↑

Properties

Label 20.0.21333423461...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{35}\cdot 31^{18}$
Root discriminant $735.28$
Ramified primes $2, 5, 31$
Class number $7895401000$ (GRH)
Class group $[2, 10, 394770050]$ (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![203649683820005, 0, 1148533543886375, 0, 607155836715875, 0, 70454737110425, 0, 3787311859300, 0, 114566349855, 0, 2101010275, 0, 23832800, 0, 163370, 0, 620, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 620*x^18 + 163370*x^16 + 23832800*x^14 + 2101010275*x^12 + 114566349855*x^10 + 3787311859300*x^8 + 70454737110425*x^6 + 607155836715875*x^4 + 1148533543886375*x^2 + 203649683820005)
 
gp: K = bnfinit(x^20 + 620*x^18 + 163370*x^16 + 23832800*x^14 + 2101010275*x^12 + 114566349855*x^10 + 3787311859300*x^8 + 70454737110425*x^6 + 607155836715875*x^4 + 1148533543886375*x^2 + 203649683820005, 1)
 

Normalized defining polynomial

\( x^{20} + 620 x^{18} + 163370 x^{16} + 23832800 x^{14} + 2101010275 x^{12} + 114566349855 x^{10} + 3787311859300 x^{8} + 70454737110425 x^{6} + 607155836715875 x^{4} + 1148533543886375 x^{2} + 203649683820005 \)

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:  \(2133342346188491938901276370239257812500000000000000000000=2^{20}\cdot 5^{35}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $735.28$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3100=2^{2}\cdot 5^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3100}(1,·)$, $\chi_{3100}(1089,·)$, $\chi_{3100}(729,·)$, $\chi_{3100}(523,·)$, $\chi_{3100}(1083,·)$, $\chi_{3100}(1721,·)$, $\chi_{3100}(281,·)$, $\chi_{3100}(27,·)$, $\chi_{3100}(2247,·)$, $\chi_{3100}(1503,·)$, $\chi_{3100}(2209,·)$, $\chi_{3100}(3067,·)$, $\chi_{3100}(743,·)$, $\chi_{3100}(1769,·)$, $\chi_{3100}(1387,·)$, $\chi_{3100}(1263,·)$, $\chi_{3100}(1461,·)$, $\chi_{3100}(249,·)$, $\chi_{3100}(2107,·)$, $\chi_{3100}(1341,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{18221149} a^{10} + \frac{10}{587779} a^{8} + \frac{1085}{587779} a^{6} + \frac{48050}{587779} a^{4} + \frac{156996}{587779} a^{2} - \frac{79316}{587779}$, $\frac{1}{18221149} a^{11} + \frac{10}{587779} a^{9} + \frac{1085}{587779} a^{7} + \frac{48050}{587779} a^{5} + \frac{156996}{587779} a^{3} - \frac{79316}{587779} a$, $\frac{1}{18221149} a^{12} - \frac{2015}{587779} a^{8} - \frac{288300}{587779} a^{6} - \frac{44029}{587779} a^{4} + \frac{37581}{587779} a^{2} - \frac{98758}{587779}$, $\frac{1}{18221149} a^{13} - \frac{2015}{587779} a^{9} - \frac{288300}{587779} a^{7} - \frac{44029}{587779} a^{5} + \frac{37581}{587779} a^{3} - \frac{98758}{587779} a$, $\frac{1}{564855619} a^{14} + \frac{10850}{587779} a^{8} - \frac{33537}{587779} a^{6} - \frac{5596533}{18221149} a^{4} - \frac{260560}{587779} a^{2} + \frac{54148}{587779}$, $\frac{1}{3751206165779} a^{15} + \frac{15}{121006650509} a^{13} + \frac{90}{3903440339} a^{11} + \frac{8525}{3903440339} a^{9} + \frac{432450}{3903440339} a^{7} - \frac{22218006972}{121006650509} a^{5} + \frac{493573221}{3903440339} a^{3} + \frac{11804959}{3903440339} a$, $\frac{1}{340763319305530139} a^{16} + \frac{9902196}{340763319305530139} a^{14} + \frac{138627024}{10992365138888069} a^{12} + \frac{107968013}{10992365138888069} a^{10} - \frac{170816580330}{354592423835099} a^{8} + \frac{4062094437039768}{10992365138888069} a^{6} - \frac{2784078425110046}{10992365138888069} a^{4} - \frac{126410924110609}{354592423835099} a^{2} - \frac{9540451979}{53394432139}$, $\frac{1}{10563662898471434309} a^{17} + \frac{17}{340763319305530139} a^{15} - \frac{214384642}{10992365138888069} a^{13} - \frac{128708398}{10992365138888069} a^{11} + \frac{430698101400}{354592423835099} a^{9} - \frac{89864249454661483}{340763319305530139} a^{7} - \frac{4690419519538512}{10992365138888069} a^{5} + \frac{120956370737120}{354592423835099} a^{3} + \frac{80951776848423}{354592423835099} a$, $\frac{1}{10563662898471434309} a^{18} - \frac{178235343}{340763319305530139} a^{14} - \frac{72267482}{10992365138888069} a^{12} - \frac{68580270}{10992365138888069} a^{10} - \frac{150845827794320273}{340763319305530139} a^{8} - \frac{1616514925643910}{10992365138888069} a^{6} - \frac{462439356270875}{10992365138888069} a^{4} + \frac{1313253022847}{354592423835099} a^{2} - \frac{26293765207}{53394432139}$, $\frac{1}{10563662898471434309} a^{19} - \frac{171}{10992365138888069} a^{15} + \frac{188082824}{10992365138888069} a^{13} + \frac{94570166}{10992365138888069} a^{11} - \frac{149490028242995603}{340763319305530139} a^{9} - \frac{5372849163231319}{10992365138888069} a^{7} - \frac{122532956655360}{354592423835099} a^{5} - \frac{90887712871911}{354592423835099} a^{3} + \frac{108258668196704}{354592423835099} a$

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

Class group and class number

$C_{2}\times C_{10}\times C_{394770050}$, which has order $7895401000$ (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:  \( 35952457733.9904 \) (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

\(\Q(\sqrt{5}) \), 4.0.1922000.2, 5.5.360750390625.1, 10.10.650704221680450439453125.3

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{20}$ R $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
5Data not computed
$31$31.10.9.5$x^{10} - 178708831$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.5$x^{10} - 178708831$$10$$1$$9$$C_{10}$$[\ ]_{10}$