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Properties

Label 20.0.52808765008...0784.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 31^{8}$
Root discriminant $13.68$
Ramified primes $2, 3, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T538

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 5, 0, 10, 0, 12, 0, 13, 0, 15, 0, 17, 0, 15, 0, 10, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 + 10*x^16 + 15*x^14 + 17*x^12 + 15*x^10 + 13*x^8 + 12*x^6 + 10*x^4 + 5*x^2 + 1)
 
gp: K = bnfinit(x^20 + 4*x^18 + 10*x^16 + 15*x^14 + 17*x^12 + 15*x^10 + 13*x^8 + 12*x^6 + 10*x^4 + 5*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} + 10 x^{16} + 15 x^{14} + 17 x^{12} + 15 x^{10} + 13 x^{8} + 12 x^{6} + 10 x^{4} + 5 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(52808765008619617910784=2^{20}\cdot 3^{10}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.68$
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:  $2, 3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{16} + \frac{2}{5} a^{14} - \frac{1}{5} a^{12} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{17} + \frac{2}{5} a^{15} - \frac{1}{5} a^{13} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{23}{5} a^{18} + \frac{78}{5} a^{16} + \frac{181}{5} a^{14} + \frac{232}{5} a^{12} + 49 a^{10} + 39 a^{8} + \frac{179}{5} a^{6} + \frac{169}{5} a^{4} + \frac{128}{5} a^{2} + \frac{41}{5} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2471.2827235 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T538:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 25600
The 88 conjugacy class representatives for t20n538 are not computed
Character table for t20n538 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.224415603.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. 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
2Data not computed
3Data not computed
$31$31.10.0.1$x^{10} - x + 11$$1$$10$$0$$C_{10}$$[\ ]^{10}$
31.10.8.2$x^{10} - 31 x^{5} + 11532$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$