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Properties

Label 20.0.65416827983...000.13
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 $13608416800$ (GRH)
Class group $[10, 1360841680]$ (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![21574938539041801, 0, 33576526013427075, 0, 6628457585928775, 0, 514849477932005, 0, 20382812663480, 0, 463900095803, 0, 6428606275, 0, 55136800, 0, 285770, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 285770*x^16 + 55136800*x^14 + 6428606275*x^12 + 463900095803*x^10 + 20382812663480*x^8 + 514849477932005*x^6 + 6628457585928775*x^4 + 33576526013427075*x^2 + 21574938539041801)
 
gp: K = bnfinit(x^20 + 820*x^18 + 285770*x^16 + 55136800*x^14 + 6428606275*x^12 + 463900095803*x^10 + 20382812663480*x^8 + 514849477932005*x^6 + 6628457585928775*x^4 + 33576526013427075*x^2 + 21574938539041801, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 285770 x^{16} + 55136800 x^{14} + 6428606275 x^{12} + 463900095803 x^{10} + 20382812663480 x^{8} + 514849477932005 x^{6} + 6628457585928775 x^{4} + 33576526013427075 x^{2} + 21574938539041801 \)

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$$\chi_{4100}(1,·)$, $\chi_{4100}(4099,·)$, $\chi_{4100}(2951,·)$, $\chi_{4100}(1609,·)$, $\chi_{4100}(2319,·)$, $\chi_{4100}(469,·)$, $\chi_{4100}(1111,·)$, $\chi_{4100}(271,·)$, $\chi_{4100}(221,·)$, $\chi_{4100}(1439,·)$, $\chi_{4100}(3741,·)$, $\chi_{4100}(2661,·)$, $\chi_{4100}(3879,·)$, $\chi_{4100}(359,·)$, $\chi_{4100}(2989,·)$, $\chi_{4100}(3631,·)$, $\chi_{4100}(3829,·)$, $\chi_{4100}(2491,·)$, $\chi_{4100}(1149,·)$, $\chi_{4100}(1781,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{167094311} a^{10} + \frac{10}{4075471} a^{8} + \frac{1435}{4075471} a^{6} + \frac{84050}{4075471} a^{4} + \frac{1723025}{4075471} a^{2} - \frac{316965}{4075471}$, $\frac{1}{167094311} a^{11} + \frac{10}{4075471} a^{9} + \frac{1435}{4075471} a^{7} + \frac{84050}{4075471} a^{5} + \frac{1723025}{4075471} a^{3} - \frac{316965}{4075471} a$, $\frac{1}{167094311} a^{12} - \frac{2665}{4075471} a^{8} - \frac{504300}{4075471} a^{6} - \frac{133707}{4075471} a^{4} - \frac{1700732}{4075471} a^{2} - \frac{459422}{4075471}$, $\frac{1}{167094311} a^{13} - \frac{2665}{4075471} a^{9} - \frac{504300}{4075471} a^{7} - \frac{133707}{4075471} a^{5} - \frac{1700732}{4075471} a^{3} - \frac{459422}{4075471} a$, $\frac{1}{6850866751} a^{14} + \frac{14350}{4075471} a^{8} - \frac{1049671}{4075471} a^{6} - \frac{65221181}{167094311} a^{4} + \frac{1975458}{4075471} a^{2} - \frac{1089228}{4075471}$, $\frac{1}{598621885835629} a^{15} + \frac{15}{14600533800869} a^{13} + \frac{90}{356110580509} a^{11} + \frac{11275}{356110580509} a^{9} + \frac{756450}{356110580509} a^{7} + \frac{2421567155201}{14600533800869} a^{5} - \frac{4869043051}{356110580509} a^{3} + \frac{141997790293}{356110580509} a$, $\frac{1}{20192114831121601799} a^{16} - \frac{179650609}{20192114831121601799} a^{14} + \frac{432267603}{492490605637112239} a^{12} - \frac{1261290485}{492490605637112239} a^{10} - \frac{3742595163695}{12011965991149079} a^{8} + \frac{149896056851195791}{492490605637112239} a^{6} + \frac{175796181518942884}{492490605637112239} a^{4} - \frac{128394407071870}{12011965991149079} a^{2} - \frac{34641986931}{137469712301}$, $\frac{1}{827876708075985673759} a^{17} + \frac{17}{20192114831121601799} a^{15} - \frac{1433364996}{492490605637112239} a^{13} - \frac{611896482}{492490605637112239} a^{11} + \frac{3813798923770}{12011965991149079} a^{9} + \frac{409841927260381518}{20192114831121601799} a^{7} + \frac{235227934013768503}{492490605637112239} a^{5} - \frac{52518608222035}{203592643917781} a^{3} - \frac{890758256869321}{12011965991149079} a$, $\frac{1}{827876708075985673759} a^{18} + \frac{286335448}{20192114831121601799} a^{14} + \frac{881697414}{492490605637112239} a^{12} - \frac{840647125}{492490605637112239} a^{10} - \frac{13573611477423308}{342239234425789861} a^{8} + \frac{175254542085688246}{492490605637112239} a^{6} - \frac{57860786434179481}{492490605637112239} a^{4} + \frac{4717336918260338}{12011965991149079} a^{2} - \frac{62230930337}{137469712301}$, $\frac{1}{827876708075985673759} a^{19} - \frac{171}{492490605637112239} a^{15} - \frac{7640302}{8073616485854299} a^{13} + \frac{16231116}{12011965991149079} a^{11} - \frac{801687248059962072}{20192114831121601799} a^{9} + \frac{2748963005035221}{8073616485854299} a^{7} + \frac{3759161395581446}{12011965991149079} a^{5} - \frac{2138861504510518}{12011965991149079} a^{3} - \frac{5332057106820576}{12011965991149079} a$

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

Class group and class number

$C_{10}\times C_{1360841680}$, which has order $13608416800$ (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:  \( 134402570544.74945 \) (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

\(\Q(\sqrt{-205}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-41}) \), \(\Q(\sqrt{5}, \sqrt{-41})\), 5.5.1103812890625.2, 10.0.255767136245282031250000000000.2, 10.10.6092014487549591064453125.1, 10.0.51153427249056406250000000000.4

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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}$