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Properties

Label 20.0.20038303066...000.37
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $462.48$
Ramified primes $2, 3, 5, 11$
Class number $4505583280$ (GRH)
Class group $[2, 2, 1126395820]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![165191136108121849, -6064244667019220, 14277619259107655, -57940495299390, 621558092253230, 2357254937444, 20204698226075, 56156873190, 473766322150, 2600095190, 8914160145, 2355870, 140041165, 428340, 1650440, -1298, 18755, 0, 110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 110*x^18 + 18755*x^16 - 1298*x^15 + 1650440*x^14 + 428340*x^13 + 140041165*x^12 + 2355870*x^11 + 8914160145*x^10 + 2600095190*x^9 + 473766322150*x^8 + 56156873190*x^7 + 20204698226075*x^6 + 2357254937444*x^5 + 621558092253230*x^4 - 57940495299390*x^3 + 14277619259107655*x^2 - 6064244667019220*x + 165191136108121849)
 
gp: K = bnfinit(x^20 + 110*x^18 + 18755*x^16 - 1298*x^15 + 1650440*x^14 + 428340*x^13 + 140041165*x^12 + 2355870*x^11 + 8914160145*x^10 + 2600095190*x^9 + 473766322150*x^8 + 56156873190*x^7 + 20204698226075*x^6 + 2357254937444*x^5 + 621558092253230*x^4 - 57940495299390*x^3 + 14277619259107655*x^2 - 6064244667019220*x + 165191136108121849, 1)
 

Normalized defining polynomial

\( x^{20} + 110 x^{18} + 18755 x^{16} - 1298 x^{15} + 1650440 x^{14} + 428340 x^{13} + 140041165 x^{12} + 2355870 x^{11} + 8914160145 x^{10} + 2600095190 x^{9} + 473766322150 x^{8} + 56156873190 x^{7} + 20204698226075 x^{6} + 2357254937444 x^{5} + 621558092253230 x^{4} - 57940495299390 x^{3} + 14277619259107655 x^{2} - 6064244667019220 x + 165191136108121849 \)

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:  \(200383030666749741651348266601562500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $462.48$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(1091,·)$, $\chi_{3300}(2689,·)$, $\chi_{3300}(2879,·)$, $\chi_{3300}(3131,·)$, $\chi_{3300}(3299,·)$, $\chi_{3300}(2341,·)$, $\chi_{3300}(2209,·)$, $\chi_{3300}(611,·)$, $\chi_{3300}(421,·)$, $\chi_{3300}(2929,·)$, $\chi_{3300}(169,·)$, $\chi_{3300}(1451,·)$, $\chi_{3300}(2161,·)$, $\chi_{3300}(371,·)$, $\chi_{3300}(1139,·)$, $\chi_{3300}(2281,·)$, $\chi_{3300}(1849,·)$, $\chi_{3300}(1019,·)$, $\chi_{3300}(959,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{11} a^{7}$, $\frac{1}{11} a^{8}$, $\frac{1}{121} a^{9} - \frac{4}{11} a^{4}$, $\frac{1}{121} a^{10}$, $\frac{1}{121} a^{11}$, $\frac{1}{121} a^{12}$, $\frac{1}{1331} a^{13} + \frac{3}{121} a^{8} + \frac{5}{11} a^{3}$, $\frac{1}{78529} a^{14} - \frac{29}{7139} a^{12} - \frac{4}{7139} a^{10} + \frac{14}{649} a^{8} + \frac{9}{649} a^{6} - \frac{225}{649} a^{4} - \frac{8}{59} a^{2} + \frac{1}{59}$, $\frac{1}{156351239} a^{15} + \frac{58}{14213749} a^{14} + \frac{10}{117469} a^{13} + \frac{1091}{1292159} a^{12} + \frac{3781}{1292159} a^{11} - \frac{8511}{14213749} a^{10} - \frac{3290}{1292159} a^{9} + \frac{7811}{1292159} a^{8} + \frac{151}{117469} a^{7} - \frac{1366}{117469} a^{6} + \frac{30101}{1292159} a^{5} + \frac{2949}{10679} a^{4} - \frac{8858}{117469} a^{3} + \frac{4846}{10679} a^{2} + \frac{4763}{10679} a + \frac{1946}{10679}$, $\frac{1}{156351239} a^{16} + \frac{4}{1292159} a^{14} - \frac{30}{240911} a^{13} - \frac{1581}{1292159} a^{12} + \frac{75}{240911} a^{11} + \frac{5268}{1292159} a^{10} + \frac{52}{21901} a^{9} - \frac{7334}{1292159} a^{8} - \frac{27}{1991} a^{7} + \frac{8123}{1292159} a^{6} - \frac{81}{1991} a^{5} - \frac{31301}{117469} a^{4} - \frac{326}{1991} a^{3} + \frac{875}{10679} a^{2} - \frac{68}{181} a + \frac{1017}{10679}$, $\frac{1}{156351239} a^{17} + \frac{23}{14213749} a^{14} - \frac{5007}{14213749} a^{13} + \frac{57445}{14213749} a^{12} + \frac{1373}{1292159} a^{11} + \frac{1796}{1292159} a^{10} + \frac{4534}{1292159} a^{9} + \frac{10917}{1292159} a^{8} + \frac{26482}{1292159} a^{7} - \frac{3923}{117469} a^{6} + \frac{488}{117469} a^{5} - \frac{24114}{117469} a^{4} + \frac{3939}{117469} a^{3} - \frac{4078}{10679} a^{2} + \frac{2389}{10679} a - \frac{4284}{10679}$, $\frac{1}{534877588619} a^{18} + \frac{9}{48625235329} a^{17} + \frac{58}{48625235329} a^{15} + \frac{25246}{4420475939} a^{14} + \frac{10247405}{48625235329} a^{13} - \frac{8629066}{4420475939} a^{12} + \frac{11941359}{4420475939} a^{11} - \frac{8537839}{4420475939} a^{10} + \frac{938241}{401861449} a^{9} + \frac{119747798}{4420475939} a^{8} + \frac{5952420}{401861449} a^{7} + \frac{74280}{401861449} a^{6} - \frac{12644782}{401861449} a^{5} - \frac{1818171}{36532859} a^{4} + \frac{187861}{619201} a^{3} + \frac{803615}{3321169} a^{2} - \frac{1208220}{3321169} a + \frac{406396}{3321169}$, $\frac{1}{746123430168958945627403939371876094031617948972634798724334765796658603} a^{19} - \frac{640926796541941987524659144280029040525647862302288756673522}{746123430168958945627403939371876094031617948972634798724334765796658603} a^{18} - \frac{26359215662793512268682852333177302037736383902068155802206212}{67829402742632631420673085397443281275601631724784981702212251436059873} a^{17} + \frac{54462957841543442862390659640093666810287330716615106711388552}{67829402742632631420673085397443281275601631724784981702212251436059873} a^{16} - \frac{140016629748298358204440644828163884920917968520418244875531649}{67829402742632631420673085397443281275601631724784981702212251436059873} a^{15} - \frac{295572777264226750374548390440751990132935295513729435191120795287}{67829402742632631420673085397443281275601631724784981702212251436059873} a^{14} - \frac{48221403201850252977365257655628866706123221044893080094674882561}{1149650893942925956282594667753275953823756469911609859359529685356947} a^{13} - \frac{189620659747511471143936181732262639078018636802503151806874725281}{104513717631175086934781333432115995802159679082873623578139062305177} a^{12} - \frac{8912049004033805058241406492663155446849812364163623773731850885879}{6166309340239330129152098672494843752327421065889543791110204676005443} a^{11} + \frac{13893102785341787422987432582161934682563997094783559776067736672476}{6166309340239330129152098672494843752327421065889543791110204676005443} a^{10} + \frac{5340435718032605747238240793255492700335483539378669803027592331301}{6166309340239330129152098672494843752327421065889543791110204676005443} a^{9} - \frac{47360946461772248020384802941023292178176887468215912985785972087424}{6166309340239330129152098672494843752327421065889543791110204676005443} a^{8} + \frac{19987437490415547364856008328476442418143893464585711050239493548624}{560573576385393648104736242954076704757038278717231253737291334182313} a^{7} + \frac{6328566007997849166551926754135173802798759572385804180824785868042}{560573576385393648104736242954076704757038278717231253737291334182313} a^{6} - \frac{1533975315778759034046163214632516237875228210375632057080262009368}{560573576385393648104736242954076704757038278717231253737291334182313} a^{5} - \frac{18073192517847143637156856308190973964826421395549433626487359568389}{50961234216853968009521476632188791341548934428839204885208303107483} a^{4} - \frac{17474249351097109029759442438177760944438281405996498587908939041097}{50961234216853968009521476632188791341548934428839204885208303107483} a^{3} + \frac{26668697090887032940565046107667139235262933405479223586125212554}{4632839474259451637229225148380799212868084948076291353200754827953} a^{2} - \frac{1714844789602956118150983400307752952202097098211416229766797521807}{4632839474259451637229225148380799212868084948076291353200754827953} a + \frac{1948529165278019386166255988027361191045299162627396008671276180644}{4632839474259451637229225148380799212868084948076291353200754827953}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1126395820}$, which has order $4505583280$ (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:  \( 655926678.0583006 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{5}, \sqrt{-33})\), 5.5.5719140625.3, 10.10.163542847442626953125.3, 10.0.89528326392656250000000000.3, 10.0.447641631963281250000000000.4

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 R R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{20}$

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
2Data not computed
3Data not computed
5Data not computed
$11$11.10.9.6$x^{10} + 216513$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.6$x^{10} + 216513$$10$$1$$9$$C_{10}$$[\ ]_{10}$