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Properties

Label 20.0.21757888750...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{32}\cdot 11^{10}$
Root discriminant $293.01$
Ramified primes $2, 5, 11$
Class number $6557121050$ (GRH)
Class group $[6557121050]$ (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![10617607463057, 1447969302080, 4500543123300, 600850423640, 859441629410, 76326522868, 97155742500, 3549040740, 7311226245, 53209300, 473353370, 44700, 23099640, -14840, 742950, -1164, 15050, -20, 180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 180*x^18 - 20*x^17 + 15050*x^16 - 1164*x^15 + 742950*x^14 - 14840*x^13 + 23099640*x^12 + 44700*x^11 + 473353370*x^10 + 53209300*x^9 + 7311226245*x^8 + 3549040740*x^7 + 97155742500*x^6 + 76326522868*x^5 + 859441629410*x^4 + 600850423640*x^3 + 4500543123300*x^2 + 1447969302080*x + 10617607463057)
 
gp: K = bnfinit(x^20 + 180*x^18 - 20*x^17 + 15050*x^16 - 1164*x^15 + 742950*x^14 - 14840*x^13 + 23099640*x^12 + 44700*x^11 + 473353370*x^10 + 53209300*x^9 + 7311226245*x^8 + 3549040740*x^7 + 97155742500*x^6 + 76326522868*x^5 + 859441629410*x^4 + 600850423640*x^3 + 4500543123300*x^2 + 1447969302080*x + 10617607463057, 1)
 

Normalized defining polynomial

\( x^{20} + 180 x^{18} - 20 x^{17} + 15050 x^{16} - 1164 x^{15} + 742950 x^{14} - 14840 x^{13} + 23099640 x^{12} + 44700 x^{11} + 473353370 x^{10} + 53209300 x^{9} + 7311226245 x^{8} + 3549040740 x^{7} + 97155742500 x^{6} + 76326522868 x^{5} + 859441629410 x^{4} + 600850423640 x^{3} + 4500543123300 x^{2} + 1447969302080 x + 10617607463057 \)

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:  \(21757888750734540800000000000000000000000000000000=2^{55}\cdot 5^{32}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $293.01$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4400=2^{4}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4400}(1,·)$, $\chi_{4400}(901,·)$, $\chi_{4400}(3521,·)$, $\chi_{4400}(3081,·)$, $\chi_{4400}(3981,·)$, $\chi_{4400}(461,·)$, $\chi_{4400}(2641,·)$, $\chi_{4400}(3541,·)$, $\chi_{4400}(441,·)$, $\chi_{4400}(2201,·)$, $\chi_{4400}(3101,·)$, $\chi_{4400}(1761,·)$, $\chi_{4400}(2661,·)$, $\chi_{4400}(1321,·)$, $\chi_{4400}(2221,·)$, $\chi_{4400}(881,·)$, $\chi_{4400}(1781,·)$, $\chi_{4400}(3961,·)$, $\chi_{4400}(1341,·)$, $\chi_{4400}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{77} a^{10} + \frac{2}{77} a^{8} + \frac{1}{77} a^{7} + \frac{3}{11} a^{6} + \frac{3}{11} a^{5} + \frac{34}{77} a^{4} - \frac{2}{11} a^{3} + \frac{3}{7} a^{2} + \frac{13}{77} a - \frac{3}{11}$, $\frac{1}{77} a^{11} + \frac{2}{77} a^{9} + \frac{1}{77} a^{8} - \frac{1}{77} a^{7} + \frac{3}{11} a^{6} + \frac{34}{77} a^{5} - \frac{2}{11} a^{4} + \frac{3}{7} a^{3} + \frac{13}{77} a^{2} + \frac{1}{77} a$, $\frac{1}{77} a^{12} + \frac{1}{77} a^{9} - \frac{5}{77} a^{8} - \frac{3}{77} a^{7} - \frac{8}{77} a^{6} + \frac{3}{11} a^{5} - \frac{5}{11} a^{4} - \frac{36}{77} a^{3} + \frac{12}{77} a^{2} - \frac{4}{77} a - \frac{5}{11}$, $\frac{1}{77} a^{13} - \frac{5}{77} a^{9} - \frac{5}{77} a^{8} + \frac{2}{77} a^{7} + \frac{3}{11} a^{5} + \frac{1}{11} a^{4} + \frac{26}{77} a^{3} - \frac{37}{77} a^{2} + \frac{18}{77} a + \frac{3}{11}$, $\frac{1}{539} a^{14} + \frac{4}{77} a^{9} + \frac{12}{539} a^{8} - \frac{4}{77} a^{7} + \frac{1}{11} a^{6} - \frac{4}{11} a^{5} + \frac{4}{11} a^{4} + \frac{24}{77} a^{3} + \frac{183}{539} a^{2} - \frac{38}{77} a + \frac{1}{11}$, $\frac{1}{539} a^{15} + \frac{12}{539} a^{9} - \frac{1}{77} a^{8} + \frac{3}{77} a^{7} - \frac{5}{11} a^{6} + \frac{3}{11} a^{5} - \frac{5}{11} a^{4} + \frac{36}{539} a^{3} - \frac{27}{77} a^{2} + \frac{32}{77} a + \frac{1}{11}$, $\frac{1}{539} a^{16} - \frac{2}{539} a^{10} - \frac{1}{77} a^{9} - \frac{1}{77} a^{8} - \frac{4}{77} a^{7} - \frac{3}{11} a^{6} + \frac{9}{49} a^{4} + \frac{1}{77} a^{3} - \frac{34}{77} a^{2} + \frac{25}{77} a - \frac{5}{11}$, $\frac{1}{539} a^{17} - \frac{2}{539} a^{11} - \frac{1}{77} a^{9} - \frac{2}{77} a^{8} + \frac{2}{77} a^{7} + \frac{3}{11} a^{6} + \frac{246}{539} a^{5} + \frac{5}{11} a^{4} + \frac{29}{77} a^{3} - \frac{19}{77} a^{2} + \frac{3}{7} a - \frac{3}{11}$, $\frac{1}{10769277642965454172676547665773} a^{18} - \frac{802257252674994060450394989}{1538468234709350596096649666539} a^{17} - \frac{67951456015827065457425747}{139860748609940963281513606049} a^{16} + \frac{641636548291596827032274919}{10769277642965454172676547665773} a^{15} + \frac{178454194972752355604621799}{1538468234709350596096649666539} a^{14} - \frac{782780649968502077426072538}{1538468234709350596096649666539} a^{13} + \frac{11373692530245964911505397449}{10769277642965454172676547665773} a^{12} + \frac{129101198904585166403062398}{219781176387050085156664238077} a^{11} + \frac{4400468321766043844238043}{855179674657782432516203261} a^{10} - \frac{64640327408821641215016863547}{979025240269586742970595242343} a^{9} - \frac{19929889545813391580346506885}{1538468234709350596096649666539} a^{8} + \frac{1161119427833154945471376226}{139860748609940963281513606049} a^{7} + \frac{2732746564504600194038307614187}{10769277642965454172676547665773} a^{6} - \frac{761172360433998024813271060053}{1538468234709350596096649666539} a^{5} + \frac{663521995792417882575477738558}{1538468234709350596096649666539} a^{4} - \frac{169134972668538673225391395270}{979025240269586742970595242343} a^{3} - \frac{301450874693544169070145100758}{1538468234709350596096649666539} a^{2} - \frac{44449099818940033586229091849}{219781176387050085156664238077} a - \frac{4203988111808068290002459032}{31397310912435726450952034011}$, $\frac{1}{17260763616895484902023264298055053571922911460568345825646610811} a^{19} + \frac{424399125816166265960119954940137}{17260763616895484902023264298055053571922911460568345825646610811} a^{18} - \frac{645564626282924227517842707604980611205596158844943560506383}{2465823373842212128860466328293579081703273065795477975092372973} a^{17} - \frac{4263229023811857575305954382682021984470456704430488393079047}{17260763616895484902023264298055053571922911460568345825646610811} a^{16} + \frac{967003984866081857327960909712248253133881531402994357031396}{1569160328808680445638478572550459415629355587324395075058782801} a^{15} - \frac{1062347739102387352566680760897144734781992657739960333964807}{2465823373842212128860466328293579081703273065795477975092372973} a^{14} - \frac{57696084683947487502548193318810148163325569603699553453695175}{17260763616895484902023264298055053571922911460568345825646610811} a^{13} - \frac{89273631338741177571359462820734982835341643394777037247965348}{17260763616895484902023264298055053571922911460568345825646610811} a^{12} + \frac{289309276078562488932345785397224652529195066368924340715220}{50322925996779839364499312822317940442923940118275060716170877} a^{11} + \frac{20321548361950182862911965185128203837577985649898742970646552}{17260763616895484902023264298055053571922911460568345825646610811} a^{10} - \frac{3945536255871237142972005098096834069193019176959847445964043}{114309692827122416569690492040099692529290804374624806792361661} a^{9} - \frac{53305842157086423530157080971355364077041936744002969284734352}{2465823373842212128860466328293579081703273065795477975092372973} a^{8} - \frac{765433095471824341513172388734741789236701176774851196464261532}{17260763616895484902023264298055053571922911460568345825646610811} a^{7} - \frac{2044778896111870425745289996731153800117412748698590007298572513}{17260763616895484902023264298055053571922911460568345825646610811} a^{6} + \frac{853208702696659225379263591891423196281798317909134527200522215}{2465823373842212128860466328293579081703273065795477975092372973} a^{5} - \frac{7701946702945091951109066905496829381977789122852071772646749694}{17260763616895484902023264298055053571922911460568345825646610811} a^{4} + \frac{1058810179221008472930890989141371815128047777787228238149850142}{17260763616895484902023264298055053571922911460568345825646610811} a^{3} - \frac{112860075978689454474626863084526627845542690958536511529851412}{2465823373842212128860466328293579081703273065795477975092372973} a^{2} - \frac{29897056309509449696480094028818862855329314657011363652794592}{352260481977458875551495189756225583100467580827925425013196139} a - \frac{6462051919296837384507272299309556033394991863828729484510158}{50322925996779839364499312822317940442923940118275060716170877}$

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

Class group and class number

$C_{6557121050}$, which has order $6557121050$ (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:  \( 42294001.73672045 \) (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

\(\Q(\sqrt{2}) \), 4.0.247808.2, 5.5.390625.1, 10.10.5000000000000000.1

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 $20$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ $20$

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
5Data not computed
11Data not computed