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Properties

Label 20.0.20038303066...000.30
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $462.48$
Ramified primes $2, 3, 5, 11$
Class number $2817907840$ (GRH)
Class group $[2, 2, 2, 2, 2, 88059620]$ (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![361518201831424, 0, 113417399828480, 0, 14817146572800, 0, -1171735656960, 0, 31603002640, 0, 754489384, 0, -16032335, 0, -111100, 0, 12870, 0, 220, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 220*x^18 + 12870*x^16 - 111100*x^14 - 16032335*x^12 + 754489384*x^10 + 31603002640*x^8 - 1171735656960*x^6 + 14817146572800*x^4 + 113417399828480*x^2 + 361518201831424)
 
gp: K = bnfinit(x^20 + 220*x^18 + 12870*x^16 - 111100*x^14 - 16032335*x^12 + 754489384*x^10 + 31603002640*x^8 - 1171735656960*x^6 + 14817146572800*x^4 + 113417399828480*x^2 + 361518201831424, 1)
 

Normalized defining polynomial

\( x^{20} + 220 x^{18} + 12870 x^{16} - 111100 x^{14} - 16032335 x^{12} + 754489384 x^{10} + 31603002640 x^{8} - 1171735656960 x^{6} + 14817146572800 x^{4} + 113417399828480 x^{2} + 361518201831424 \)

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:  \(200383030666749741651348266601562500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $462.48$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(1579,·)$, $\chi_{3300}(2821,·)$, $\chi_{3300}(961,·)$, $\chi_{3300}(1099,·)$, $\chi_{3300}(1741,·)$, $\chi_{3300}(1681,·)$, $\chi_{3300}(1811,·)$, $\chi_{3300}(139,·)$, $\chi_{3300}(1691,·)$, $\chi_{3300}(2719,·)$, $\chi_{3300}(929,·)$, $\chi_{3300}(2659,·)$, $\chi_{3300}(389,·)$, $\chi_{3300}(1769,·)$, $\chi_{3300}(1451,·)$, $\chi_{3300}(749,·)$, $\chi_{3300}(431,·)$, $\chi_{3300}(1271,·)$, $\chi_{3300}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{8} - \frac{1}{32} a^{6} + \frac{1}{64} a^{4} + \frac{1}{16} a^{2}$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} + \frac{1}{128} a^{5} + \frac{1}{32} a^{3}$, $\frac{1}{1408} a^{10} + \frac{7}{128} a^{6} + \frac{1}{64} a^{4} - \frac{5}{16} a^{2}$, $\frac{1}{5632} a^{11} + \frac{1}{512} a^{9} + \frac{5}{512} a^{7} - \frac{13}{512} a^{5} + \frac{3}{128} a^{3} - \frac{1}{8} a$, $\frac{1}{214016} a^{12} - \frac{3}{11264} a^{10} - \frac{1}{1024} a^{8} + \frac{391}{19456} a^{6} + \frac{9}{256} a^{4} + \frac{9}{19}$, $\frac{1}{214016} a^{13} - \frac{1}{11264} a^{11} + \frac{1}{1024} a^{9} + \frac{581}{19456} a^{7} + \frac{5}{512} a^{5} + \frac{3}{128} a^{3} + \frac{53}{152} a$, $\frac{1}{214016} a^{14} + \frac{1}{5632} a^{10} - \frac{21}{4864} a^{8} + \frac{1}{1024} a^{6} + \frac{5}{256} a^{4} + \frac{125}{304} a^{2}$, $\frac{1}{428032} a^{15} - \frac{1}{11264} a^{11} - \frac{1}{4864} a^{9} - \frac{35}{2048} a^{7} + \frac{5}{128} a^{5} + \frac{481}{2432} a^{3} - \frac{3}{8} a$, $\frac{1}{65060864} a^{16} - \frac{13}{16265216} a^{14} + \frac{3}{32530432} a^{12} + \frac{283}{1478656} a^{10} + \frac{25971}{5914624} a^{8} - \frac{35479}{739328} a^{6} - \frac{32085}{369664} a^{4} - \frac{6641}{23104} a^{2} + \frac{115}{361}$, $\frac{1}{109822738432} a^{17} - \frac{13693}{27455684608} a^{15} - \frac{97277}{54911369216} a^{13} + \frac{118633}{27455684608} a^{11} + \frac{28650611}{9983885312} a^{9} - \frac{22791855}{1247985664} a^{7} + \frac{35836875}{623992832} a^{5} - \frac{2183053}{38999552} a^{3} + \frac{31241}{152342} a$, $\frac{1}{454850683676919898824008464192820324335616} a^{18} + \frac{821665682590107221855791577956195}{113712670919229974706002116048205081083904} a^{16} - \frac{113730205683347312391092382714606013}{227425341838459949412004232096410162167808} a^{14} + \frac{31199508625219473341162746797052297}{113712670919229974706002116048205081083904} a^{12} - \frac{139935869089557214777248231677163409807}{454850683676919898824008464192820324335616} a^{10} - \frac{5832483491708673537585099572608988143}{5168757769055907941181914365827503685632} a^{8} - \frac{43041418813625217068404272784899176309}{2584378884527953970590957182913751842816} a^{6} - \frac{12444846238379317769100509862099166429}{161523680282997123161934823932109490176} a^{4} - \frac{950553271995688529289082041051469}{78868984513182189043913488248100337} a^{2} - \frac{179587818737124159498774966802687}{373786656460579094994850655204267}$, $\frac{1}{3638805469415359190592067713542562594684928} a^{19} - \frac{2528717804964688437563329364017}{909701367353839797648016928385640648671232} a^{17} + \frac{1971390510347296723253977674474067811}{1819402734707679595296033856771281297342464} a^{15} + \frac{2045803974097438914289475715282038033}{909701367353839797648016928385640648671232} a^{13} - \frac{89125535092683038099205155877967871183}{3638805469415359190592067713542562594684928} a^{11} + \frac{144865231462362414276192853509996985811}{41350062152447263529455314926620029485056} a^{9} + \frac{115070154635261964319623318649842839619}{20675031076223631764727657463310014742528} a^{7} - \frac{2822765522133700498896189094434803531}{323047360565994246323869647864218980352} a^{5} - \frac{80231168724477646331964919840675009}{382757536215632993274727070929169408} a^{3} - \frac{216929350574722459946220063869320219}{1261903752210915024702615811969605392} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{88059620}$, which has order $2817907840$ (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:  \( 1206432246219.7283 \) (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

\(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-15}, \sqrt{-33})\), 5.5.5719140625.4, 10.0.89528326392656250000000000.4, 10.0.39740911928558349609375.12, 10.10.1842146633593750000000000.3

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$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}$
5Data not computed
11Data not computed