↑

Properties

Label 20.0.76861758362...0000.2
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 $3200800000$ (GRH)
Class group $[2, 2, 2, 2, 10, 10, 2000500]$ (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![5674608900000, 0, 26481508200000, 0, 31350462750000, 0, 6929405820000, 0, 661086315000, 0, 33380552000, 0, 956784000, 0, 15692200, 0, 141050, 0, 620, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 620*x^18 + 141050*x^16 + 15692200*x^14 + 956784000*x^12 + 33380552000*x^10 + 661086315000*x^8 + 6929405820000*x^6 + 31350462750000*x^4 + 26481508200000*x^2 + 5674608900000)
 
gp: K = bnfinit(x^20 + 620*x^18 + 141050*x^16 + 15692200*x^14 + 956784000*x^12 + 33380552000*x^10 + 661086315000*x^8 + 6929405820000*x^6 + 31350462750000*x^4 + 26481508200000*x^2 + 5674608900000, 1)
 

Normalized defining polynomial

\( x^{20} + 620 x^{18} + 141050 x^{16} + 15692200 x^{14} + 956784000 x^{12} + 33380552000 x^{10} + 661086315000 x^{8} + 6929405820000 x^{6} + 31350462750000 x^{4} + 26481508200000 x^{2} + 5674608900000 \)

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$$\chi_{2480}(123,·)$, $\chi_{2480}(1,·)$, $\chi_{2480}(387,·)$, $\chi_{2480}(147,·)$, $\chi_{2480}(2081,·)$, $\chi_{2480}(969,·)$, $\chi_{2480}(523,·)$, $\chi_{2480}(721,·)$, $\chi_{2480}(2123,·)$, $\chi_{2480}(867,·)$, $\chi_{2480}(2329,·)$, $\chi_{2480}(729,·)$, $\chi_{2480}(1883,·)$, $\chi_{2480}(481,·)$, $\chi_{2480}(1827,·)$, $\chi_{2480}(1769,·)$, $\chi_{2480}(1521,·)$, $\chi_{2480}(1267,·)$, $\chi_{2480}(249,·)$, $\chi_{2480}(1083,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4}$, $\frac{1}{10} a^{5}$, $\frac{1}{50} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{50} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{500} a^{8} - \frac{1}{25} a^{4} + \frac{1}{5}$, $\frac{1}{500} a^{9} - \frac{1}{25} a^{5} + \frac{1}{5} a$, $\frac{1}{77500} a^{10} - \frac{1}{125} a^{6} - \frac{9}{25} a^{2}$, $\frac{1}{232500} a^{11} + \frac{1}{1500} a^{9} - \frac{7}{750} a^{7} - \frac{7}{150} a^{5} + \frac{7}{25} a^{3} + \frac{1}{15} a$, $\frac{1}{6975000} a^{12} - \frac{1}{697500} a^{10} - \frac{13}{22500} a^{8} + \frac{11}{1125} a^{6} + \frac{8}{375} a^{4} + \frac{14}{225} a^{2} - \frac{9}{25}$, $\frac{1}{20925000} a^{13} - \frac{1}{2092500} a^{11} + \frac{8}{16875} a^{9} + \frac{11}{3375} a^{7} + \frac{61}{2250} a^{5} - \frac{211}{675} a^{3} - \frac{4}{75} a$, $\frac{1}{62775000} a^{14} - \frac{1}{62775000} a^{12} - \frac{4}{1569375} a^{10} - \frac{8}{50625} a^{8} - \frac{31}{3375} a^{6} - \frac{29}{10125} a^{4} - \frac{56}{225} a^{2} + \frac{6}{25}$, $\frac{1}{188325000} a^{15} - \frac{1}{188325000} a^{13} - \frac{4}{4708125} a^{11} - \frac{437}{607500} a^{9} - \frac{197}{20250} a^{7} + \frac{376}{30375} a^{5} + \frac{214}{675} a^{3} - \frac{8}{25} a$, $\frac{1}{1723173750000} a^{16} + \frac{23}{6892695000} a^{14} + \frac{6221}{172317375000} a^{12} + \frac{3619}{689269500} a^{10} - \frac{14993}{185287500} a^{8} + \frac{3521}{2223450} a^{6} - \frac{57611}{6176250} a^{4} - \frac{838}{2745} a^{2} - \frac{3671}{7625}$, $\frac{1}{5169521250000} a^{17} + \frac{23}{20678085000} a^{15} + \frac{6221}{516952125000} a^{13} + \frac{3619}{2067808500} a^{11} + \frac{177791}{277931250} a^{9} + \frac{4799}{667035} a^{7} - \frac{461143}{9264375} a^{5} - \frac{1387}{8235} a^{3} - \frac{3257}{7625} a$, $\frac{1}{527034661588641270026250000} a^{18} + \frac{66151300313807}{527034661588641270026250000} a^{16} - \frac{309106987601297029}{52703466158864127002625000} a^{14} + \frac{465808058453027893}{13175866539716031750656250} a^{12} - \frac{3450978793145181829}{878391102647735450043750} a^{10} + \frac{137361631773410503097}{170011181157626216137500} a^{8} - \frac{9899340361079947411}{1889013123973624623750} a^{6} - \frac{711544824397683214}{104945173554090256875} a^{4} + \frac{533935564620279379}{2332114967868672375} a^{2} - \frac{16398713372268608}{259123885318741375}$, $\frac{1}{1581103984765923810078750000} a^{19} + \frac{66151300313807}{1581103984765923810078750000} a^{17} - \frac{309106987601297029}{158110398476592381007875000} a^{15} + \frac{465808058453027893}{39527599619148095251968750} a^{13} - \frac{3450978793145181829}{2635173307943206350131250} a^{11} + \frac{119345998522165733843}{127508385868219662103125} a^{9} - \frac{23839801420276219943}{2833519685960436935625} a^{7} - \frac{30807738243940638353}{629671041324541541250} a^{5} - \frac{1331756409674658521}{6996344903606017125} a^{3} + \frac{11808687897159889}{259123885318741375} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{10}\times C_{2000500}$, which has order $3200800000$ (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

\(\Q(\sqrt{10}) \), 4.0.246016000.10, 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R $20$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ $20$ $20$ $20$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$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}$