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Properties

Label 20.0.44322362984...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}$
Root discriminant $340.67$
Ramified primes $2, 3, 5, 31$
Class number $430890240$ (GRH)
Class group $[2, 2, 2, 2, 2, 202, 66660]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![265070961, 0, 217939545108, 0, 412083596367, 0, 246505701708, 0, 56627249607, 0, 6019595028, 0, 307624392, 0, 7247988, 0, 83997, 0, 468, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 468*x^18 + 83997*x^16 + 7247988*x^14 + 307624392*x^12 + 6019595028*x^10 + 56627249607*x^8 + 246505701708*x^6 + 412083596367*x^4 + 217939545108*x^2 + 265070961)
 
gp: K = bnfinit(x^20 + 468*x^18 + 83997*x^16 + 7247988*x^14 + 307624392*x^12 + 6019595028*x^10 + 56627249607*x^8 + 246505701708*x^6 + 412083596367*x^4 + 217939545108*x^2 + 265070961, 1)
 

Normalized defining polynomial

\( x^{20} + 468 x^{18} + 83997 x^{16} + 7247988 x^{14} + 307624392 x^{12} + 6019595028 x^{10} + 56627249607 x^{8} + 246505701708 x^{6} + 412083596367 x^{4} + 217939545108 x^{2} + 265070961 \)

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:  \(443223629840246396758117587996045905756160000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $340.67$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(1349,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(371,·)$, $\chi_{3720}(1999,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2479,·)$, $\chi_{3720}(1759,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(3611,·)$, $\chi_{3720}(869,·)$, $\chi_{3720}(1639,·)$, $\chi_{3720}(3371,·)$, $\chi_{3720}(1709,·)$, $\chi_{3720}(3439,·)$, $\chi_{3720}(3251,·)$, $\chi_{3720}(1331,·)$, $\chi_{3720}(1589,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(3629,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{405} a^{8} + \frac{2}{45} a^{4} + \frac{1}{5}$, $\frac{1}{405} a^{9} + \frac{2}{45} a^{5} + \frac{1}{5} a$, $\frac{1}{1215} a^{10} + \frac{2}{135} a^{6} + \frac{1}{15} a^{2}$, $\frac{1}{1215} a^{11} + \frac{2}{135} a^{7} + \frac{1}{15} a^{3}$, $\frac{1}{3645} a^{12} + \frac{2}{45} a^{4} - \frac{2}{5}$, $\frac{1}{3645} a^{13} + \frac{2}{45} a^{5} - \frac{2}{5} a$, $\frac{1}{10935} a^{14} + \frac{2}{135} a^{6} - \frac{2}{15} a^{2}$, $\frac{1}{10935} a^{15} + \frac{2}{135} a^{7} - \frac{2}{15} a^{3}$, $\frac{1}{164025} a^{16} - \frac{1}{18225} a^{12} + \frac{1}{2025} a^{8} + \frac{1}{25} a^{4} + \frac{6}{25}$, $\frac{1}{10989675} a^{17} - \frac{4}{244215} a^{15} + \frac{49}{1221075} a^{13} - \frac{2}{9045} a^{11} + \frac{1}{1675} a^{9} - \frac{13}{1809} a^{7} + \frac{244}{15075} a^{5} - \frac{53}{335} a^{3} + \frac{236}{1675} a$, $\frac{1}{482156647054269185451070128450479625} a^{18} - \frac{255319374629668448695077623687}{160718882351423061817023376150159875} a^{16} + \frac{212016949826182351202416185911}{5952551198200854141371236153709625} a^{14} - \frac{554395929676263043496273879213}{17857653594602562424113708461128875} a^{12} - \frac{1794886947154498819850248655584}{5952551198200854141371236153709625} a^{10} - \frac{59725844740482426313795326821}{73488286397541409152731310539625} a^{8} - \frac{2138844428927097270578054932487}{220464859192624227458193931618875} a^{6} + \frac{272768173658262789164174144648}{24496095465847136384243770179875} a^{4} - \frac{10506258243879558037177768173329}{73488286397541409152731310539625} a^{2} + \frac{46298572862110522141958531144}{365613365161897557973787614625}$, $\frac{1}{482156647054269185451070128450479625} a^{19} - \frac{6702286319578109272902045742}{160718882351423061817023376150159875} a^{17} + \frac{562928176446167542012491034808}{17857653594602562424113708461128875} a^{15} + \frac{203267023464367670499202374238}{1984183732733618047123745384569875} a^{13} + \frac{325670570784507016397719509241}{5952551198200854141371236153709625} a^{11} - \frac{1071490027553417124671268154522}{1984183732733618047123745384569875} a^{9} - \frac{2943193832283271898120387684662}{220464859192624227458193931618875} a^{7} + \frac{4326853992553280599303359888712}{220464859192624227458193931618875} a^{5} + \frac{2469598883017807434512608314432}{24496095465847136384243770179875} a^{3} - \frac{11712649174598684065586652557957}{24496095465847136384243770179875} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{202}\times C_{66660}$, which has order $430890240$ (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:  \( 2425171378.8887568 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{155}) \), \(\Q(\sqrt{-186}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-30}, \sqrt{155})\), 5.5.923521.1, 10.10.84606790914147200000.1, 10.0.210528769967490760704.1, 10.0.21222658262851891200000.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$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}$