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Properties

Label 20.0.32635629350...9237.1
Degree $20$
Signature $[0, 10]$
Discriminant $13^{15}\cdot 41^{16}$
Root discriminant $133.56$
Ramified primes $13, 41$
Class number $1421461$ (GRH)
Class group $[1421461]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4782969, -5019165, 12269070, -22614228, 32100489, -7998318, -5545938, 5280269, -2500979, 62094, 882024, -475614, 6036, 63644, -11039, -4296, 1263, 135, -57, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 57*x^18 + 135*x^17 + 1263*x^16 - 4296*x^15 - 11039*x^14 + 63644*x^13 + 6036*x^12 - 475614*x^11 + 882024*x^10 + 62094*x^9 - 2500979*x^8 + 5280269*x^7 - 5545938*x^6 - 7998318*x^5 + 32100489*x^4 - 22614228*x^3 + 12269070*x^2 - 5019165*x + 4782969)
 
gp: K = bnfinit(x^20 - x^19 - 57*x^18 + 135*x^17 + 1263*x^16 - 4296*x^15 - 11039*x^14 + 63644*x^13 + 6036*x^12 - 475614*x^11 + 882024*x^10 + 62094*x^9 - 2500979*x^8 + 5280269*x^7 - 5545938*x^6 - 7998318*x^5 + 32100489*x^4 - 22614228*x^3 + 12269070*x^2 - 5019165*x + 4782969, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 57 x^{18} + 135 x^{17} + 1263 x^{16} - 4296 x^{15} - 11039 x^{14} + 63644 x^{13} + 6036 x^{12} - 475614 x^{11} + 882024 x^{10} + 62094 x^{9} - 2500979 x^{8} + 5280269 x^{7} - 5545938 x^{6} - 7998318 x^{5} + 32100489 x^{4} - 22614228 x^{3} + 12269070 x^{2} - 5019165 x + 4782969 \)

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:  \(3263562935079536383073550022036789161199237=13^{15}\cdot 41^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.56$
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:  $13, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(533=13\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{533}(1,·)$, $\chi_{533}(324,·)$, $\chi_{533}(385,·)$, $\chi_{533}(467,·)$, $\chi_{533}(493,·)$, $\chi_{533}(18,·)$, $\chi_{533}(83,·)$, $\chi_{533}(469,·)$, $\chi_{533}(346,·)$, $\chi_{533}(411,·)$, $\chi_{533}(92,·)$, $\chi_{533}(428,·)$, $\chi_{533}(365,·)$, $\chi_{533}(174,·)$, $\chi_{533}(242,·)$, $\chi_{533}(51,·)$, $\chi_{533}(502,·)$, $\chi_{533}(57,·)$, $\chi_{533}(508,·)$, $\chi_{533}(447,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} - \frac{1}{27} a^{8} + \frac{1}{27} a^{6} - \frac{10}{81} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{243} a^{12} - \frac{1}{243} a^{10} - \frac{1}{81} a^{8} - \frac{4}{243} a^{6} - \frac{11}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{11} - \frac{13}{243} a^{7} - \frac{2}{243} a^{5} - \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{729} a^{14} + \frac{1}{729} a^{12} + \frac{1}{243} a^{11} + \frac{4}{729} a^{10} - \frac{1}{243} a^{9} - \frac{37}{729} a^{8} - \frac{1}{81} a^{7} + \frac{8}{729} a^{6} - \frac{4}{243} a^{5} - \frac{94}{729} a^{4} - \frac{11}{243} a^{3} + \frac{13}{81} a^{2} + \frac{2}{27} a$, $\frac{1}{2187} a^{15} + \frac{4}{2187} a^{13} + \frac{1}{729} a^{12} + \frac{1}{2187} a^{11} - \frac{4}{729} a^{10} + \frac{8}{2187} a^{9} + \frac{2}{243} a^{8} - \frac{4}{2187} a^{7} - \frac{13}{729} a^{6} - \frac{289}{2187} a^{5} - \frac{62}{729} a^{4} - \frac{23}{243} a^{3} + \frac{8}{81} a^{2} + \frac{2}{9} a$, $\frac{1}{2187} a^{16} + \frac{1}{2187} a^{14} + \frac{1}{729} a^{13} - \frac{2}{2187} a^{12} + \frac{2}{729} a^{11} - \frac{4}{2187} a^{10} + \frac{107}{2187} a^{8} - \frac{31}{729} a^{7} - \frac{70}{2187} a^{6} - \frac{86}{729} a^{5} + \frac{106}{729} a^{4} + \frac{2}{243} a^{3} - \frac{13}{81} a^{2} + \frac{4}{27} a$, $\frac{1}{6561} a^{17} + \frac{1}{2187} a^{14} + \frac{4}{2187} a^{13} + \frac{1}{2187} a^{12} - \frac{23}{6561} a^{11} + \frac{13}{2187} a^{10} - \frac{4}{729} a^{9} + \frac{98}{2187} a^{8} + \frac{35}{2187} a^{7} - \frac{46}{2187} a^{6} + \frac{895}{6561} a^{5} - \frac{265}{2187} a^{4} + \frac{110}{729} a^{3} + \frac{22}{243} a^{2} - \frac{8}{27} a$, $\frac{1}{59049} a^{18} + \frac{1}{19683} a^{17} - \frac{11}{19683} a^{14} + \frac{4}{2187} a^{13} - \frac{104}{59049} a^{12} - \frac{119}{19683} a^{11} + \frac{28}{6561} a^{10} - \frac{49}{19683} a^{8} - \frac{95}{2187} a^{7} - \frac{2399}{59049} a^{6} - \frac{1160}{19683} a^{5} - \frac{56}{729} a^{4} - \frac{172}{2187} a^{3} - \frac{157}{729} a^{2} - \frac{4}{27} a$, $\frac{1}{553741165762884521219804496917354293948790139} a^{19} - \frac{183455183998146558508236814499208043912}{553741165762884521219804496917354293948790139} a^{18} - \frac{7500943391429346798558023560671457875859}{184580388587628173739934832305784764649596713} a^{17} - \frac{3636752227538803089235677670603290512402}{20508932065292019304437203589531640516621857} a^{16} - \frac{40893794128975946432803965520701763792997}{184580388587628173739934832305784764649596713} a^{15} - \frac{55579250850355766706072594812712827156908}{184580388587628173739934832305784764649596713} a^{14} + \frac{407702953617032204201484808896183629996236}{553741165762884521219804496917354293948790139} a^{13} + \frac{1052127312018352348217419603890645623583791}{553741165762884521219804496917354293948790139} a^{12} - \frac{415651838332447328391689152362969599098888}{184580388587628173739934832305784764649596713} a^{11} - \frac{224216847485833365959957814895715974729270}{61526796195876057913311610768594921549865571} a^{10} + \frac{692454450689322805613815946366801220934286}{184580388587628173739934832305784764649596713} a^{9} - \frac{754731271717849371690285775090175901124649}{184580388587628173739934832305784764649596713} a^{8} + \frac{6540945770996649005402314023354601168178857}{553741165762884521219804496917354293948790139} a^{7} + \frac{21816530289337077774342350238760075276959092}{553741165762884521219804496917354293948790139} a^{6} + \frac{379050149424239583432871563756903863345363}{184580388587628173739934832305784764649596713} a^{5} + \frac{2151612946122467699342667548287175475949153}{20508932065292019304437203589531640516621857} a^{4} - \frac{431345010075409778793771516576295372923026}{20508932065292019304437203589531640516621857} a^{3} + \frac{660537057859294665040765007919972800676220}{6836310688430673101479067863177213505540619} a^{2} - \frac{5196876756192068665059015546696045665287}{84398897388033001252827998310829796364699} a + \frac{143049521713355575851805055433836903280}{9377655265337000139203110923425532929411}$

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

Class group and class number

$C_{1421461}$, which has order $1421461$ (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:  \( 12581669852.706907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 5.5.2825761.1, 10.10.2964746843096023453.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ $20$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
13Data not computed
41Data not computed