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Properties

Label 20.0.36528267636...5625.8
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 11^{16}\cdot 13^{10}$
Root discriminant $95.09$
Ramified primes $3, 5, 11, 13$
Class number $1806200$ (GRH)
Class group $[5, 361240]$ (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![76208864731, -41930249223, 56142388702, -24494214376, 18098949794, -6502554995, 3446562112, -1046061945, 438995984, -114345518, 39843720, -8957981, 2652441, -511863, 129865, -21030, 4535, -576, 101, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 101*x^18 - 576*x^17 + 4535*x^16 - 21030*x^15 + 129865*x^14 - 511863*x^13 + 2652441*x^12 - 8957981*x^11 + 39843720*x^10 - 114345518*x^9 + 438995984*x^8 - 1046061945*x^7 + 3446562112*x^6 - 6502554995*x^5 + 18098949794*x^4 - 24494214376*x^3 + 56142388702*x^2 - 41930249223*x + 76208864731)
 
gp: K = bnfinit(x^20 - 8*x^19 + 101*x^18 - 576*x^17 + 4535*x^16 - 21030*x^15 + 129865*x^14 - 511863*x^13 + 2652441*x^12 - 8957981*x^11 + 39843720*x^10 - 114345518*x^9 + 438995984*x^8 - 1046061945*x^7 + 3446562112*x^6 - 6502554995*x^5 + 18098949794*x^4 - 24494214376*x^3 + 56142388702*x^2 - 41930249223*x + 76208864731, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 101 x^{18} - 576 x^{17} + 4535 x^{16} - 21030 x^{15} + 129865 x^{14} - 511863 x^{13} + 2652441 x^{12} - 8957981 x^{11} + 39843720 x^{10} - 114345518 x^{9} + 438995984 x^{8} - 1046061945 x^{7} + 3446562112 x^{6} - 6502554995 x^{5} + 18098949794 x^{4} - 24494214376 x^{3} + 56142388702 x^{2} - 41930249223 x + 76208864731 \)

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:  \(3652826763616533724927636470896103515625=3^{10}\cdot 5^{10}\cdot 11^{16}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.09$
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:  $3, 5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2145=3\cdot 5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2145}(896,·)$, $\chi_{2145}(1,·)$, $\chi_{2145}(196,·)$, $\chi_{2145}(389,·)$, $\chi_{2145}(584,·)$, $\chi_{2145}(586,·)$, $\chi_{2145}(779,·)$, $\chi_{2145}(1676,·)$, $\chi_{2145}(1871,·)$, $\chi_{2145}(1169,·)$, $\chi_{2145}(2066,·)$, $\chi_{2145}(1171,·)$, $\chi_{2145}(664,·)$, $\chi_{2145}(1754,·)$, $\chi_{2145}(859,·)$, $\chi_{2145}(1054,·)$, $\chi_{2145}(1951,·)$, $\chi_{2145}(1444,·)$, $\chi_{2145}(2029,·)$, $\chi_{2145}(311,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{331} a^{18} + \frac{122}{331} a^{17} - \frac{107}{331} a^{16} - \frac{36}{331} a^{15} - \frac{83}{331} a^{14} + \frac{147}{331} a^{13} + \frac{70}{331} a^{12} + \frac{46}{331} a^{11} + \frac{138}{331} a^{10} - \frac{67}{331} a^{9} + \frac{138}{331} a^{8} - \frac{34}{331} a^{7} + \frac{151}{331} a^{6} - \frac{65}{331} a^{5} - \frac{95}{331} a^{4} - \frac{72}{331} a^{3} + \frac{4}{331} a^{2} + \frac{21}{331} a + \frac{149}{331}$, $\frac{1}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{19} + \frac{25269925934931451209119296826431942127029310754645544387954188907244964952}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{18} - \frac{485325253714218669860358881631837369762676337595680162491448164408858094863}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{17} + \frac{11674347857614436678816323920254852237558489744839817369152748717646459283659}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{16} - \frac{9396503300394381649195217333411587505326337660246670789748681425528751704046}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{15} + \frac{16987488194017769932855663578726378780091675250323314126584645642205767227062}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{14} - \frac{8931231001173742295374981390337999752944811715502363227645798195478637711060}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{13} - \frac{19799075174965494714930947433453055226940339440636894039084494049370000828904}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{12} + \frac{345180854694618687595149216327326457220062454049958193052086696222109600467}{741494887514256346655301394532892144893432125839097903993483713167882429147} a^{11} + \frac{20079018420665823056518894409743458319329036065568331165495849100205246205179}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{10} - \frac{23334455012247020135205310600614348124681238987930242484535983080921390886753}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{9} - \frac{7618533655196708276380873383852278281993385799547268880838551591368578836984}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{8} + \frac{15828212011203144819644190251450443031833538096618597423500306846697029982478}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{7} - \frac{8489435557062664395982514585665150762949545016128303769580164205281185989808}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{6} - \frac{1387883835786197034903941290774644303675756830666753597569018265167438952}{32238908152793754202404408457951832386670961993004256695368857094255757789} a^{5} + \frac{4437697476820416581497139901987955324533065348002689950602570889198446886651}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{4} + \frac{14661797788288145317933598123842196081077147909863009477110761394548967307659}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{3} - \frac{21939276260604407198210123921223764179800824661642505335095189187167815799193}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a^{2} + \frac{15413717027449283777613872703508130643727940772928886090559469065910659879174}{49680157463455175225905193433703773707859952431219559567563408782248122752849} a - \frac{13088540179590484325823928040381761549309240615452316834641580720472764285082}{49680157463455175225905193433703773707859952431219559567563408782248122752849}$

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

Class group and class number

$C_{5}\times C_{361240}$, which has order $1806200$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\zeta_{11})^+\), 10.0.60438619802379121875.1, 10.0.19340358336761319.3, 10.10.669871503125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
13Data not computed