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Properties

Label 20.0.13795520063...0000.1
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 $104512512$ (GRH)
Class group $[2, 4, 4, 3266016]$ (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, 622098868750, 0, 578280596250, 0, 200996738750, 0, 30396100750, 0, 2353148375, 0, 102805450, 0, 2635925, 0, 39165, 0, 310, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 310*x^18 + 39165*x^16 + 2635925*x^14 + 102805450*x^12 + 2353148375*x^10 + 30396100750*x^8 + 200996738750*x^6 + 578280596250*x^4 + 622098868750*x^2 + 201503753125)
 
gp: K = bnfinit(x^20 + 310*x^18 + 39165*x^16 + 2635925*x^14 + 102805450*x^12 + 2353148375*x^10 + 30396100750*x^8 + 200996738750*x^6 + 578280596250*x^4 + 622098868750*x^2 + 201503753125, 1)
 

Normalized defining polynomial

\( x^{20} + 310 x^{18} + 39165 x^{16} + 2635925 x^{14} + 102805450 x^{12} + 2353148375 x^{10} + 30396100750 x^{8} + 200996738750 x^{6} + 578280596250 x^{4} + 622098868750 x^{2} + 201503753125 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{150375} a^{14} - \frac{91}{150375} a^{12} - \frac{187}{30075} a^{10} + \frac{271}{30075} a^{8} + \frac{364}{6015} a^{6} - \frac{394}{6015} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{150375} a^{15} - \frac{91}{150375} a^{13} - \frac{187}{30075} a^{11} + \frac{271}{30075} a^{9} + \frac{364}{6015} a^{7} - \frac{394}{6015} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{751875} a^{16} - \frac{53}{50125} a^{12} - \frac{154}{10025} a^{10} + \frac{1}{10025} a^{8} + \frac{177}{2005} a^{6} - \frac{11}{2005} a^{4} + \frac{1}{3}$, $\frac{1}{751875} a^{17} - \frac{53}{50125} a^{13} - \frac{154}{10025} a^{11} + \frac{1}{10025} a^{9} + \frac{177}{2005} a^{7} - \frac{11}{2005} a^{5} + \frac{1}{3} a$, $\frac{1}{29088309386297645588004817997881875} a^{18} + \frac{1291575925047991920822747416}{29088309386297645588004817997881875} a^{16} - \frac{18801918126461379154775103523}{5817661877259529117600963599576375} a^{14} - \frac{2994169127760719309609805064936}{1163532375451905823520192719915275} a^{12} + \frac{14070586982288422648093001410441}{1163532375451905823520192719915275} a^{10} - \frac{17794332535783871855774693614843}{1163532375451905823520192719915275} a^{8} - \frac{56710615204502850026283573857}{580315399227883203750719561055} a^{6} + \frac{29867148163557756934325762564}{580315399227883203750719561055} a^{4} + \frac{6743371733954357272567060}{96478038109373766209595937} a^{2} + \frac{27211480718336161926590999}{96478038109373766209595937}$, $\frac{1}{29088309386297645588004817997881875} a^{19} + \frac{1291575925047991920822747416}{29088309386297645588004817997881875} a^{17} - \frac{18801918126461379154775103523}{5817661877259529117600963599576375} a^{15} - \frac{2994169127760719309609805064936}{1163532375451905823520192719915275} a^{13} + \frac{14070586982288422648093001410441}{1163532375451905823520192719915275} a^{11} - \frac{17794332535783871855774693614843}{1163532375451905823520192719915275} a^{9} - \frac{56710615204502850026283573857}{580315399227883203750719561055} a^{7} + \frac{29867148163557756934325762564}{580315399227883203750719561055} a^{5} + \frac{6743371733954357272567060}{96478038109373766209595937} a^{3} + \frac{27211480718336161926590999}{96478038109373766209595937} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{3266016}$, which has order $104512512$ (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 $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
401Data not computed