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Properties

Label 20.0.65416827983...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{34}\cdot 41^{18}$
Root discriminant $872.54$
Ramified primes $2, 5, 41$
Class number $13948321000$ (GRH)
Class group $[5, 10, 278966420]$ (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![71877073801, 0, 876479585827850, 0, 743596568481275, 0, 136377368788730, 0, 8608657192805, 0, 267724300303, 0, 4657110050, 0, 47070050, 0, 271420, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 271420*x^16 + 47070050*x^14 + 4657110050*x^12 + 267724300303*x^10 + 8608657192805*x^8 + 136377368788730*x^6 + 743596568481275*x^4 + 876479585827850*x^2 + 71877073801)
 
gp: K = bnfinit(x^20 + 820*x^18 + 271420*x^16 + 47070050*x^14 + 4657110050*x^12 + 267724300303*x^10 + 8608657192805*x^8 + 136377368788730*x^6 + 743596568481275*x^4 + 876479585827850*x^2 + 71877073801, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 271420 x^{16} + 47070050 x^{14} + 4657110050 x^{12} + 267724300303 x^{10} + 8608657192805 x^{8} + 136377368788730 x^{6} + 743596568481275 x^{4} + 876479585827850 x^{2} + 71877073801 \)

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:  \(65416827983112661350313274854125976562500000000000000000000=2^{20}\cdot 5^{34}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $872.54$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4100=2^{2}\cdot 5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4100}(1,·)$, $\chi_{4100}(2051,·)$, $\chi_{4100}(2049,·)$, $\chi_{4100}(269,·)$, $\chi_{4100}(2319,·)$, $\chi_{4100}(4099,·)$, $\chi_{4100}(1439,·)$, $\chi_{4100}(1691,·)$, $\chi_{4100}(221,·)$, $\chi_{4100}(1829,·)$, $\chi_{4100}(3489,·)$, $\chi_{4100}(611,·)$, $\chi_{4100}(2661,·)$, $\chi_{4100}(3879,·)$, $\chi_{4100}(2409,·)$, $\chi_{4100}(359,·)$, $\chi_{4100}(3741,·)$, $\chi_{4100}(1781,·)$, $\chi_{4100}(3831,·)$, $\chi_{4100}(2271,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41} a^{10}$, $\frac{1}{533} a^{11} + \frac{6}{13} a^{9} - \frac{6}{13} a^{7} + \frac{6}{13} a^{5} - \frac{6}{13} a^{3} + \frac{6}{13} a$, $\frac{1}{533} a^{12} - \frac{1}{533} a^{10} - \frac{6}{13} a^{8} + \frac{6}{13} a^{6} - \frac{6}{13} a^{4} + \frac{6}{13} a^{2}$, $\frac{1}{533} a^{13} + \frac{6}{13} a$, $\frac{1}{533} a^{14} + \frac{6}{13} a^{2}$, $\frac{1}{533} a^{15} + \frac{6}{13} a^{3}$, $\frac{1}{1918267} a^{16} - \frac{1581}{1918267} a^{14} + \frac{231}{1918267} a^{12} + \frac{4774}{1918267} a^{10} + \frac{6934}{46787} a^{8} + \frac{3752}{46787} a^{6} + \frac{12010}{46787} a^{4} - \frac{963}{46787} a^{2} - \frac{1598}{3599}$, $\frac{1}{1918267} a^{17} - \frac{1581}{1918267} a^{15} + \frac{231}{1918267} a^{13} + \frac{1175}{1918267} a^{11} - \frac{14660}{46787} a^{9} - \frac{21441}{46787} a^{7} - \frac{9584}{46787} a^{5} + \frac{1587}{3599} a^{3} + \frac{4419}{46787} a$, $\frac{1}{99590878139500166828121177665554528864471728992392477} a^{18} + \frac{18048620669478023562316063699490665262291433356}{99590878139500166828121177665554528864471728992392477} a^{16} + \frac{70058080441444783552672260935810376324916955437576}{99590878139500166828121177665554528864471728992392477} a^{14} - \frac{78648442226569643759495872772463096422674723468665}{99590878139500166828121177665554528864471728992392477} a^{12} + \frac{1156303926339576490945798636763234663141104159097977}{99590878139500166828121177665554528864471728992392477} a^{10} - \frac{280083529394435737156559713142637092851798398967915}{2429045808280491873856614089403768996694432414448597} a^{8} - \frac{460982175364495736289323328841950687942828038252851}{2429045808280491873856614089403768996694432414448597} a^{6} - \frac{830743296913750280330725186681571270289801050216843}{2429045808280491873856614089403768996694432414448597} a^{4} + \frac{248706318293847063448536486850857084724701781171169}{2429045808280491873856614089403768996694432414448597} a^{2} - \frac{5146749771944177257359562281885109285513066014005}{14373052120002910496192982777537094654996641505613}$, $\frac{1}{2053862679870911940496343046996731048772000467010110053171} a^{19} - \frac{9630816761674030947863356186773144155168298320830}{50094211704168583914544952365773928018829279683173415931} a^{17} - \frac{20145108072263937485819361336821741240410298599982696}{50094211704168583914544952365773928018829279683173415931} a^{15} + \frac{466631127184546223605718276863463524405901932840728}{50094211704168583914544952365773928018829279683173415931} a^{13} + \frac{18151693081493924935859258210285007867874221799031687}{50094211704168583914544952365773928018829279683173415931} a^{11} - \frac{21991005729539652142831925915675592217615268046939482261}{50094211704168583914544952365773928018829279683173415931} a^{9} - \frac{106590048066523432782144514498161221668574067160134316}{1221810041565087412549876886970095805337299504467644291} a^{7} - \frac{107989805701721946750715842180931375561859402035278173}{1221810041565087412549876886970095805337299504467644291} a^{5} - \frac{246033167331825009589078664017534902021110165874722400}{1221810041565087412549876886970095805337299504467644291} a^{3} + \frac{43812002593310968010481778903470309728256383932403697}{93985387812699031734605914382315061949023038805203407} a$

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

Class group and class number

$C_{5}\times C_{10}\times C_{278966420}$, which has order $13948321000$ (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:  \( -\frac{1610923731792264272750}{13030038445900174051808492311005563} a^{19} - \frac{32246229925055846933725}{317805815753662781751426641731843} a^{17} - \frac{10685200070973460451359107}{317805815753662781751426641731843} a^{15} - \frac{1855544680213243356161085230}{317805815753662781751426641731843} a^{13} - \frac{183875161386985217580677234380}{317805815753662781751426641731843} a^{11} - \frac{10588204230708320836522956371550}{317805815753662781751426641731843} a^{9} - \frac{8317870723519443856830866339375}{7751361359845433701254308334923} a^{7} - \frac{10152111187914316379890486389383}{596258566141956438558023718071} a^{5} - \frac{720704083314924817645281912977695}{7751361359845433701254308334923} a^{3} - \frac{855989765403458592357095935105395}{7751361359845433701254308334923} a \) (order $4$)
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:  \( 793721345751.2314 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{-205}) \), \(\Q(i, \sqrt{205})\), 5.5.1103812890625.2, 10.0.1247644567050156250000000000.2, 10.10.249772593989533233642578125.3, 10.0.255767136245282031250000000000.2

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.10.17.4$x^{10} - 5 x^{8} + 105$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.4$x^{10} - 5 x^{8} + 105$$10$$1$$17$$C_{10}$$[2]_{2}$
$41$41.10.9.3$x^{10} - 53136$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.3$x^{10} - 53136$$10$$1$$9$$C_{10}$$[\ ]_{10}$