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Properties

Label 20.0.18530919141...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{14}\cdot 6029^{7}$
Root discriminant $129.84$
Ramified primes $2, 5, 6029$
Class number $13415216$ (GRH)
Class group $[2, 6707608]$ (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![219147162389, 0, 240992815830, 0, 107663590980, 0, 25954091375, 0, 3757874380, 0, 342629406, 0, 19971630, 0, 736170, 0, 16430, 0, 200, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 200*x^18 + 16430*x^16 + 736170*x^14 + 19971630*x^12 + 342629406*x^10 + 3757874380*x^8 + 25954091375*x^6 + 107663590980*x^4 + 240992815830*x^2 + 219147162389)
 
gp: K = bnfinit(x^20 + 200*x^18 + 16430*x^16 + 736170*x^14 + 19971630*x^12 + 342629406*x^10 + 3757874380*x^8 + 25954091375*x^6 + 107663590980*x^4 + 240992815830*x^2 + 219147162389, 1)
 

Normalized defining polynomial

\( x^{20} + 200 x^{18} + 16430 x^{16} + 736170 x^{14} + 19971630 x^{12} + 342629406 x^{10} + 3757874380 x^{8} + 25954091375 x^{6} + 107663590980 x^{4} + 240992815830 x^{2} + 219147162389 \)

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:  \(1853091914135141046229493177600000000000000=2^{20}\cdot 5^{14}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.84$
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:  $2, 5, 6029$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{10} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{2}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{10} - \frac{2}{5} a^{4} - \frac{2}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{11} - \frac{2}{5} a^{5} - \frac{2}{5} a$, $\frac{1}{150725} a^{16} + \frac{6229}{150725} a^{14} + \frac{4372}{150725} a^{12} - \frac{29513}{150725} a^{10} - \frac{1786}{6029} a^{8} + \frac{49568}{150725} a^{6} - \frac{43523}{150725} a^{4} - \frac{6}{25} a^{2} + \frac{9}{25}$, $\frac{1}{150725} a^{17} + \frac{6229}{150725} a^{15} + \frac{4372}{150725} a^{13} - \frac{29513}{150725} a^{11} - \frac{1786}{6029} a^{9} + \frac{49568}{150725} a^{7} - \frac{43523}{150725} a^{5} - \frac{6}{25} a^{3} + \frac{9}{25} a$, $\frac{1}{24208495805919174659076521636220126175} a^{18} - \frac{5905738556812347109685473426566}{4841699161183834931815304327244025235} a^{16} - \frac{525492712158113288796727733718357144}{24208495805919174659076521636220126175} a^{14} - \frac{567417268020990762413093671235281931}{24208495805919174659076521636220126175} a^{12} + \frac{700485605916885402865748113736295979}{1862191985070705743005886279709240475} a^{10} + \frac{9089011849804155466323154685519960593}{24208495805919174659076521636220126175} a^{8} - \frac{754210193953107470083973786522811372}{4841699161183834931815304327244025235} a^{6} + \frac{1119836693460257784962799989732892}{4015341815544729583525712661506075} a^{4} + \frac{1897778180649062633587934083070473}{4015341815544729583525712661506075} a^{2} - \frac{266348354987827568578274134259}{666004613624934414252067119175}$, $\frac{1}{24208495805919174659076521636220126175} a^{19} - \frac{5905738556812347109685473426566}{4841699161183834931815304327244025235} a^{17} - \frac{525492712158113288796727733718357144}{24208495805919174659076521636220126175} a^{15} - \frac{567417268020990762413093671235281931}{24208495805919174659076521636220126175} a^{13} + \frac{700485605916885402865748113736295979}{1862191985070705743005886279709240475} a^{11} + \frac{9089011849804155466323154685519960593}{24208495805919174659076521636220126175} a^{9} - \frac{754210193953107470083973786522811372}{4841699161183834931815304327244025235} a^{7} + \frac{1119836693460257784962799989732892}{4015341815544729583525712661506075} a^{5} + \frac{1897778180649062633587934083070473}{4015341815544729583525712661506075} a^{3} - \frac{266348354987827568578274134259}{666004613624934414252067119175} a$

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

Class group and class number

$C_{2}\times C_{6707608}$, which has order $13415216$ (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:  \( -1 \) (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:  \( 341439.528105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$5.4.2.1$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}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed