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Properties

Label 20.0.83033970074...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 41^{19}$
Root discriminant $394.42$
Ramified primes $2, 3, 5, 41$
Class number $3134988288$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 510252]$ (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![5515361128125, 0, 22379201943750, 0, 21957374750625, 0, 5504713171875, 0, 606522532500, 0, 34317553500, 0, 1043541225, 0, 16992450, 0, 145755, 0, 615, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 615*x^18 + 145755*x^16 + 16992450*x^14 + 1043541225*x^12 + 34317553500*x^10 + 606522532500*x^8 + 5504713171875*x^6 + 21957374750625*x^4 + 22379201943750*x^2 + 5515361128125)
 
gp: K = bnfinit(x^20 + 615*x^18 + 145755*x^16 + 16992450*x^14 + 1043541225*x^12 + 34317553500*x^10 + 606522532500*x^8 + 5504713171875*x^6 + 21957374750625*x^4 + 22379201943750*x^2 + 5515361128125, 1)
 

Normalized defining polynomial

\( x^{20} + 615 x^{18} + 145755 x^{16} + 16992450 x^{14} + 1043541225 x^{12} + 34317553500 x^{10} + 606522532500 x^{8} + 5504713171875 x^{6} + 21957374750625 x^{4} + 22379201943750 x^{2} + 5515361128125 \)

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:  \(8303397007448741046809295126072534048000000000000000=2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $394.42$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2460=2^{2}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{2460}(1,·)$, $\chi_{2460}(323,·)$, $\chi_{2460}(1861,·)$, $\chi_{2460}(961,·)$, $\chi_{2460}(2123,·)$, $\chi_{2460}(143,·)$, $\chi_{2460}(2387,·)$, $\chi_{2460}(409,·)$, $\chi_{2460}(1727,·)$, $\chi_{2460}(863,·)$, $\chi_{2460}(1187,·)$, $\chi_{2460}(1607,·)$, $\chi_{2460}(1009,·)$, $\chi_{2460}(1907,·)$, $\chi_{2460}(1909,·)$, $\chi_{2460}(1849,·)$, $\chi_{2460}(443,·)$, $\chi_{2460}(769,·)$, $\chi_{2460}(1021,·)$, $\chi_{2460}(2101,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{135} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{135} a^{5} + \frac{1}{3} a$, $\frac{1}{1215} a^{6} + \frac{1}{405} a^{4} - \frac{4}{27} a^{2}$, $\frac{1}{1215} a^{7} + \frac{1}{405} a^{5} - \frac{1}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{18225} a^{8} - \frac{4}{27} a^{2}$, $\frac{1}{164025} a^{9} + \frac{1}{3645} a^{7} - \frac{4}{1215} a^{5} - \frac{4}{243} a^{3} + \frac{1}{9} a$, $\frac{1}{492075} a^{10} - \frac{1}{54675} a^{8} - \frac{1}{3645} a^{6} - \frac{11}{3645} a^{4} - \frac{4}{27} a^{2}$, $\frac{1}{492075} a^{11} - \frac{1}{3645} a^{7} - \frac{2}{3645} a^{5} - \frac{4}{81} a^{3}$, $\frac{1}{22143375} a^{12} + \frac{1}{1476225} a^{10} + \frac{4}{164025} a^{8} - \frac{7}{32805} a^{6} + \frac{22}{10935} a^{4} + \frac{13}{81} a^{2}$, $\frac{1}{22143375} a^{13} + \frac{1}{1476225} a^{11} + \frac{11}{32805} a^{7} - \frac{23}{10935} a^{5} + \frac{10}{243} a^{3} + \frac{2}{9} a$, $\frac{1}{597871125} a^{14} + \frac{4}{199290375} a^{12} - \frac{7}{13286025} a^{10} - \frac{101}{4428675} a^{8} - \frac{79}{295245} a^{6} - \frac{211}{98415} a^{4} - \frac{100}{729} a^{2} + \frac{1}{3}$, $\frac{1}{597871125} a^{15} + \frac{4}{199290375} a^{13} - \frac{7}{13286025} a^{11} + \frac{7}{4428675} a^{9} + \frac{2}{295245} a^{7} - \frac{292}{98415} a^{5} - \frac{40}{729} a^{3} + \frac{4}{9} a$, $\frac{1}{13479004513125} a^{16} + \frac{134}{299533433625} a^{14} - \frac{731}{99844477875} a^{12} - \frac{269}{19968895575} a^{10} - \frac{54649}{2218766175} a^{8} + \frac{35267}{147917745} a^{6} + \frac{160541}{147917745} a^{4} + \frac{122339}{1095687} a^{2} + \frac{85}{4509}$, $\frac{1}{13479004513125} a^{17} + \frac{134}{299533433625} a^{15} - \frac{731}{99844477875} a^{13} - \frac{269}{19968895575} a^{11} - \frac{541}{2218766175} a^{9} - \frac{9179}{29583549} a^{7} - \frac{326431}{147917745} a^{5} + \frac{9614}{1095687} a^{3} + \frac{586}{4509} a$, $\frac{1}{1088837313884352241875} a^{18} + \frac{2192383}{72589154258956816125} a^{16} + \frac{1717183661}{8065461584328535125} a^{14} - \frac{139763581036}{8065461584328535125} a^{12} + \frac{101351356712}{537697438955235675} a^{10} - \frac{1616456522299}{59744159883915075} a^{8} - \frac{4259176604683}{11948831976783015} a^{6} - \frac{4359551747933}{3982943992261005} a^{4} - \frac{1226118600896}{29503288831563} a^{2} + \frac{15561340019}{121412711241}$, $\frac{1}{1088837313884352241875} a^{19} + \frac{2192383}{72589154258956816125} a^{17} + \frac{1717183661}{8065461584328535125} a^{15} - \frac{139763581036}{8065461584328535125} a^{13} + \frac{101351356712}{537697438955235675} a^{11} - \frac{159503987407}{59744159883915075} a^{9} - \frac{981033401176}{11948831976783015} a^{7} - \frac{1527538990288}{796588798452201} a^{5} + \frac{1202135623924}{29503288831563} a^{3} + \frac{29051641268}{121412711241} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{510252}$, which has order $3134988288$ (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:  \( 3541438824.6395073 \) (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

\(\Q(\sqrt{205}) \), 4.0.1240578000.2, 5.5.2825761.1, 10.10.1023068544981128125.1

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 ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
5Data not computed
41Data not computed