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Properties

Label 5.5.650892809785201.1
Degree $5$
Signature $[5, 0]$
Discriminant $5051^{4}$
Root discriminant $917.70$
Ramified prime $5051$
Class number $1451$
Class group $[1451]$
Galois group $C_5$ (as 5T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1920311, -558681, 55965, -2020, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 2020*x^3 + 55965*x^2 - 558681*x + 1920311)
 
gp: K = bnfinit(x^5 - x^4 - 2020*x^3 + 55965*x^2 - 558681*x + 1920311, 1)
 

Normalized defining polynomial

\( x^{5} - x^{4} - 2020 x^{3} + 55965 x^{2} - 558681 x + 1920311 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $5$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(650892809785201=5051^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $917.70$
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:  $5051$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(5051\)
Dirichlet character group:    $\lbrace$$\chi_{5051}(1,·)$, $\chi_{5051}(3002,·)$, $\chi_{5051}(1020,·)$, $\chi_{5051}(1134,·)$, $\chi_{5051}(4945,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{665} a^{4} - \frac{13}{133} a^{3} + \frac{29}{133} a^{2} + \frac{27}{133} a - \frac{4}{35}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{1451}$, which has order $1451$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $4$
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:  \( \frac{72}{665} a^{4} + \frac{128}{133} a^{3} - \frac{27837}{133} a^{2} + \frac{530220}{133} a - \frac{731998}{35} \),  \( \frac{8}{665} a^{4} + \frac{29}{133} a^{3} - \frac{2694}{133} a^{2} + \frac{37589}{133} a - \frac{39582}{35} \),  \( \frac{17}{665} a^{4} + \frac{45}{133} a^{3} - \frac{6157}{133} a^{2} + \frac{104598}{133} a - \frac{134013}{35} \),  \( \frac{8}{665} a^{4} + \frac{29}{133} a^{3} - \frac{2694}{133} a^{2} + \frac{37722}{133} a - \frac{40247}{35} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1436.69320773 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5$ (as 5T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 5
The 5 conjugacy class representatives for $C_5$
Character table for $C_5$

Intermediate fields

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }$

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
5051Data not computed