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Properties

Label 20.0.16210890375...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{34}\cdot 11^{10}$
Root discriminant $144.71$
Ramified primes $2, 5, 11$
Class number $1312648$ (GRH)
Class group $[1312648]$ (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![12604500001, 1956265360, 6316888360, 425458240, 1552432110, 81077744, 263561040, 17089940, 30361415, 2022520, 2552682, 255180, 175630, 9980, 7190, 356, 480, -20, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 20*x^17 + 480*x^16 + 356*x^15 + 7190*x^14 + 9980*x^13 + 175630*x^12 + 255180*x^11 + 2552682*x^10 + 2022520*x^9 + 30361415*x^8 + 17089940*x^7 + 263561040*x^6 + 81077744*x^5 + 1552432110*x^4 + 425458240*x^3 + 6316888360*x^2 + 1956265360*x + 12604500001)
 
gp: K = bnfinit(x^20 - 10*x^18 - 20*x^17 + 480*x^16 + 356*x^15 + 7190*x^14 + 9980*x^13 + 175630*x^12 + 255180*x^11 + 2552682*x^10 + 2022520*x^9 + 30361415*x^8 + 17089940*x^7 + 263561040*x^6 + 81077744*x^5 + 1552432110*x^4 + 425458240*x^3 + 6316888360*x^2 + 1956265360*x + 12604500001, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} - 20 x^{17} + 480 x^{16} + 356 x^{15} + 7190 x^{14} + 9980 x^{13} + 175630 x^{12} + 255180 x^{11} + 2552682 x^{10} + 2022520 x^{9} + 30361415 x^{8} + 17089940 x^{7} + 263561040 x^{6} + 81077744 x^{5} + 1552432110 x^{4} + 425458240 x^{3} + 6316888360 x^{2} + 1956265360 x + 12604500001 \)

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:  \(16210890375625000000000000000000000000000000=2^{30}\cdot 5^{34}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.71$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2200=2^{3}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2200}(1,·)$, $\chi_{2200}(901,·)$, $\chi_{2200}(769,·)$, $\chi_{2200}(329,·)$, $\chi_{2200}(461,·)$, $\chi_{2200}(2069,·)$, $\chi_{2200}(1209,·)$, $\chi_{2200}(1629,·)$, $\chi_{2200}(1761,·)$, $\chi_{2200}(1189,·)$, $\chi_{2200}(881,·)$, $\chi_{2200}(1321,·)$, $\chi_{2200}(749,·)$, $\chi_{2200}(1649,·)$, $\chi_{2200}(1781,·)$, $\chi_{2200}(2089,·)$, $\chi_{2200}(441,·)$, $\chi_{2200}(21,·)$, $\chi_{2200}(1341,·)$, $\chi_{2200}(309,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{50} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{11}{25} a^{5} - \frac{1}{10} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{50}$, $\frac{1}{50} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{11}{25} a^{6} - \frac{1}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{50} a$, $\frac{1}{50} a^{12} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{11}{25} a^{7} - \frac{1}{10} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{50} a^{2} + \frac{2}{5}$, $\frac{1}{50} a^{13} - \frac{1}{5} a^{9} - \frac{11}{25} a^{8} - \frac{1}{10} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{50} a^{3} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{50} a^{14} - \frac{11}{25} a^{9} - \frac{1}{10} a^{8} + \frac{2}{5} a^{7} - \frac{1}{50} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{200} a^{15} + \frac{1}{200} a^{12} - \frac{1}{100} a^{11} - \frac{1}{200} a^{10} + \frac{19}{40} a^{9} - \frac{1}{4} a^{8} - \frac{3}{50} a^{7} - \frac{21}{200} a^{6} + \frac{17}{200} a^{5} + \frac{1}{40} a^{4} + \frac{3}{20} a^{3} - \frac{31}{200} a^{2} + \frac{31}{100} a - \frac{1}{200}$, $\frac{1}{1400} a^{16} + \frac{1}{700} a^{15} + \frac{1}{350} a^{14} - \frac{3}{1400} a^{13} - \frac{1}{175} a^{12} - \frac{13}{1400} a^{11} - \frac{1}{200} a^{10} - \frac{21}{50} a^{9} - \frac{121}{350} a^{8} - \frac{689}{1400} a^{7} - \frac{689}{1400} a^{6} - \frac{281}{1400} a^{5} + \frac{32}{175} a^{4} + \frac{313}{1400} a^{3} - \frac{34}{175} a^{2} - \frac{389}{1400} a - \frac{331}{700}$, $\frac{1}{1400} a^{17} - \frac{11}{1400} a^{14} - \frac{1}{700} a^{13} + \frac{3}{1400} a^{12} - \frac{9}{1400} a^{11} - \frac{1}{100} a^{10} + \frac{33}{350} a^{9} + \frac{559}{1400} a^{8} - \frac{431}{1400} a^{7} + \frac{313}{1400} a^{6} - \frac{81}{700} a^{5} + \frac{361}{1400} a^{4} - \frac{29}{700} a^{3} - \frac{81}{280} a^{2} + \frac{18}{175} a - \frac{159}{350}$, $\frac{1}{1043756443161598600} a^{18} + \frac{44080697335077}{1043756443161598600} a^{17} - \frac{250425166249731}{1043756443161598600} a^{16} - \frac{229439559260587}{104375644316159860} a^{15} + \frac{7571591437528231}{1043756443161598600} a^{14} - \frac{8742261119731}{74554031654399900} a^{13} + \frac{1427261788698231}{149108063308799800} a^{12} - \frac{1268227496410181}{521878221580799300} a^{11} - \frac{876611329921048}{130469555395199825} a^{10} + \frac{83118598785627857}{260939110790399650} a^{9} + \frac{128368623003160979}{521878221580799300} a^{8} - \frac{393300168755066719}{1043756443161598600} a^{7} - \frac{491193215503767641}{1043756443161598600} a^{6} - \frac{18876661756294667}{1043756443161598600} a^{5} + \frac{5965400590592687}{521878221580799300} a^{4} - \frac{56325291690201357}{260939110790399650} a^{3} - \frac{26083263656589}{4877366556829900} a^{2} + \frac{27611282690268149}{1043756443161598600} a + \frac{59951572187965499}{1043756443161598600}$, $\frac{1}{26165389781978204052278772524186507066951882758600} a^{19} + \frac{7666874076034597865160943893259}{26165389781978204052278772524186507066951882758600} a^{18} + \frac{1237262361074062835705921727743843468125361729}{26165389781978204052278772524186507066951882758600} a^{17} + \frac{162411188096492018403471414969654275456615277}{467239103249610786647835223646187626195569334975} a^{16} + \frac{35711178533047431003748032516515535028885170511}{26165389781978204052278772524186507066951882758600} a^{15} - \frac{97205724608694011558838133230307846640721391259}{13082694890989102026139386262093253533475941379300} a^{14} - \frac{37504535795655210379386491954833541419450438231}{26165389781978204052278772524186507066951882758600} a^{13} + \frac{3790894415636012551692399566767986341606144229}{1868956412998443146591340894584750504782277339900} a^{12} + \frac{10252440835712367888924493860045258868099443691}{1868956412998443146591340894584750504782277339900} a^{11} - \frac{4588449422611838506797718885407754905583175277}{1308269489098910202613938626209325353347594137930} a^{10} + \frac{3543407498482180267818432867231495641877103089273}{13082694890989102026139386262093253533475941379300} a^{9} + \frac{12818515957770462445912925568300529749550025665427}{26165389781978204052278772524186507066951882758600} a^{8} + \frac{5751289562491021507753022469899338871336705044043}{26165389781978204052278772524186507066951882758600} a^{7} + \frac{4548003808200469699797370536822478823242345366517}{26165389781978204052278772524186507066951882758600} a^{6} - \frac{107169472560021509101171171517020869038654668993}{654134744549455101306969313104662676673797068965} a^{5} - \frac{245039162632978278613936624355435134068421837797}{13082694890989102026139386262093253533475941379300} a^{4} + \frac{5644383470011241250019582577937206466172999595419}{13082694890989102026139386262093253533475941379300} a^{3} + \frac{1491169336614351604590562606691689791129012589131}{26165389781978204052278772524186507066951882758600} a^{2} + \frac{7609305644947417597767352708565239350512921965277}{26165389781978204052278772524186507066951882758600} a + \frac{1653488326678646706731515546515191030393095174819}{13082694890989102026139386262093253533475941379300}$

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

Class group and class number

$C_{1312648}$, which has order $1312648$ (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:  \( 19344397.966990974 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{10}, \sqrt{-22})\), 5.5.390625.1, 10.10.25000000000000000.1, 10.0.122872161865234375.1, 10.0.805255000000000000000.4

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$5$$15$$C_{10}$$[3]^{5}$
5Data not computed
11Data not computed