Properties

Label 5.1.162409.1
Degree $5$
Signature $[1, 2]$
Discriminant $13^{2}\cdot 31^{2}$
Root discriminant $11.02$
Ramified primes $13, 31$
Class number $1$
Class group Trivial
Galois group $A_5$ (as 5T4)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^5 + 2*x^3 - 5*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^5 + 2*x^3 - 5*x^2 + 2*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, -5, 2, 0, 1]);
 

Normalized defining polynomial

\( x^{5} + 2 x^{3} - 5 x^{2} + 2 x - 1 \)

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:  \(162409=13^{2}\cdot 31^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.02$
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:  $13, 31$
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:  \( a^{4} + 2 a^{2} - 5 a + 2 \),  \( a - 1 \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2.5379106436 \)
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: 6.2.156075049.3
Degree 10 sibling: 10.2.25347992633041.1
Degree 12 sibling: Deg 12
Degree 15 sibling: Deg 15
Degree 20 sibling: Deg 20
Degree 30 sibling: Deg 30

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }$ R ${\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
3.162409.12t33.a.a$3$ $ 13^{2} \cdot 31^{2}$ $x^{5} + 2 x^{3} - 5 x^{2} + 2 x - 1$ $A_5$ (as 5T4) $1$ $-1$
3.162409.12t33.a.b$3$ $ 13^{2} \cdot 31^{2}$ $x^{5} + 2 x^{3} - 5 x^{2} + 2 x - 1$ $A_5$ (as 5T4) $1$ $-1$
* 4.162409.5t4.a.a$4$ $ 13^{2} \cdot 31^{2}$ $x^{5} + 2 x^{3} - 5 x^{2} + 2 x - 1$ $A_5$ (as 5T4) $1$ $0$
5.156075049.6t12.a.a$5$ $ 13^{2} \cdot 31^{4}$ $x^{5} + 2 x^{3} - 5 x^{2} + 2 x - 1$ $A_5$ (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 *.