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Properties

Label 20.0.13795520063...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 401^{11}$
Root discriminant $180.71$
Ramified primes $2, 5, 401$
Class number $98564736$ (GRH)
Class group $[2, 4, 4, 3080148]$ (GRH)
Galois group 20T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![201503753125, 0, 243211512500, 0, 120600750000, 0, 32309321875, 0, 5152499125, 0, 506522125, 0, 30700600, 0, 1117300, 0, 23145, 0, 245, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 245*x^18 + 23145*x^16 + 1117300*x^14 + 30700600*x^12 + 506522125*x^10 + 5152499125*x^8 + 32309321875*x^6 + 120600750000*x^4 + 243211512500*x^2 + 201503753125)
 
gp: K = bnfinit(x^20 + 245*x^18 + 23145*x^16 + 1117300*x^14 + 30700600*x^12 + 506522125*x^10 + 5152499125*x^8 + 32309321875*x^6 + 120600750000*x^4 + 243211512500*x^2 + 201503753125, 1)
 

Normalized defining polynomial

\( x^{20} + 245 x^{18} + 23145 x^{16} + 1117300 x^{14} + 30700600 x^{12} + 506522125 x^{10} + 5152499125 x^{8} + 32309321875 x^{6} + 120600750000 x^{4} + 243211512500 x^{2} + 201503753125 \)

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:  \(1379552006303010922834201740832000000000000000=2^{20}\cdot 5^{15}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $180.71$
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:  $2, 5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{150375} a^{14} - \frac{557}{150375} a^{12} + \frac{218}{30075} a^{10} - \frac{298}{30075} a^{8} - \frac{239}{6015} a^{6} - \frac{41}{6015} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{150375} a^{15} - \frac{557}{150375} a^{13} + \frac{218}{30075} a^{11} - \frac{298}{30075} a^{9} - \frac{239}{6015} a^{7} - \frac{41}{6015} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{361651875} a^{16} + \frac{1}{4822025} a^{14} + \frac{16544}{24110125} a^{12} - \frac{28609}{4822025} a^{10} - \frac{98}{192881} a^{8} + \frac{594}{964405} a^{6} + \frac{585}{14837} a^{4} - \frac{186}{481} a^{2} + \frac{199}{1443}$, $\frac{1}{361651875} a^{17} + \frac{1}{4822025} a^{15} + \frac{16544}{24110125} a^{13} - \frac{28609}{4822025} a^{11} - \frac{98}{192881} a^{9} + \frac{594}{964405} a^{7} + \frac{585}{14837} a^{5} - \frac{186}{481} a^{3} + \frac{199}{1443} a$, $\frac{1}{182971049925155271525793125} a^{18} + \frac{102528527806134406}{182971049925155271525793125} a^{16} - \frac{13016641051537388621}{36594209985031054305158625} a^{14} + \frac{107893613488366399687927}{36594209985031054305158625} a^{12} + \frac{67281085674674429702693}{7318841997006210861031725} a^{10} - \frac{126402816801406126463749}{7318841997006210861031725} a^{8} + \frac{341100432362142860021}{3650295260352224868345} a^{6} + \frac{44252548706420325536}{3650295260352224868345} a^{4} - \frac{367221549438256238}{1820596139826546069} a^{2} - \frac{27057814013811305}{1820596139826546069}$, $\frac{1}{182971049925155271525793125} a^{19} + \frac{102528527806134406}{182971049925155271525793125} a^{17} - \frac{13016641051537388621}{36594209985031054305158625} a^{15} + \frac{107893613488366399687927}{36594209985031054305158625} a^{13} + \frac{67281085674674429702693}{7318841997006210861031725} a^{11} - \frac{126402816801406126463749}{7318841997006210861031725} a^{9} + \frac{341100432362142860021}{3650295260352224868345} a^{7} + \frac{44252548706420325536}{3650295260352224868345} a^{5} - \frac{367221549438256238}{1820596139826546069} a^{3} - \frac{27057814013811305}{1820596139826546069} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{3080148}$, which has order $98564736$ (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:  \( 2526424.45141 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 28 conjugacy class representatives for t20n138
Character table for t20n138 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
401Data not computed