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Properties

Label 20.0.11659991713...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $898.12$
Ramified primes $2, 5, 11$
Class number $49459158180$ (GRH)
Class group $[6, 8243193030]$ (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![103893806112800, 0, 95835622256000, 0, 34121257060000, 0, 6083933240000, 0, 601169648200, 0, 34241740720, 0, 1127002800, 0, 20823000, 0, 202400, 0, 880, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 880*x^18 + 202400*x^16 + 20823000*x^14 + 1127002800*x^12 + 34241740720*x^10 + 601169648200*x^8 + 6083933240000*x^6 + 34121257060000*x^4 + 95835622256000*x^2 + 103893806112800)
 
gp: K = bnfinit(x^20 + 880*x^18 + 202400*x^16 + 20823000*x^14 + 1127002800*x^12 + 34241740720*x^10 + 601169648200*x^8 + 6083933240000*x^6 + 34121257060000*x^4 + 95835622256000*x^2 + 103893806112800, 1)
 

Normalized defining polynomial

\( x^{20} + 880 x^{18} + 202400 x^{16} + 20823000 x^{14} + 1127002800 x^{12} + 34241740720 x^{10} + 601169648200 x^{8} + 6083933240000 x^{6} + 34121257060000 x^{4} + 95835622256000 x^{2} + 103893806112800 \)

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:  \(116599917138248602348421120000000000000000000000000000000000=2^{55}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $898.12$
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$$\chi_{4400}(1,·)$, $\chi_{4400}(4361,·)$, $\chi_{4400}(1509,·)$, $\chi_{4400}(1229,·)$, $\chi_{4400}(589,·)$, $\chi_{4400}(81,·)$, $\chi_{4400}(469,·)$, $\chi_{4400}(1241,·)$, $\chi_{4400}(2201,·)$, $\chi_{4400}(3429,·)$, $\chi_{4400}(549,·)$, $\chi_{4400}(2789,·)$, $\chi_{4400}(3441,·)$, $\chi_{4400}(2281,·)$, $\chi_{4400}(2669,·)$, $\chi_{4400}(2749,·)$, $\chi_{4400}(2161,·)$, $\chi_{4400}(1521,·)$, $\chi_{4400}(3721,·)$, $\chi_{4400}(3709,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{220} a^{10}$, $\frac{1}{220} a^{11}$, $\frac{1}{440} a^{12}$, $\frac{1}{440} a^{13}$, $\frac{1}{7480} a^{14} + \frac{1}{7480} a^{12} - \frac{2}{935} a^{10} - \frac{1}{34} a^{8} - \frac{1}{34} a^{6} - \frac{1}{34} a^{4} + \frac{8}{17} a^{2} + \frac{8}{17}$, $\frac{1}{7480} a^{15} + \frac{1}{7480} a^{13} - \frac{2}{935} a^{11} - \frac{1}{34} a^{9} - \frac{1}{34} a^{7} - \frac{1}{34} a^{5} + \frac{8}{17} a^{3} + \frac{8}{17} a$, $\frac{1}{165356718691120} a^{16} - \frac{1227660999}{82678359345560} a^{14} + \frac{8536742011}{10334794918195} a^{12} - \frac{29618817383}{20669589836390} a^{10} + \frac{13854561603}{187905362149} a^{8} + \frac{380329267}{1888496102} a^{6} + \frac{8763298753}{187905362149} a^{4} - \frac{70990474455}{187905362149} a^{2} - \frac{6274209640}{187905362149}$, $\frac{1}{29929566083092720} a^{17} - \frac{1715779776}{110035169423135} a^{15} - \frac{71669102368}{110035169423135} a^{13} - \frac{457968470357}{440140677692540} a^{11} + \frac{527964489423}{8002557776228} a^{9} + \frac{2701553631}{20106929086} a^{7} - \frac{9237385306}{2000639444057} a^{5} + \frac{925598027015}{2000639444057} a^{3} - \frac{6748760733810}{34010870548969} a$, $\frac{1}{25237872969642581091345813321039579440} a^{18} - \frac{653251244226636451921}{630946824241064527283645333025989486} a^{16} + \frac{670427270030594073809966961084667}{12618936484821290545672906660519789720} a^{14} - \frac{1265860948074382623288812452703049}{1261893648482129054567290666051978972} a^{12} - \frac{2611491858588253965862157941090595}{1261893648482129054567290666051978972} a^{10} + \frac{1857297390104754840248889293976048}{28679401101866569421983878773908613} a^{8} - \frac{4445737003054473587730704294156385}{28679401101866569421983878773908613} a^{6} - \frac{2398002924181064208858023462162630}{28679401101866569421983878773908613} a^{4} + \frac{4489543072113563012669734628877336}{28679401101866569421983878773908613} a^{2} + \frac{61979606440751292004449585868}{875412872069429181709467927533}$, $\frac{1}{4568055007505307177533592211108163878640} a^{19} - \frac{653251244226636451921}{114201375187632679438339805277704096966} a^{17} + \frac{101891842923677309680811892045468007}{2284027503752653588766796105554081939320} a^{15} - \frac{54042402013355047293556813045426997}{51909715994378490653790820580774589530} a^{13} - \frac{12121238692929405072940365871246786}{285503437969081698595849513194260242415} a^{11} + \frac{980841803429654934448330736737390645}{20763886397751396261516328232309835812} a^{9} + \frac{88340560679455015718687727033195010}{5190971599437849065379082058077458953} a^{7} + \frac{893411525610592368913109013534629929}{5190971599437849065379082058077458953} a^{5} - \frac{2082358642987236223751686620860825857}{5190971599437849065379082058077458953} a^{3} + \frac{72051814615444398117288929744170}{158449729844566681889413694883473} a$

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

Class group and class number

$C_{6}\times C_{8243193030}$, which has order $49459158180$ (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:  \( 59880888363.52121 \) (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

\(\Q(\sqrt{2}) \), 4.0.6195200.5, 5.5.5719140625.3, 10.10.1071794405000000000000000.3

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 $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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