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Properties

Label 20.0.94964489254...000.15
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $177.37$
Ramified primes $2, 3, 5, 7, 11$
Class number $12730080$ (GRH)
Class group $[2, 4, 1591260]$ (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![1707629255641, -532283743532, 901238922048, -540414511564, 179590101080, -10964008446, 4121797687, -2839812368, 1557827049, -14347420, 11666666, -7782400, 3898134, -13868, 10402, -6936, 3470, -4, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 4*x^17 + 3470*x^16 - 6936*x^15 + 10402*x^14 - 13868*x^13 + 3898134*x^12 - 7782400*x^11 + 11666666*x^10 - 14347420*x^9 + 1557827049*x^8 - 2839812368*x^7 + 4121797687*x^6 - 10964008446*x^5 + 179590101080*x^4 - 540414511564*x^3 + 901238922048*x^2 - 532283743532*x + 1707629255641)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 4*x^17 + 3470*x^16 - 6936*x^15 + 10402*x^14 - 13868*x^13 + 3898134*x^12 - 7782400*x^11 + 11666666*x^10 - 14347420*x^9 + 1557827049*x^8 - 2839812368*x^7 + 4121797687*x^6 - 10964008446*x^5 + 179590101080*x^4 - 540414511564*x^3 + 901238922048*x^2 - 532283743532*x + 1707629255641, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} - 4 x^{17} + 3470 x^{16} - 6936 x^{15} + 10402 x^{14} - 13868 x^{13} + 3898134 x^{12} - 7782400 x^{11} + 11666666 x^{10} - 14347420 x^{9} + 1557827049 x^{8} - 2839812368 x^{7} + 4121797687 x^{6} - 10964008446 x^{5} + 179590101080 x^{4} - 540414511564 x^{3} + 901238922048 x^{2} - 532283743532 x + 1707629255641 \)

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:  \(949644892545940254829738055123773440000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.37$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(1,·)$, $\chi_{4620}(839,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(2059,·)$, $\chi_{4620}(461,·)$, $\chi_{4620}(4241,·)$, $\chi_{4620}(2899,·)$, $\chi_{4620}(3739,·)$, $\chi_{4620}(2141,·)$, $\chi_{4620}(799,·)$, $\chi_{4620}(419,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(41,·)$, $\chi_{4620}(1259,·)$, $\chi_{4620}(2099,·)$, $\chi_{4620}(3319,·)$, $\chi_{4620}(3359,·)$, $\chi_{4620}(2941,·)$, $\chi_{4620}(3401,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{1}{11}$, $\frac{1}{11} a^{12} + \frac{1}{11} a$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{2}$, $\frac{1}{11} a^{14} + \frac{1}{11} a^{3}$, $\frac{1}{451} a^{15} + \frac{19}{451} a^{14} - \frac{7}{451} a^{13} + \frac{6}{451} a^{12} + \frac{7}{451} a^{11} + \frac{8}{451} a^{10} + \frac{146}{451} a^{9} - \frac{47}{451} a^{8} + \frac{80}{451} a^{7} + \frac{107}{451} a^{6} - \frac{129}{451} a^{5} - \frac{13}{451} a^{4} + \frac{5}{41} a^{3} - \frac{4}{11} a^{2} - \frac{46}{451} a - \frac{73}{451}$, $\frac{1}{14464344187051} a^{16} - \frac{7151951533}{14464344187051} a^{15} - \frac{365845729701}{14464344187051} a^{14} - \frac{32102927214}{14464344187051} a^{13} + \frac{481106960993}{14464344187051} a^{12} - \frac{144007219402}{14464344187051} a^{11} + \frac{243007443526}{14464344187051} a^{10} + \frac{7118916537512}{14464344187051} a^{9} + \frac{6287060637881}{14464344187051} a^{8} - \frac{2191841487595}{14464344187051} a^{7} - \frac{53971990797}{14464344187051} a^{6} + \frac{92748229585}{1314940380641} a^{5} + \frac{16187235125}{1314940380641} a^{4} - \frac{2450020443554}{14464344187051} a^{3} + \frac{2055349744652}{14464344187051} a^{2} + \frac{5404806681646}{14464344187051} a + \frac{2716374123253}{14464344187051}$, $\frac{1}{14464344187051} a^{17} - \frac{10490303790}{14464344187051} a^{15} + \frac{2447589520}{352788882611} a^{14} + \frac{42168333890}{1314940380641} a^{13} + \frac{591437829864}{14464344187051} a^{12} - \frac{591698881544}{14464344187051} a^{11} + \frac{154557910642}{14464344187051} a^{10} + \frac{5090633381105}{14464344187051} a^{9} - \frac{883382585808}{14464344187051} a^{8} + \frac{4501361659341}{14464344187051} a^{7} - \frac{1065192144133}{14464344187051} a^{6} + \frac{267327991466}{14464344187051} a^{5} - \frac{5582318273616}{14464344187051} a^{4} + \frac{2020740013261}{14464344187051} a^{3} - \frac{6314783171892}{14464344187051} a^{2} + \frac{3456376332947}{14464344187051} a - \frac{4611934952623}{14464344187051}$, $\frac{1}{14464344187051} a^{18} - \frac{12830181707}{14464344187051} a^{15} + \frac{219115000758}{14464344187051} a^{14} + \frac{67840173224}{14464344187051} a^{13} - \frac{437682754812}{14464344187051} a^{12} + \frac{603196394236}{14464344187051} a^{11} - \frac{1529357532}{1314940380641} a^{10} + \frac{1585878103}{32071716601} a^{9} - \frac{168013794165}{1314940380641} a^{8} + \frac{30813772920}{352788882611} a^{7} - \frac{79278989685}{1314940380641} a^{6} - \frac{21852291765}{1314940380641} a^{5} + \frac{156881365672}{14464344187051} a^{4} + \frac{854225444561}{14464344187051} a^{3} - \frac{18407098381}{14464344187051} a^{2} + \frac{4298344105885}{14464344187051} a - \frac{1628688525853}{14464344187051}$, $\frac{1}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{19} + \frac{61942060481385781821678913383023768023783913486548655348}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{18} + \frac{18195711777563385004026022711636885945104964276040324050}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{17} + \frac{25258925055436235412871122914002440130165574058584449632}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{16} + \frac{41459054464717341340663755820861476491446032408913324409020981220}{48575939020871590511411418740492742386157069118270606988380766246653} a^{15} + \frac{62205290738112632674374600362815716934769778161449785020817810361828}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{14} - \frac{31095731165938136469061410923245201438359301263235562359813186134826}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{13} + \frac{144598841740693391354607876954221154315111553054180601911637762064565}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{12} + \frac{102931582643538859726396691176485916367300355708688825197334426287119}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{11} + \frac{312550781622996493772240841913657869158890992254315371927785781306}{7216381184918839388613226287390274367788300733756387290956787890301} a^{10} + \frac{85530913530402619879210476740853200352257057929565557647137039138535}{295871628581672414933142277783001249079320330084011878929228303502341} a^{9} - \frac{153579558736946461783380551033963078216651568049598198763692363971851}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{8} + \frac{304852635045005807667844617811735503186733105685209019053077046349532}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{7} - \frac{763426949504513893962892125886973307262990093791485272026718180595100}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{6} + \frac{375495724102552398816841826237274330650981584907402611726613332601802}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{5} + \frac{1565346625400451439063459774178099948455704648520542404931651375751849}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{4} - \frac{1401561663529818537232556828144808501456361627993397072911254249961}{79380193034107233274745489161293018045671308071320260200524666793311} a^{3} + \frac{173109776027454110412508321017927895704077465126964040905386983548869}{3254587914398396564264565055613013739872523630924130668221511338525751} a^{2} - \frac{1060586660472351131242178002907775875785929205478834715993876696227544}{3254587914398396564264565055613013739872523630924130668221511338525751} a + \frac{826084624432449211045757752538011791468783109336199703154426580657985}{3254587914398396564264565055613013739872523630924130668221511338525751}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{1591260}$, which has order $12730080$ (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:  \( 11184526.893275889 \) (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

\(\Q(\sqrt{-231}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{55}, \sqrt{-105})\), \(\Q(\zeta_{11})^+\), 10.0.9630096522760791.1, 10.10.7545432611200000.1, 10.0.2801482624803139200000.1

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 R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$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}$
7Data 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}$