↑

Properties

Label 20.0.19331544372...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 11^{16}\cdot 13^{10}$
Root discriminant $82.09$
Ramified primes $5, 11, 13$
Class number $331922$ (GRH)
Class group $[331922]$ (GRH)
Galois group $C_{20}$ (as 20T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4751163121, -1149290321, 3808215652, -973042478, 1487095911, -374838878, 364805340, -85243529, 61958107, -12647278, 7455206, -1272388, 651922, -86676, 42001, -3824, 1959, -97, 61, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 61*x^18 - 97*x^17 + 1959*x^16 - 3824*x^15 + 42001*x^14 - 86676*x^13 + 651922*x^12 - 1272388*x^11 + 7455206*x^10 - 12647278*x^9 + 61958107*x^8 - 85243529*x^7 + 364805340*x^6 - 374838878*x^5 + 1487095911*x^4 - 973042478*x^3 + 3808215652*x^2 - 1149290321*x + 4751163121)
 
gp: K = bnfinit(x^20 - x^19 + 61*x^18 - 97*x^17 + 1959*x^16 - 3824*x^15 + 42001*x^14 - 86676*x^13 + 651922*x^12 - 1272388*x^11 + 7455206*x^10 - 12647278*x^9 + 61958107*x^8 - 85243529*x^7 + 364805340*x^6 - 374838878*x^5 + 1487095911*x^4 - 973042478*x^3 + 3808215652*x^2 - 1149290321*x + 4751163121, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 61 x^{18} - 97 x^{17} + 1959 x^{16} - 3824 x^{15} + 42001 x^{14} - 86676 x^{13} + 651922 x^{12} - 1272388 x^{11} + 7455206 x^{10} - 12647278 x^{9} + 61958107 x^{8} - 85243529 x^{7} + 364805340 x^{6} - 374838878 x^{5} + 1487095911 x^{4} - 973042478 x^{3} + 3808215652 x^{2} - 1149290321 x + 4751163121 \)

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:  \(193315443721344440894830801055908203125=5^{15}\cdot 11^{16}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.09$
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:  $5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(715=5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{715}(1,·)$, $\chi_{715}(196,·)$, $\chi_{715}(456,·)$, $\chi_{715}(521,·)$, $\chi_{715}(586,·)$, $\chi_{715}(12,·)$, $\chi_{715}(14,·)$, $\chi_{715}(207,·)$, $\chi_{715}(144,·)$, $\chi_{715}(339,·)$, $\chi_{715}(532,·)$, $\chi_{715}(597,·)$, $\chi_{715}(599,·)$, $\chi_{715}(664,·)$, $\chi_{715}(38,·)$, $\chi_{715}(103,·)$, $\chi_{715}(168,·)$, $\chi_{715}(298,·)$, $\chi_{715}(493,·)$, $\chi_{715}(467,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} + \frac{2}{19} a^{14} + \frac{7}{19} a^{13} - \frac{6}{19} a^{12} - \frac{9}{19} a^{11} - \frac{5}{19} a^{10} - \frac{8}{19} a^{9} + \frac{9}{19} a^{8} - \frac{1}{19} a^{7} - \frac{9}{19} a^{6} - \frac{8}{19} a^{5} - \frac{9}{19} a^{3} + \frac{5}{19} a^{2} - \frac{5}{19} a + \frac{4}{19}$, $\frac{1}{19} a^{16} + \frac{3}{19} a^{14} - \frac{1}{19} a^{13} + \frac{3}{19} a^{12} - \frac{6}{19} a^{11} + \frac{2}{19} a^{10} + \frac{6}{19} a^{9} - \frac{7}{19} a^{7} - \frac{9}{19} a^{6} - \frac{3}{19} a^{5} - \frac{9}{19} a^{4} + \frac{4}{19} a^{3} + \frac{4}{19} a^{2} - \frac{5}{19} a - \frac{8}{19}$, $\frac{1}{1691} a^{17} - \frac{42}{1691} a^{16} - \frac{43}{1691} a^{15} + \frac{370}{1691} a^{14} - \frac{524}{1691} a^{13} - \frac{464}{1691} a^{12} + \frac{421}{1691} a^{11} - \frac{8}{89} a^{10} + \frac{135}{1691} a^{9} - \frac{592}{1691} a^{8} + \frac{293}{1691} a^{7} - \frac{503}{1691} a^{6} - \frac{674}{1691} a^{5} + \frac{6}{19} a^{4} - \frac{320}{1691} a^{3} + \frac{34}{1691} a^{2} - \frac{841}{1691} a - \frac{30}{89}$, $\frac{1}{221521} a^{18} - \frac{65}{221521} a^{17} + \frac{2169}{221521} a^{16} - \frac{2913}{221521} a^{15} - \frac{312}{221521} a^{14} + \frac{51460}{221521} a^{13} + \frac{37081}{221521} a^{12} + \frac{14373}{221521} a^{11} - \frac{18174}{221521} a^{10} - \frac{51668}{221521} a^{9} + \frac{41410}{221521} a^{8} + \frac{22128}{221521} a^{7} - \frac{29511}{221521} a^{6} + \frac{4226}{11659} a^{5} + \frac{55661}{221521} a^{4} + \frac{81264}{221521} a^{3} - \frac{61965}{221521} a^{2} - \frac{91231}{221521} a + \frac{3983}{11659}$, $\frac{1}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{19} + \frac{15354946000840923503615014267975087892011171070204381321766089226}{11545817302317106616107430149778630196466048626825895373908234519498901} a^{18} - \frac{30245197562062407743416299783700069814260001374071457350990933503574}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{17} - \frac{3736754340802770620125049244276231931800911406276738246229096992097237}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{16} + \frac{1541380884033942254751361271525643603531331349330277760510740054828987}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{15} + \frac{90577592796558478465485817319676254407700038576946747168114564311948405}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{14} + \frac{92552595250621035282091393558523057281907690877071730466538518490241148}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{13} - \frac{80121101368106371768891297939699119214169680440564834314235107437206379}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{12} + \frac{85203900152034446348007774593387757300354429678094111354976871214845374}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{11} - \frac{34842325884765264905273566012966581354323325970541170653017386454385967}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{10} - \frac{83971622104989613259482479761780361969344832048144548541875554757312941}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{9} + \frac{45010992567885122745736497195548274293995161660435400211212752356186188}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{8} + \frac{83915118188492809552556138330365938985064902425208058384898869944076906}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{7} - \frac{74039142280254043239903746641326348057940304940544157939734826562202526}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{6} + \frac{49061248426894181798323818895840903370828254187391312130383478684811267}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{5} + \frac{64426160859921893901009896921511615180913736685290977305063707492615395}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{4} - \frac{3294746237975463310957352344442142993977142859399021782079764382000306}{11545817302317106616107430149778630196466048626825895373908234519498901} a^{3} - \frac{67532261098369811359360765498967862603664562329316283550571819159502875}{219370528744025025706041172845793973732854923909692012104256455870479119} a^{2} + \frac{60243094183889085850988726766255048787354674482225322085034485739399092}{219370528744025025706041172845793973732854923909692012104256455870479119} a - \frac{84566366818552174475948714655731097870893972548706332139515239193452834}{219370528744025025706041172845793973732854923909692012104256455870479119}$

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

Class group and class number

$C_{331922}$, which has order $331922$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), 4.0.21125.1, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 $20$ $20$ R $20$ R R $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
13Data not computed