^{2} + 110 x + 116$ $A_5$ (as 5T4) $1$ $-1$
3.40000.12t33.e.b$3$ $2^{6} \cdot 5^{4}$ $x^{5} - 10 x^{3} - 20 x^{2} + 110 x + 116$ $A_5$ (as 5T4) $1$ $-1$
* 4.25000000.5t4.c.a$4$ $2^{6} \cdot 5^{8}$ $x^{5} - 10 x^{3} - 20 x^{2} + 110 x + 116$ $A_5$ (as 5T4) $1$ $0$
5.25000000.6t12.c.a$5$ $2^{6} \cdot 5^{8}$ $x^{5} - 10 x^{3} - 20 x^{2} + 110 x + 116$ $A_5$ (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 *.