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Properties

Label 20.0.20128753876...9769.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 19^{10}$
Root discriminant $65.34$
Ramified primes $3, 11, 19$
Class number $44110$ (GRH)
Class group $[44110]$ (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![185992621, -93204582, 187924326, -94586934, 96425775, -46229958, 32582283, -14323286, 7888313, -3119270, 1417274, -496518, 190380, -58074, 18820, -4868, 1305, -268, 57, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 57*x^18 - 268*x^17 + 1305*x^16 - 4868*x^15 + 18820*x^14 - 58074*x^13 + 190380*x^12 - 496518*x^11 + 1417274*x^10 - 3119270*x^9 + 7888313*x^8 - 14323286*x^7 + 32582283*x^6 - 46229958*x^5 + 96425775*x^4 - 94586934*x^3 + 187924326*x^2 - 93204582*x + 185992621)
 
gp: K = bnfinit(x^20 - 8*x^19 + 57*x^18 - 268*x^17 + 1305*x^16 - 4868*x^15 + 18820*x^14 - 58074*x^13 + 190380*x^12 - 496518*x^11 + 1417274*x^10 - 3119270*x^9 + 7888313*x^8 - 14323286*x^7 + 32582283*x^6 - 46229958*x^5 + 96425775*x^4 - 94586934*x^3 + 187924326*x^2 - 93204582*x + 185992621, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 57 x^{18} - 268 x^{17} + 1305 x^{16} - 4868 x^{15} + 18820 x^{14} - 58074 x^{13} + 190380 x^{12} - 496518 x^{11} + 1417274 x^{10} - 3119270 x^{9} + 7888313 x^{8} - 14323286 x^{7} + 32582283 x^{6} - 46229958 x^{5} + 96425775 x^{4} - 94586934 x^{3} + 187924326 x^{2} - 93204582 x + 185992621 \)

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:  \(2012875387628441371387683404151209769=3^{10}\cdot 11^{18}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.34$
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:  $3, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(627=3\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{627}(512,·)$, $\chi_{627}(1,·)$, $\chi_{627}(322,·)$, $\chi_{627}(134,·)$, $\chi_{627}(265,·)$, $\chi_{627}(398,·)$, $\chi_{627}(400,·)$, $\chi_{627}(590,·)$, $\chi_{627}(37,·)$, $\chi_{627}(227,·)$, $\chi_{627}(229,·)$, $\chi_{627}(362,·)$, $\chi_{627}(493,·)$, $\chi_{627}(305,·)$, $\chi_{627}(626,·)$, $\chi_{627}(115,·)$, $\chi_{627}(248,·)$, $\chi_{627}(569,·)$, $\chi_{627}(58,·)$, $\chi_{627}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{56443886891869865288553228781212796515897735827511208151498} a^{19} + \frac{2118448718550882212049578751644206103620832462217347360281}{56443886891869865288553228781212796515897735827511208151498} a^{18} - \frac{2794201434621478472940088644300002729982068699443491870972}{28221943445934932644276614390606398257948867913755604075749} a^{17} - \frac{3817042776353978288765342391133099972513959818033679273337}{56443886891869865288553228781212796515897735827511208151498} a^{16} - \frac{2880252957667890163073681404028824093421878873702102356489}{28221943445934932644276614390606398257948867913755604075749} a^{15} - \frac{202592427891689596150381995636787307136628080073638691106}{28221943445934932644276614390606398257948867913755604075749} a^{14} - \frac{6122701896932955414158174061718666928752131044719260674405}{56443886891869865288553228781212796515897735827511208151498} a^{13} + \frac{6843886573578178824970290117689722484900602916760015646367}{28221943445934932644276614390606398257948867913755604075749} a^{12} - \frac{6907831766173871883262258658777261150267159952183350741552}{28221943445934932644276614390606398257948867913755604075749} a^{11} - \frac{3887789184361893384873573722393675001500132920774213965526}{28221943445934932644276614390606398257948867913755604075749} a^{10} + \frac{10633417284258446585291137065832449691595374043926688807120}{28221943445934932644276614390606398257948867913755604075749} a^{9} + \frac{17083208627724508180604058495299459150571098489807142123565}{56443886891869865288553228781212796515897735827511208151498} a^{8} - \frac{19537543888774527441466783190304283086450374427611889816077}{56443886891869865288553228781212796515897735827511208151498} a^{7} + \frac{19183638421910484174950297150632956904796941457232236130563}{56443886891869865288553228781212796515897735827511208151498} a^{6} + \frac{27008802134868546604205539728687341011557740501346297497153}{56443886891869865288553228781212796515897735827511208151498} a^{5} - \frac{8890096784885728301854923496078505753911355859753015059482}{28221943445934932644276614390606398257948867913755604075749} a^{4} + \frac{18495245716599174445040414117276671233258809597111390869821}{56443886891869865288553228781212796515897735827511208151498} a^{3} - \frac{16828968540752987093957663638544883066095735620597003267207}{56443886891869865288553228781212796515897735827511208151498} a^{2} + \frac{8084152855137962442368502254018098555594338097185300605755}{28221943445934932644276614390606398257948867913755604075749} a + \frac{90881100517188757618774537499404213650945768320451494003}{430869365586792864798116250238265622258761342194742046958}$

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

Class group and class number

$C_{44110}$, which has order $44110$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-627}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-19}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\), 10.0.1418758396496190387.1, 10.0.530773810885219.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
19Data not computed