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Properties

Label 20.0.58299958569...0000.2
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 $50431501000$ (GRH)
Class group $[10, 5043150100]$ (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![1417582740640, 0, 6279892643200, 0, 8121911556000, 0, 3803674688000, 0, 802430109800, 0, 79169797520, 0, 3314018400, 0, 46800600, 0, 296560, 0, 880, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 880*x^18 + 296560*x^16 + 46800600*x^14 + 3314018400*x^12 + 79169797520*x^10 + 802430109800*x^8 + 3803674688000*x^6 + 8121911556000*x^4 + 6279892643200*x^2 + 1417582740640)
 
gp: K = bnfinit(x^20 + 880*x^18 + 296560*x^16 + 46800600*x^14 + 3314018400*x^12 + 79169797520*x^10 + 802430109800*x^8 + 3803674688000*x^6 + 8121911556000*x^4 + 6279892643200*x^2 + 1417582740640, 1)
 

Normalized defining polynomial

\( x^{20} + 880 x^{18} + 296560 x^{16} + 46800600 x^{14} + 3314018400 x^{12} + 79169797520 x^{10} + 802430109800 x^{8} + 3803674688000 x^{6} + 8121911556000 x^{4} + 6279892643200 x^{2} + 1417582740640 \)

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$$\chi_{4400}(1,·)$, $\chi_{4400}(387,·)$, $\chi_{4400}(2161,·)$, $\chi_{4400}(9,·)$, $\chi_{4400}(2763,·)$, $\chi_{4400}(81,·)$, $\chi_{4400}(547,·)$, $\chi_{4400}(729,·)$, $\chi_{4400}(3803,·)$, $\chi_{4400}(3483,·)$, $\chi_{4400}(523,·)$, $\chi_{4400}(3427,·)$, $\chi_{4400}(1521,·)$, $\chi_{4400}(489,·)$, $\chi_{4400}(43,·)$, $\chi_{4400}(307,·)$, $\chi_{4400}(3441,·)$, $\chi_{4400}(2867,·)$, $\chi_{4400}(169,·)$, $\chi_{4400}(1849,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{44} a^{10}$, $\frac{1}{44} a^{11}$, $\frac{1}{88} a^{12}$, $\frac{1}{3784} a^{13} - \frac{4}{473} a^{11} - \frac{7}{172} a^{9} + \frac{10}{43} a^{7} - \frac{17}{86} a^{5} - \frac{2}{43} a^{3} - \frac{16}{43} a$, $\frac{1}{3784} a^{14} + \frac{1}{344} a^{12} + \frac{9}{1892} a^{10} - \frac{3}{172} a^{8} - \frac{17}{86} a^{6} - \frac{2}{43} a^{4} - \frac{16}{43} a^{2}$, $\frac{1}{3784} a^{15} + \frac{13}{1892} a^{11} - \frac{3}{43} a^{9} + \frac{21}{86} a^{7} + \frac{11}{86} a^{5} + \frac{6}{43} a^{3} + \frac{4}{43} a$, $\frac{1}{31687216} a^{16} + \frac{493}{15843608} a^{14} + \frac{66109}{15843608} a^{12} - \frac{19207}{3960902} a^{10} - \frac{11437}{180041} a^{8} - \frac{4478}{180041} a^{6} + \frac{38577}{180041} a^{4} + \frac{74545}{180041} a^{2} - \frac{1756}{4187}$, $\frac{1}{6305755984} a^{17} + \frac{89097}{788219498} a^{15} + \frac{401069}{3152877992} a^{13} + \frac{1749435}{1576438996} a^{11} - \frac{10081987}{143312636} a^{9} - \frac{7231240}{35828159} a^{7} + \frac{14861451}{71656318} a^{5} - \frac{14123572}{35828159} a^{3} + \frac{5170803}{35828159} a$, $\frac{1}{258069908135528562810412254056770140417705488} a^{18} + \frac{10029456258299591933217266720241659}{777319000408218562681964620652922109691884} a^{16} + \frac{11759227283298663637949730074793901832285}{129034954067764281405206127028385070208852744} a^{14} - \frac{7220180164793899812926354486975299027905}{16129369258470535175650765878548133776106593} a^{12} - \frac{119821025968373244554852834570289893241219}{64517477033882140702603063514192535104426372} a^{10} + \frac{3882838855274113385547908200896263358133}{5865225184898376427509369410381139554947852} a^{8} - \frac{114895018129385329502283051552871836094028}{1466306296224594106877342352595284888736963} a^{6} + \frac{26657895565520269820612707539261305988033}{2932612592449188213754684705190569777473926} a^{4} - \frac{433128384690698518141450786984935443306625}{1466306296224594106877342352595284888736963} a^{2} - \frac{1073234754906981370240227932872656210}{3985058598086685745866548406160833613}$, $\frac{1}{258069908135528562810412254056770140417705488} a^{19} + \frac{177910331249632396434180647737037}{3109276001632874250727858482611688438767536} a^{17} - \frac{10266618932799947624880977229308605726917}{129034954067764281405206127028385070208852744} a^{15} + \frac{122677785706017612606105979345540234436}{1466306296224594106877342352595284888736963} a^{13} - \frac{19895322804274968791668480864324037771821}{64517477033882140702603063514192535104426372} a^{11} - \frac{134486458091558678757420797107400446030106}{1466306296224594106877342352595284888736963} a^{9} + \frac{157157246823855722850615682131816225768557}{1466306296224594106877342352595284888736963} a^{7} - \frac{67516666479661008023470676434451701426647}{1466306296224594106877342352595284888736963} a^{5} + \frac{692499067627104597315110038807249055895559}{1466306296224594106877342352595284888736963} a^{3} - \frac{41590906128259870644503069824117263181}{793026661019250463427443132826005888987} a$

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

Class group and class number

$C_{10}\times C_{5043150100}$, which has order $50431501000$ (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

\(\Q(\sqrt{10}) \), 4.0.30976000.4, 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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