# Properties

 Label 5.1.5025.1 Degree $5$ Signature $[1, 2]$ Discriminant $5025$ Root discriminant $5.50$ Ramified primes $3, 5, 67$ Class number $1$ Class group trivial Galois group $S_5$ (as 5T5)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

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

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

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: $$5025$$$$\medspace = 3\cdot 5^{2}\cdot 67$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $5.50$ 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: $3, 5, 67$ 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 $$a$$)

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

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: $$a$$,  $$a^{2} - a$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.708326486922$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{2}\cdot 0.708326486922 \cdot 1}{2\sqrt{5025}}\approx 0.3944801781318$

## Galois group

$S_5$ (as 5T5):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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: 6.2.5075375625.4 Degree 10 siblings: 10.2.205050250625625.1, 10.2.126884390625.1 Degree 12 sibling: Deg 12 Degree 15 sibling: Deg 15 Degree 20 siblings: Deg 20, Deg 20, Deg 20 Degree 24 sibling: Deg 24 Degree 30 siblings: Deg 30, Deg 30, Deg 30 Degree 40 sibling: Deg 40

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.5.0.1}{5} }$ R R ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.5.0.1}{5} }$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.5.0.1}{5} }$ ${\href{/padicField/41.5.0.1}{5} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2} 3.3.0.1x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
$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}$
$67$$\Q_{67}$$x + 4$$1$$1$$0Trivial[\ ] 67.2.1.1x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.201.2t1.a.a$1$ $3 \cdot 67$ $$\Q(\sqrt{201})$$ $C_2$ (as 2T1) $1$ $1$
4.203015025.10t12.a.a$4$ $3^{3} \cdot 5^{2} \cdot 67^{3}$ 5.1.5025.1 $S_5$ (as 5T5) $1$ $0$
* 4.5025.5t5.a.a$4$ $3 \cdot 5^{2} \cdot 67$ 5.1.5025.1 $S_5$ (as 5T5) $1$ $0$
5.25250625.10t13.a.a$5$ $3^{2} \cdot 5^{4} \cdot 67^{2}$ 5.1.5025.1 $S_5$ (as 5T5) $1$ $1$
5.5075375625.6t14.a.a$5$ $3^{3} \cdot 5^{4} \cdot 67^{3}$ 5.1.5025.1 $S_5$ (as 5T5) $1$ $1$
6.5075375625.20t30.a.a$6$ $3^{3} \cdot 5^{4} \cdot 67^{3}$ 5.1.5025.1 $S_5$ (as 5T5) $1$ $-2$

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 *.