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Properties

Label 20.20.1214806479...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{36}\cdot 5^{13}\cdot 3469^{4}$
Root discriminant $50.61$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T755

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 0, -3375, 0, 18925, 0, -45450, 0, 57035, 0, -40265, 0, 16421, 0, -3908, 0, 529, 0, -37, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 37*x^18 + 529*x^16 - 3908*x^14 + 16421*x^12 - 40265*x^10 + 57035*x^8 - 45450*x^6 + 18925*x^4 - 3375*x^2 + 125)
 
gp: K = bnfinit(x^20 - 37*x^18 + 529*x^16 - 3908*x^14 + 16421*x^12 - 40265*x^10 + 57035*x^8 - 45450*x^6 + 18925*x^4 - 3375*x^2 + 125, 1)
 

Normalized defining polynomial

\( x^{20} - 37 x^{18} + 529 x^{16} - 3908 x^{14} + 16421 x^{12} - 40265 x^{10} + 57035 x^{8} - 45450 x^{6} + 18925 x^{4} - 3375 x^{2} + 125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12148064793879306567680000000000000=2^{36}\cdot 5^{13}\cdot 3469^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.61$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} - \frac{1}{5} a^{5}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{5}$, $\frac{1}{25} a^{16} - \frac{2}{25} a^{14} - \frac{1}{25} a^{12} + \frac{2}{25} a^{10} + \frac{1}{25} a^{8}$, $\frac{1}{25} a^{17} - \frac{2}{25} a^{15} - \frac{1}{25} a^{13} + \frac{2}{25} a^{11} + \frac{1}{25} a^{9}$, $\frac{1}{75925} a^{18} - \frac{88}{15185} a^{16} + \frac{948}{15185} a^{14} - \frac{771}{15185} a^{12} - \frac{636}{15185} a^{10} - \frac{858}{75925} a^{8} - \frac{2652}{15185} a^{6} - \frac{6321}{15185} a^{4} + \frac{621}{3037} a^{2} - \frac{1364}{3037}$, $\frac{1}{75925} a^{19} - \frac{88}{15185} a^{17} + \frac{948}{15185} a^{15} - \frac{771}{15185} a^{13} - \frac{636}{15185} a^{11} - \frac{858}{75925} a^{9} - \frac{2652}{15185} a^{7} - \frac{6321}{15185} a^{5} + \frac{621}{3037} a^{3} - \frac{1364}{3037} a$

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:  $19$
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:  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:  \( 55441443074.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T755:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n755 are not computed
Character table for t20n755 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
3469Data not computed