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Properties

Label 20.0.12383472844...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 5^{34}$
Root discriminant $179.74$
Ramified primes $2, 3, 5$
Class number $41110100$ (GRH)
Class group $[2, 20555050]$ (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![47239200, 0, 314928000, 0, 656100000, 0, 612360000, 0, 285913800, 0, 68603760, 0, 8650800, 0, 567000, 0, 18000, 0, 240, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 240*x^18 + 18000*x^16 + 567000*x^14 + 8650800*x^12 + 68603760*x^10 + 285913800*x^8 + 612360000*x^6 + 656100000*x^4 + 314928000*x^2 + 47239200)
 
gp: K = bnfinit(x^20 + 240*x^18 + 18000*x^16 + 567000*x^14 + 8650800*x^12 + 68603760*x^10 + 285913800*x^8 + 612360000*x^6 + 656100000*x^4 + 314928000*x^2 + 47239200, 1)
 

Normalized defining polynomial

\( x^{20} + 240 x^{18} + 18000 x^{16} + 567000 x^{14} + 8650800 x^{12} + 68603760 x^{10} + 285913800 x^{8} + 612360000 x^{6} + 656100000 x^{4} + 314928000 x^{2} + 47239200 \)

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:  \(1238347284480000000000000000000000000000000000=2^{55}\cdot 3^{10}\cdot 5^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.74$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1200=2^{4}\cdot 3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1200}(1,·)$, $\chi_{1200}(389,·)$, $\chi_{1200}(961,·)$, $\chi_{1200}(841,·)$, $\chi_{1200}(269,·)$, $\chi_{1200}(721,·)$, $\chi_{1200}(149,·)$, $\chi_{1200}(121,·)$, $\chi_{1200}(601,·)$, $\chi_{1200}(989,·)$, $\chi_{1200}(481,·)$, $\chi_{1200}(869,·)$, $\chi_{1200}(361,·)$, $\chi_{1200}(749,·)$, $\chi_{1200}(29,·)$, $\chi_{1200}(241,·)$, $\chi_{1200}(629,·)$, $\chi_{1200}(1081,·)$, $\chi_{1200}(509,·)$, $\chi_{1200}(1109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{18} a^{4}$, $\frac{1}{18} a^{5}$, $\frac{1}{378} a^{6} - \frac{3}{7}$, $\frac{1}{378} a^{7} - \frac{3}{7} a$, $\frac{1}{2268} a^{8} + \frac{2}{21} a^{2}$, $\frac{1}{2268} a^{9} + \frac{2}{21} a^{3}$, $\frac{1}{34020} a^{10} - \frac{1}{63} a^{4}$, $\frac{1}{34020} a^{11} - \frac{1}{63} a^{5}$, $\frac{1}{1428840} a^{12} - \frac{1}{1323} a^{6} + \frac{10}{49}$, $\frac{1}{1428840} a^{13} - \frac{1}{1323} a^{7} + \frac{10}{49} a$, $\frac{1}{4286520} a^{14} + \frac{1}{5292} a^{8} + \frac{8}{49} a^{2}$, $\frac{1}{4286520} a^{15} + \frac{1}{5292} a^{9} + \frac{8}{49} a^{3}$, $\frac{1}{180033840} a^{16} - \frac{1}{30005640} a^{14} + \frac{1}{5000940} a^{12} - \frac{2}{138915} a^{10} - \frac{5}{55566} a^{8} + \frac{5}{9261} a^{6} - \frac{79}{3087} a^{4} - \frac{29}{343} a^{2} - \frac{169}{343}$, $\frac{1}{180033840} a^{17} - \frac{1}{30005640} a^{15} + \frac{1}{5000940} a^{13} - \frac{2}{138915} a^{11} - \frac{5}{55566} a^{9} + \frac{5}{9261} a^{7} - \frac{79}{3087} a^{5} - \frac{29}{343} a^{3} - \frac{169}{343} a$, $\frac{1}{3780710640} a^{18} + \frac{1}{630118440} a^{16} + \frac{1}{17503290} a^{14} + \frac{2}{8751645} a^{12} + \frac{1}{11668860} a^{10} - \frac{19}{194481} a^{8} + \frac{5}{43218} a^{6} - \frac{71}{43218} a^{4} + \frac{652}{7203} a^{2} + \frac{521}{2401}$, $\frac{1}{3780710640} a^{19} + \frac{1}{630118440} a^{17} + \frac{1}{17503290} a^{15} + \frac{2}{8751645} a^{13} + \frac{1}{11668860} a^{11} - \frac{19}{194481} a^{9} + \frac{5}{43218} a^{7} - \frac{71}{43218} a^{5} + \frac{652}{7203} a^{3} + \frac{521}{2401} a$

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

Class group and class number

$C_{2}\times C_{20555050}$, which has order $41110100$ (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:  \( 42294001.73672045 \) (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

\(\Q(\sqrt{2}) \), 4.0.460800.2, 5.5.390625.1, 10.10.5000000000000000.1

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 R R R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ $20$ $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ $20$

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
2Data not computed
3Data not computed
5Data not computed