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Properties

Label 20.0.92738759037...5681.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $39.66$
Ramified primes $3, 7, 11$
Class number $66$ (GRH)
Class group $[66]$ (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![4489, -28073, 147488, -237335, 399607, -203544, 361505, -146169, 202840, -53723, 67715, -13925, 16258, -2333, 2619, -286, 307, -21, 22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 22*x^18 - 21*x^17 + 307*x^16 - 286*x^15 + 2619*x^14 - 2333*x^13 + 16258*x^12 - 13925*x^11 + 67715*x^10 - 53723*x^9 + 202840*x^8 - 146169*x^7 + 361505*x^6 - 203544*x^5 + 399607*x^4 - 237335*x^3 + 147488*x^2 - 28073*x + 4489)
 
gp: K = bnfinit(x^20 - x^19 + 22*x^18 - 21*x^17 + 307*x^16 - 286*x^15 + 2619*x^14 - 2333*x^13 + 16258*x^12 - 13925*x^11 + 67715*x^10 - 53723*x^9 + 202840*x^8 - 146169*x^7 + 361505*x^6 - 203544*x^5 + 399607*x^4 - 237335*x^3 + 147488*x^2 - 28073*x + 4489, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 22 x^{18} - 21 x^{17} + 307 x^{16} - 286 x^{15} + 2619 x^{14} - 2333 x^{13} + 16258 x^{12} - 13925 x^{11} + 67715 x^{10} - 53723 x^{9} + 202840 x^{8} - 146169 x^{7} + 361505 x^{6} - 203544 x^{5} + 399607 x^{4} - 237335 x^{3} + 147488 x^{2} - 28073 x + 4489 \)

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:  \(92738759037689478010716606945681=3^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.66$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(71,·)$, $\chi_{231}(139,·)$, $\chi_{231}(76,·)$, $\chi_{231}(13,·)$, $\chi_{231}(83,·)$, $\chi_{231}(148,·)$, $\chi_{231}(218,·)$, $\chi_{231}(155,·)$, $\chi_{231}(92,·)$, $\chi_{231}(160,·)$, $\chi_{231}(230,·)$, $\chi_{231}(167,·)$, $\chi_{231}(41,·)$, $\chi_{231}(113,·)$, $\chi_{231}(62,·)$, $\chi_{231}(118,·)$, $\chi_{231}(169,·)$, $\chi_{231}(190,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} + \frac{5}{23} a^{11} - \frac{9}{23} a^{10} - \frac{6}{23} a^{9} - \frac{10}{23} a^{8} - \frac{5}{23} a^{7} + \frac{8}{23} a^{6} - \frac{9}{23} a^{5} + \frac{9}{23} a^{4} - \frac{1}{23} a^{3} - \frac{4}{23} a^{2} + \frac{10}{23} a + \frac{3}{23}$, $\frac{1}{23} a^{13} - \frac{11}{23} a^{11} - \frac{7}{23} a^{10} - \frac{3}{23} a^{9} - \frac{1}{23} a^{8} + \frac{10}{23} a^{7} - \frac{3}{23} a^{6} + \frac{8}{23} a^{5} + \frac{1}{23} a^{3} + \frac{7}{23} a^{2} - \frac{1}{23} a + \frac{8}{23}$, $\frac{1}{23} a^{14} + \frac{2}{23} a^{11} - \frac{10}{23} a^{10} + \frac{2}{23} a^{9} - \frac{8}{23} a^{8} + \frac{11}{23} a^{7} + \frac{4}{23} a^{6} - \frac{7}{23} a^{5} + \frac{8}{23} a^{4} - \frac{4}{23} a^{3} + \frac{1}{23} a^{2} + \frac{3}{23} a + \frac{10}{23}$, $\frac{1}{23} a^{15} + \frac{3}{23} a^{11} - \frac{3}{23} a^{10} + \frac{4}{23} a^{9} + \frac{8}{23} a^{8} - \frac{9}{23} a^{7} + \frac{3}{23} a^{5} + \frac{1}{23} a^{4} + \frac{3}{23} a^{3} + \frac{11}{23} a^{2} - \frac{10}{23} a - \frac{6}{23}$, $\frac{1}{23} a^{16} + \frac{5}{23} a^{11} + \frac{8}{23} a^{10} + \frac{3}{23} a^{9} - \frac{2}{23} a^{8} - \frac{8}{23} a^{7} + \frac{2}{23} a^{6} + \frac{5}{23} a^{5} - \frac{1}{23} a^{4} - \frac{9}{23} a^{3} + \frac{2}{23} a^{2} + \frac{10}{23} a - \frac{9}{23}$, $\frac{1}{23} a^{17} + \frac{6}{23} a^{11} + \frac{2}{23} a^{10} + \frac{5}{23} a^{9} - \frac{4}{23} a^{8} + \frac{4}{23} a^{7} + \frac{11}{23} a^{6} - \frac{2}{23} a^{5} - \frac{8}{23} a^{4} + \frac{7}{23} a^{3} + \frac{7}{23} a^{2} + \frac{10}{23} a + \frac{8}{23}$, $\frac{1}{1541} a^{18} + \frac{29}{1541} a^{17} + \frac{21}{1541} a^{16} + \frac{6}{1541} a^{15} + \frac{18}{1541} a^{14} - \frac{14}{1541} a^{13} - \frac{12}{1541} a^{12} + \frac{54}{1541} a^{11} + \frac{592}{1541} a^{10} + \frac{24}{67} a^{9} + \frac{726}{1541} a^{8} - \frac{453}{1541} a^{7} + \frac{444}{1541} a^{6} + \frac{280}{1541} a^{5} - \frac{672}{1541} a^{4} - \frac{259}{1541} a^{3} - \frac{515}{1541} a^{2} - \frac{262}{1541} a + \frac{1}{23}$, $\frac{1}{87070153861328478955417633542374969071} a^{19} + \frac{10259237727482110124465250589743969}{87070153861328478955417633542374969071} a^{18} - \frac{45151679603784639398970053088034302}{3785658863536020824148592762711955177} a^{17} + \frac{83638254171126767414982178031595534}{87070153861328478955417633542374969071} a^{16} - \frac{17006491509087408963966206700108331}{3785658863536020824148592762711955177} a^{15} + \frac{1007730697696585305777760056430250668}{87070153861328478955417633542374969071} a^{14} - \frac{68128446460249410542750087513016186}{87070153861328478955417633542374969071} a^{13} - \frac{960693771952475354236888799277825694}{87070153861328478955417633542374969071} a^{12} + \frac{10721384145867809556303920199065437998}{87070153861328478955417633542374969071} a^{11} + \frac{13701187748502660925703234201529421678}{87070153861328478955417633542374969071} a^{10} + \frac{16986730692542573174597604576919745482}{87070153861328478955417633542374969071} a^{9} + \frac{897635929516458700636450912194234346}{3785658863536020824148592762711955177} a^{8} + \frac{27152221983666109358969913582512867894}{87070153861328478955417633542374969071} a^{7} + \frac{24786555008262689670884921039398963732}{87070153861328478955417633542374969071} a^{6} + \frac{13165439731853115123514548354249051916}{87070153861328478955417633542374969071} a^{5} + \frac{17854580698979691458255167977925632001}{87070153861328478955417633542374969071} a^{4} - \frac{7733174635439686911454765370825542616}{87070153861328478955417633542374969071} a^{3} - \frac{22132313031218817132204385274398391507}{87070153861328478955417633542374969071} a^{2} - \frac{1985796216255954286751126427263376996}{87070153861328478955417633542374969071} a - \frac{126913091203078961698273565207433834}{1299554535243708641125636321527984613}$

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

Class group and class number

$C_{66}$, which has order $66$ (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:  \( -\frac{146502643753380322695060734927274}{3785658863536020824148592762711955177} a^{19} + \frac{133951941684383513223963556238640}{3785658863536020824148592762711955177} a^{18} - \frac{3234124095413111427168297445087474}{3785658863536020824148592762711955177} a^{17} + \frac{2822284336483241836995089013923606}{3785658863536020824148592762711955177} a^{16} - \frac{45213197011202564070835838839430508}{3785658863536020824148592762711955177} a^{15} + \frac{38467301227840170446024993466208526}{3785658863536020824148592762711955177} a^{14} - \frac{386947710222035953251306267513258946}{3785658863536020824148592762711955177} a^{13} + \frac{314428810631822861789079513921677620}{3785658863536020824148592762711955177} a^{12} - \frac{2408901470370388096390985965437547125}{3785658863536020824148592762711955177} a^{11} + \frac{1877123282674590923990021017670046654}{3785658863536020824148592762711955177} a^{10} - \frac{10084454820466429151532851151374502894}{3785658863536020824148592762711955177} a^{9} + \frac{7249657667885609098786555915271123568}{3785658863536020824148592762711955177} a^{8} - \frac{30370924162614788659699236119553309982}{3785658863536020824148592762711955177} a^{7} + \frac{19603057084266659388839531430167129194}{3785658863536020824148592762711955177} a^{6} - \frac{54864135446055304788368278136737361784}{3785658863536020824148592762711955177} a^{5} + \frac{26929939487259024573031322556249146630}{3785658863536020824148592762711955177} a^{4} - \frac{61595627150858414899398486874812707462}{3785658863536020824148592762711955177} a^{3} + \frac{29589483083216569005728129101156438720}{3785658863536020824148592762711955177} a^{2} - \frac{23865122466474675354892031992650708282}{3785658863536020824148592762711955177} a + \frac{68064334284838084185442333211952382}{56502371097552549614158100935999331} \) (order $6$)
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:  \( 1415140.16249 \) (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

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{-3}, \sqrt{77})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, 10.0.52089208083.1, 10.0.9630096522760791.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed