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Properties

Label 20.0.76861758362...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{15}\cdot 31^{18}$
Root discriminant $494.64$
Ramified primes $2, 5, 31$
Class number $3981314304$ (GRH)
Class group $[2, 2, 2, 2, 2, 66, 1885092]$ (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![96100000, 0, 4036200000, 0, 12637150000, 0, 14972380000, 0, 8029155000, 0, 1934152000, 0, 184016000, 0, 6813800, 0, 103850, 0, 620, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 620*x^18 + 103850*x^16 + 6813800*x^14 + 184016000*x^12 + 1934152000*x^10 + 8029155000*x^8 + 14972380000*x^6 + 12637150000*x^4 + 4036200000*x^2 + 96100000)
 
gp: K = bnfinit(x^20 + 620*x^18 + 103850*x^16 + 6813800*x^14 + 184016000*x^12 + 1934152000*x^10 + 8029155000*x^8 + 14972380000*x^6 + 12637150000*x^4 + 4036200000*x^2 + 96100000, 1)
 

Normalized defining polynomial

\( x^{20} + 620 x^{18} + 103850 x^{16} + 6813800 x^{14} + 184016000 x^{12} + 1934152000 x^{10} + 8029155000 x^{8} + 14972380000 x^{6} + 12637150000 x^{4} + 4036200000 x^{2} + 96100000 \)

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:  \(768617583627855357889739724818288214016000000000000000=2^{55}\cdot 5^{15}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $494.64$
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:  \(2480=2^{4}\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{2480}(1,·)$, $\chi_{2480}(643,·)$, $\chi_{2480}(2081,·)$, $\chi_{2480}(2323,·)$, $\chi_{2480}(969,·)$, $\chi_{2480}(587,·)$, $\chi_{2480}(721,·)$, $\chi_{2480}(1363,·)$, $\chi_{2480}(2329,·)$, $\chi_{2480}(729,·)$, $\chi_{2480}(1627,·)$, $\chi_{2480}(481,·)$, $\chi_{2480}(1763,·)$, $\chi_{2480}(1769,·)$, $\chi_{2480}(1387,·)$, $\chi_{2480}(27,·)$, $\chi_{2480}(1521,·)$, $\chi_{2480}(883,·)$, $\chi_{2480}(249,·)$, $\chi_{2480}(2107,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4}$, $\frac{1}{10} a^{5}$, $\frac{1}{50} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{50} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{500} a^{8} + \frac{1}{25} a^{4} + \frac{1}{5}$, $\frac{1}{500} a^{9} + \frac{1}{25} a^{5} + \frac{1}{5} a$, $\frac{1}{232500} a^{10} - \frac{1}{1500} a^{8} + \frac{1}{375} a^{6} - \frac{1}{75} a^{4} + \frac{7}{25} a^{2} - \frac{1}{15}$, $\frac{1}{232500} a^{11} - \frac{1}{1500} a^{9} + \frac{1}{375} a^{7} - \frac{1}{75} a^{5} + \frac{7}{25} a^{3} - \frac{1}{15} a$, $\frac{1}{2325000} a^{12} - \frac{1}{3750} a^{8} - \frac{1}{375} a^{4} + \frac{1}{3} a^{2} - \frac{4}{75}$, $\frac{1}{2325000} a^{13} - \frac{1}{3750} a^{9} - \frac{1}{375} a^{5} + \frac{1}{3} a^{3} - \frac{4}{75} a$, $\frac{1}{86025000} a^{14} + \frac{1}{5735000} a^{12} + \frac{1}{860250} a^{10} - \frac{11}{18500} a^{8} - \frac{49}{13875} a^{6} + \frac{13}{1110} a^{4} + \frac{68}{2775} a^{2} + \frac{67}{185}$, $\frac{1}{86025000} a^{15} + \frac{1}{5735000} a^{13} + \frac{1}{860250} a^{11} - \frac{11}{18500} a^{9} - \frac{49}{13875} a^{7} + \frac{13}{1110} a^{5} + \frac{68}{2775} a^{3} + \frac{67}{185} a$, $\frac{1}{262376250000} a^{16} - \frac{7}{5247525000} a^{14} - \frac{347}{3279703125} a^{12} + \frac{41}{26237625} a^{10} + \frac{24847}{28212500} a^{8} - \frac{5779}{1692750} a^{6} - \frac{55366}{1410625} a^{4} - \frac{9077}{56425} a^{2} + \frac{38407}{282125}$, $\frac{1}{262376250000} a^{17} - \frac{7}{5247525000} a^{15} - \frac{347}{3279703125} a^{13} + \frac{41}{26237625} a^{11} + \frac{24847}{28212500} a^{9} - \frac{5779}{1692750} a^{7} - \frac{55366}{1410625} a^{5} - \frac{9077}{56425} a^{3} + \frac{38407}{282125} a$, $\frac{1}{1592830327608750000} a^{18} + \frac{70181}{66367930317031250} a^{16} - \frac{739184471}{159283032760875000} a^{14} + \frac{425031433}{4304946831375000} a^{12} + \frac{28183627031}{15928303276087500} a^{10} - \frac{13045179571}{85636039118750} a^{8} - \frac{254205731608}{25690811735625} a^{6} - \frac{6260181884}{8563603911875} a^{4} + \frac{767744543446}{5138162347125} a^{2} - \frac{67678413437}{1712720782375}$, $\frac{1}{1592830327608750000} a^{19} + \frac{70181}{66367930317031250} a^{17} - \frac{739184471}{159283032760875000} a^{15} + \frac{425031433}{4304946831375000} a^{13} + \frac{28183627031}{15928303276087500} a^{11} - \frac{13045179571}{85636039118750} a^{9} - \frac{254205731608}{25690811735625} a^{7} - \frac{6260181884}{8563603911875} a^{5} + \frac{767744543446}{5138162347125} a^{3} - \frac{67678413437}{1712720782375} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{66}\times C_{1885092}$, which has order $3981314304$ (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:  \( 2725468819.573966 \) (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

\(\Q(\sqrt{10}) \), 4.0.246016000.8, 5.5.923521.1, 10.10.87336042233958400000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $20$ $20$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ $20$ $20$ $20$ $20$ R ${\href{/LocalNumberField/37.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$5$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
$31$31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$