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Properties

Label 20.0.87965623143...5625.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{35}\cdot 13^{15}$
Root discriminant $198.25$
Ramified primes $3, 5, 13$
Class number $44096104$ (GRH)
Class group $[2, 22048052]$ (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![6583472493451, -937131504260, 2128501694900, -312318598760, 401126058450, -71053686994, 48310631580, -7781677350, 3902735955, -511086650, 236068876, -24656430, 11380600, -860280, 420700, -17594, 10760, -150, 160, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 160*x^18 - 150*x^17 + 10760*x^16 - 17594*x^15 + 420700*x^14 - 860280*x^13 + 11380600*x^12 - 24656430*x^11 + 236068876*x^10 - 511086650*x^9 + 3902735955*x^8 - 7781677350*x^7 + 48310631580*x^6 - 71053686994*x^5 + 401126058450*x^4 - 312318598760*x^3 + 2128501694900*x^2 - 937131504260*x + 6583472493451)
 
gp: K = bnfinit(x^20 + 160*x^18 - 150*x^17 + 10760*x^16 - 17594*x^15 + 420700*x^14 - 860280*x^13 + 11380600*x^12 - 24656430*x^11 + 236068876*x^10 - 511086650*x^9 + 3902735955*x^8 - 7781677350*x^7 + 48310631580*x^6 - 71053686994*x^5 + 401126058450*x^4 - 312318598760*x^3 + 2128501694900*x^2 - 937131504260*x + 6583472493451, 1)
 

Normalized defining polynomial

\( x^{20} + 160 x^{18} - 150 x^{17} + 10760 x^{16} - 17594 x^{15} + 420700 x^{14} - 860280 x^{13} + 11380600 x^{12} - 24656430 x^{11} + 236068876 x^{10} - 511086650 x^{9} + 3902735955 x^{8} - 7781677350 x^{7} + 48310631580 x^{6} - 71053686994 x^{5} + 401126058450 x^{4} - 312318598760 x^{3} + 2128501694900 x^{2} - 937131504260 x + 6583472493451 \)

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:  \(8796562314350871340409503318369388580322265625=3^{10}\cdot 5^{35}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $198.25$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(975=3\cdot 5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{975}(64,·)$, $\chi_{975}(1,·)$, $\chi_{975}(259,·)$, $\chi_{975}(196,·)$, $\chi_{975}(454,·)$, $\chi_{975}(391,·)$, $\chi_{975}(649,·)$, $\chi_{975}(586,·)$, $\chi_{975}(844,·)$, $\chi_{975}(781,·)$, $\chi_{975}(83,·)$, $\chi_{975}(278,·)$, $\chi_{975}(473,·)$, $\chi_{975}(668,·)$, $\chi_{975}(863,·)$, $\chi_{975}(47,·)$, $\chi_{975}(242,·)$, $\chi_{975}(437,·)$, $\chi_{975}(632,·)$, $\chi_{975}(827,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{14} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{3}{7} a$, $\frac{1}{14} a^{8} - \frac{1}{2} a^{3} - \frac{1}{14} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{9} - \frac{1}{2} a^{4} - \frac{1}{14} a^{3} - \frac{1}{2} a$, $\frac{1}{28} a^{10} - \frac{1}{28} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{28} a^{11} - \frac{1}{28} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{196} a^{12} - \frac{1}{196} a^{11} + \frac{3}{196} a^{10} + \frac{1}{98} a^{9} + \frac{1}{49} a^{8} + \frac{41}{196} a^{6} + \frac{15}{196} a^{5} - \frac{31}{196} a^{4} + \frac{27}{98} a^{3} + \frac{31}{196} a^{2} + \frac{11}{28} a - \frac{1}{4}$, $\frac{1}{196} a^{13} + \frac{1}{98} a^{11} - \frac{1}{98} a^{10} + \frac{3}{98} a^{9} + \frac{1}{49} a^{8} - \frac{1}{196} a^{7} - \frac{3}{14} a^{6} - \frac{4}{49} a^{5} - \frac{17}{49} a^{4} - \frac{13}{196} a^{3} - \frac{22}{49} a^{2} + \frac{5}{14} a - \frac{1}{2}$, $\frac{1}{1372} a^{14} - \frac{1}{1372} a^{13} + \frac{1}{1372} a^{12} + \frac{9}{686} a^{11} + \frac{3}{343} a^{10} - \frac{1}{343} a^{9} + \frac{5}{1372} a^{8} - \frac{41}{1372} a^{7} + \frac{279}{1372} a^{6} + \frac{103}{686} a^{5} - \frac{509}{1372} a^{4} - \frac{423}{1372} a^{3} + \frac{309}{1372} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{2744} a^{15} + \frac{5}{2744} a^{12} + \frac{11}{686} a^{11} + \frac{15}{2744} a^{10} + \frac{71}{2744} a^{9} - \frac{23}{686} a^{8} + \frac{3}{196} a^{7} - \frac{89}{2744} a^{6} - \frac{513}{2744} a^{5} + \frac{139}{2744} a^{4} - \frac{121}{343} a^{3} - \frac{223}{2744} a^{2} - \frac{1}{7} a - \frac{23}{56}$, $\frac{1}{2744} a^{16} + \frac{5}{2744} a^{13} + \frac{1}{1372} a^{12} - \frac{41}{2744} a^{11} + \frac{43}{2744} a^{10} + \frac{5}{686} a^{9} + \frac{5}{196} a^{8} - \frac{89}{2744} a^{7} + \frac{509}{2744} a^{6} - \frac{393}{2744} a^{5} - \frac{142}{343} a^{4} + \frac{57}{2744} a^{3} - \frac{37}{196} a^{2} - \frac{19}{56} a - \frac{1}{2}$, $\frac{1}{19208} a^{17} + \frac{1}{19208} a^{15} - \frac{3}{19208} a^{14} - \frac{9}{9604} a^{13} + \frac{3}{4802} a^{12} + \frac{125}{19208} a^{11} + \frac{261}{19208} a^{10} - \frac{79}{19208} a^{9} - \frac{501}{19208} a^{8} + \frac{319}{19208} a^{7} + \frac{379}{9604} a^{6} + \frac{37}{392} a^{5} - \frac{4691}{9604} a^{4} + \frac{3427}{9604} a^{3} + \frac{363}{1372} a^{2} - \frac{15}{196} a - \frac{3}{56}$, $\frac{1}{486942008} a^{18} - \frac{255}{243471004} a^{17} + \frac{84057}{486942008} a^{16} - \frac{82497}{486942008} a^{15} - \frac{5447}{17390786} a^{14} - \frac{91283}{243471004} a^{13} - \frac{1214447}{486942008} a^{12} + \frac{3803759}{486942008} a^{11} + \frac{87113}{9937592} a^{10} - \frac{10273479}{486942008} a^{9} + \frac{331929}{9937592} a^{8} + \frac{335373}{243471004} a^{7} - \frac{2256847}{486942008} a^{6} - \frac{14122987}{121735502} a^{5} + \frac{44460281}{121735502} a^{4} - \frac{60769847}{121735502} a^{3} - \frac{6208261}{34781572} a^{2} - \frac{4236129}{9937592} a - \frac{161319}{354914}$, $\frac{1}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{19} - \frac{462601971392207495019709421149160255777088505373321789438979266245847}{2995821680308668740637820035852316071214170108005918646060430078170700374187208} a^{18} + \frac{415506842130961950951363377870325501874114394274416285929695398607862482587}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{17} - \frac{1556628896140230143221169927643102017505340226191272092472451473500760828489}{10485375881080340592232370125483106249249595378020715261211505273597451309655228} a^{16} - \frac{382419430571707806274459566915484278380596463167254272738245460368678252588}{2621343970270085148058092531370776562312398844505178815302876318399362827413807} a^{15} + \frac{1102243150819094770473301782863084667755142334860160706918537989714979915319}{5242687940540170296116185062741553124624797689010357630605752636798725654827614} a^{14} + \frac{577362683969863621622831664824822904085914947867523781674282445371750362779}{2995821680308668740637820035852316071214170108005918646060430078170700374187208} a^{13} - \frac{48176902951646893854210482743274307251747203239136701563013518920700960714853}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{12} + \frac{5679691141907293251091873056551265839305914477502999179392302119462194264439}{1497910840154334370318910017926158035607085054002959323030215039085350187093604} a^{11} - \frac{95387088530166973313123016210486600913266455300582847599320094946057965777345}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{10} + \frac{649243117248808589702150634636405689935170446756600989330452431339634850046529}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{9} + \frac{612586639080315126534638876482474744286482378759316822028688478122786155852277}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{8} + \frac{593737507306820467350052866732252909196229555780437037781007639072886905973003}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{7} - \frac{814312540151527327253564365366770111392138188078921940583728723999567177626577}{5242687940540170296116185062741553124624797689010357630605752636798725654827614} a^{6} + \frac{4606912805018487993101651496309579410073629693748880432363384156214017035337961}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{5} + \frac{8293463892752699722773282362821098226788280507121409069620255555679558784406641}{20970751762160681184464740250966212498499190756041430522423010547194902619310456} a^{4} - \frac{1629753317671487833108865666916915983408283386123768736464483550221477162017573}{5242687940540170296116185062741553124624797689010357630605752636798725654827614} a^{3} + \frac{262021325549849610762275759709786694426750714283915385968215813445205351129153}{748955420077167185159455008963079017803542527001479661515107519542675093546802} a^{2} + \frac{138482687044690961719105365120082772700043213801373639876799740555186388439797}{427974525758381248662545719407473724459167158286559806580061439738671482026744} a + \frac{86142153792368548613061956321676244923391524832246958576983469365504833085}{307232251082829324237290537980957447565805569480660306231199884952384409208}$

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

Class group and class number

$C_{2}\times C_{22048052}$, which has order $44096104$ (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:  \( 208779686.22504243 \) (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

\(\Q(\sqrt{65}) \), 4.0.2471625.1, 5.5.390625.1, 10.10.283274078369140625.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
3Data not computed
5Data not computed
13Data not computed