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Properties

Label 20.0.58299958569...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{35}\cdot 11^{18}$
Root discriminant $973.38$
Ramified primes $2, 5, 11$
Class number $64742292360$ (GRH)
Class group $[3, 3, 6, 1198931340]$ (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![29061634614163360, 0, 59652203230294400, 0, 16694916231268000, 0, 1163689163929600, 0, 39102956021800, 0, 761619477520, 0, 9223517600, 0, 70613400, 0, 332640, 0, 880, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 880*x^18 + 332640*x^16 + 70613400*x^14 + 9223517600*x^12 + 761619477520*x^10 + 39102956021800*x^8 + 1163689163929600*x^6 + 16694916231268000*x^4 + 59652203230294400*x^2 + 29061634614163360)
 
gp: K = bnfinit(x^20 + 880*x^18 + 332640*x^16 + 70613400*x^14 + 9223517600*x^12 + 761619477520*x^10 + 39102956021800*x^8 + 1163689163929600*x^6 + 16694916231268000*x^4 + 59652203230294400*x^2 + 29061634614163360, 1)
 

Normalized defining polynomial

\( x^{20} + 880 x^{18} + 332640 x^{16} + 70613400 x^{14} + 9223517600 x^{12} + 761619477520 x^{10} + 39102956021800 x^{8} + 1163689163929600 x^{6} + 16694916231268000 x^{4} + 59652203230294400 x^{2} + 29061634614163360 \)

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:  \(582999585691243011742105600000000000000000000000000000000000=2^{55}\cdot 5^{35}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $973.38$
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$$\chi_{4400}(2587,·)$, $\chi_{4400}(1,·)$, $\chi_{4400}(1283,·)$, $\chi_{4400}(2161,·)$, $\chi_{4400}(9,·)$, $\chi_{4400}(2507,·)$, $\chi_{4400}(81,·)$, $\chi_{4400}(1603,·)$, $\chi_{4400}(729,·)$, $\chi_{4400}(667,·)$, $\chi_{4400}(2723,·)$, $\chi_{4400}(2243,·)$, $\chi_{4400}(3441,·)$, $\chi_{4400}(489,·)$, $\chi_{4400}(1521,·)$, $\chi_{4400}(563,·)$, $\chi_{4400}(1227,·)$, $\chi_{4400}(169,·)$, $\chi_{4400}(1849,·)$, $\chi_{4400}(2747,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{44} a^{10}$, $\frac{1}{44} a^{11}$, $\frac{1}{88} a^{12}$, $\frac{1}{88} a^{13}$, $\frac{1}{3256} a^{14} + \frac{1}{407} a^{12} - \frac{5}{814} a^{10} - \frac{3}{37} a^{8} + \frac{5}{74} a^{6} - \frac{1}{37} a^{4} + \frac{14}{37} a^{2} + \frac{4}{37}$, $\frac{1}{172568} a^{15} + \frac{933}{172568} a^{13} + \frac{164}{21571} a^{11} - \frac{715}{7844} a^{9} - \frac{386}{1961} a^{7} + \frac{221}{1961} a^{5} - \frac{393}{1961} a^{3} - \frac{662}{1961} a$, $\frac{1}{14840848} a^{16} + \frac{69}{7420424} a^{14} + \frac{8543}{3710212} a^{12} + \frac{7375}{1855106} a^{10} + \frac{9824}{84323} a^{8} + \frac{40395}{168646} a^{6} + \frac{41583}{168646} a^{4} + \frac{9792}{84323} a^{2} + \frac{747}{1591}$, $\frac{1}{14840848} a^{17} - \frac{17}{7420424} a^{15} + \frac{21171}{7420424} a^{13} - \frac{20833}{1855106} a^{11} + \frac{16463}{337292} a^{9} + \frac{11232}{84323} a^{7} + \frac{3571}{168646} a^{5} - \frac{40733}{84323} a^{3} + \frac{12200}{84323} a$, $\frac{1}{11736098857387766351566097598005149128365233387408} a^{18} + \frac{1088873796342830689287925612740504152833}{110717913748941191995906581113256123852502201768} a^{16} - \frac{2602212000170665659933447932708656035627581}{66682379871521399724807372715938347320257007883} a^{14} - \frac{14544040838133795704392246473090160806289127915}{2934024714346941587891524399501287282091308346852} a^{12} + \frac{7030068123575365350159886392821192590221987155}{1467012357173470793945762199750643641045654173426} a^{10} - \frac{1193428141603924405316859888578856147098697759}{66682379871521399724807372715938347320257007883} a^{8} - \frac{3769718951143060963949715434825139787311554861}{66682379871521399724807372715938347320257007883} a^{6} + \frac{13165403810807520238510386791243164819358026287}{133364759743042799449614745431876694640514015766} a^{4} + \frac{26860019443985425108722396294208147022992552245}{66682379871521399724807372715938347320257007883} a^{2} - \frac{490598732021913242934794330052652968232899423}{1258158110783422636317120239923365043778434111}$, $\frac{1}{271303397286232994749153478173085032400419100216710736} a^{19} - \frac{425797774753440045819203332927444427131952835}{24663945207839363159013952561189548400038100019700976} a^{17} - \frac{2925780850216854023234031138911941852765811719}{67825849321558248687288369543271258100104775054177684} a^{15} + \frac{393739438665039828450661782593684770466549247984039}{135651698643116497374576739086542516200209550108355368} a^{13} - \frac{177637586173899193092433290653477421794499032923435}{16956462330389562171822092385817814525026193763544421} a^{11} + \frac{190898801307687645788850887710231423177864743893797}{3082993150979920394876744070148693550004762502462622} a^{9} + \frac{260092036397707383537347552359084612671993949972257}{1541496575489960197438372035074346775002381251231311} a^{7} - \frac{650466734003512198141297485062971326685760531847925}{3082993150979920394876744070148693550004762502462622} a^{5} - \frac{596408197423638129078276834621224367090410290886927}{1541496575489960197438372035074346775002381251231311} a^{3} - \frac{229588750900446532019822501893913392938917511049573}{1541496575489960197438372035074346775002381251231311} a$

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

Class group and class number

$C_{3}\times C_{3}\times C_{6}\times C_{1198931340}$, which has order $64742292360$ (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:  \( 94527714076.15956 \) (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

\(\Q(\sqrt{10}) \), 4.0.30976000.2, 5.5.5719140625.3, 10.10.5358972025000000000000000.2

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R $20$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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