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Properties

Label 20.2.12888021157...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{20}\cdot 5^{15}\cdot 3319^{5}$
Root discriminant $50.76$
Ramified primes $2, 5, 3319$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10371875, 0, -25518750, 0, -21549375, 0, -7156250, 0, -686500, 0, 109000, 0, 24025, 0, 25, 0, -245, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 245*x^16 + 25*x^14 + 24025*x^12 + 109000*x^10 - 686500*x^8 - 7156250*x^6 - 21549375*x^4 - 25518750*x^2 - 10371875)
 
gp: K = bnfinit(x^20 - 5*x^18 - 245*x^16 + 25*x^14 + 24025*x^12 + 109000*x^10 - 686500*x^8 - 7156250*x^6 - 21549375*x^4 - 25518750*x^2 - 10371875, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 245 x^{16} + 25 x^{14} + 24025 x^{12} + 109000 x^{10} - 686500 x^{8} - 7156250 x^{6} - 21549375 x^{4} - 25518750 x^{2} - 10371875 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-12888021157371923168000000000000000=-\,2^{20}\cdot 5^{15}\cdot 3319^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.76$
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:  $2, 5, 3319$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{1875} a^{16} + \frac{1}{375} a^{14} - \frac{1}{375} a^{12} + \frac{1}{75} a^{10} + \frac{1}{15} a^{6} - \frac{1}{15} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{1875} a^{17} + \frac{1}{375} a^{15} - \frac{1}{375} a^{13} + \frac{1}{75} a^{11} + \frac{1}{15} a^{7} - \frac{1}{15} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{912021248598342523125} a^{18} + \frac{8366412832884986}{182404249719668504625} a^{16} + \frac{73579291016189636}{36480849943933700925} a^{14} - \frac{117346546975521503}{60801416573222834875} a^{12} + \frac{259933065451257464}{36480849943933700925} a^{10} - \frac{240696117343231843}{36480849943933700925} a^{8} - \frac{445321446993160334}{7296169988786740185} a^{6} - \frac{6923128468595914}{486411332585782679} a^{4} - \frac{615016621055085812}{1459233997757348037} a^{2} - \frac{335044697714663981}{1459233997757348037}$, $\frac{1}{912021248598342523125} a^{19} + \frac{8366412832884986}{182404249719668504625} a^{17} + \frac{73579291016189636}{36480849943933700925} a^{15} - \frac{117346546975521503}{60801416573222834875} a^{13} + \frac{259933065451257464}{36480849943933700925} a^{11} - \frac{240696117343231843}{36480849943933700925} a^{9} - \frac{445321446993160334}{7296169988786740185} a^{7} - \frac{6923128468595914}{486411332585782679} a^{5} - \frac{615016621055085812}{1459233997757348037} a^{3} - \frac{335044697714663981}{1459233997757348037} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  \( 317982652.694 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

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 t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.34424253125.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
3319Data not computed