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Properties

Label 20.0.10975122531...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $126.48$
Ramified primes $2, 5, 41$
Class number $5123360$ (GRH)
Class group $[2, 2561680]$ (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![81, 0, 442026, 0, 5678089, 0, 16100059, 0, 14712760, 0, 5146456, 0, 881785, 0, 81286, 0, 4075, 0, 103, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 103*x^18 + 4075*x^16 + 81286*x^14 + 881785*x^12 + 5146456*x^10 + 14712760*x^8 + 16100059*x^6 + 5678089*x^4 + 442026*x^2 + 81)
 
gp: K = bnfinit(x^20 + 103*x^18 + 4075*x^16 + 81286*x^14 + 881785*x^12 + 5146456*x^10 + 14712760*x^8 + 16100059*x^6 + 5678089*x^4 + 442026*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{20} + 103 x^{18} + 4075 x^{16} + 81286 x^{14} + 881785 x^{12} + 5146456 x^{10} + 14712760 x^{8} + 16100059 x^{6} + 5678089 x^{4} + 442026 x^{2} + 81 \)

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:  \(1097512253107525471809057479895040000000000=2^{20}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.48$
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:  \(820=2^{2}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{820}(1,·)$, $\chi_{820}(769,·)$, $\chi_{820}(201,·)$, $\chi_{820}(139,·)$, $\chi_{820}(141,·)$, $\chi_{820}(269,·)$, $\chi_{820}(209,·)$, $\chi_{820}(739,·)$, $\chi_{820}(409,·)$, $\chi_{820}(461,·)$, $\chi_{820}(271,·)$, $\chi_{820}(221,·)$, $\chi_{820}(351,·)$, $\chi_{820}(291,·)$, $\chi_{820}(31,·)$, $\chi_{820}(491,·)$, $\chi_{820}(59,·)$, $\chi_{820}(119,·)$, $\chi_{820}(379,·)$, $\chi_{820}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{9} a^{4} - \frac{7}{27} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} + \frac{1}{27} a^{6} + \frac{8}{81} a^{4} - \frac{13}{27} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{729} a^{12} - \frac{4}{729} a^{10} - \frac{4}{243} a^{8} - \frac{13}{729} a^{6} - \frac{8}{729} a^{4} - \frac{32}{81} a^{2} + \frac{4}{9}$, $\frac{1}{729} a^{13} - \frac{4}{729} a^{11} - \frac{1}{243} a^{9} - \frac{40}{729} a^{7} + \frac{19}{729} a^{5} + \frac{4}{27} a^{3} - \frac{1}{9} a$, $\frac{1}{6561} a^{14} + \frac{1}{6561} a^{12} - \frac{5}{6561} a^{10} + \frac{8}{6561} a^{8} + \frac{251}{6561} a^{6} - \frac{841}{6561} a^{4} - \frac{115}{729} a^{2} + \frac{20}{81}$, $\frac{1}{6561} a^{15} + \frac{1}{6561} a^{13} - \frac{5}{6561} a^{11} + \frac{8}{6561} a^{9} + \frac{251}{6561} a^{7} - \frac{841}{6561} a^{5} - \frac{115}{729} a^{3} + \frac{20}{81} a$, $\frac{1}{59049} a^{16} - \frac{2}{19683} a^{12} + \frac{256}{59049} a^{10} - \frac{607}{19683} a^{6} + \frac{5395}{59049} a^{4} - \frac{731}{6561} a^{2} + \frac{34}{729}$, $\frac{1}{59049} a^{17} - \frac{2}{19683} a^{13} + \frac{256}{59049} a^{11} - \frac{607}{19683} a^{7} + \frac{5395}{59049} a^{5} - \frac{731}{6561} a^{3} + \frac{34}{729} a$, $\frac{1}{13289610632427} a^{18} + \frac{12886757}{13289610632427} a^{16} + \frac{240907417}{4429870210809} a^{14} + \frac{645447412}{13289610632427} a^{12} - \frac{71008381651}{13289610632427} a^{10} - \frac{59090565649}{4429870210809} a^{8} - \frac{541769617820}{13289610632427} a^{6} + \frac{1693759226801}{13289610632427} a^{4} - \frac{524494231486}{1476623403603} a^{2} + \frac{69823685774}{164069267067}$, $\frac{1}{13289610632427} a^{19} + \frac{12886757}{13289610632427} a^{17} + \frac{240907417}{4429870210809} a^{15} + \frac{645447412}{13289610632427} a^{13} - \frac{71008381651}{13289610632427} a^{11} - \frac{4400809960}{4429870210809} a^{9} + \frac{442645984582}{13289610632427} a^{7} - \frac{767279779204}{13289610632427} a^{5} - \frac{50516348848}{1476623403603} a^{3} + \frac{33363848648}{164069267067} a$

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

Class group and class number

$C_{2}\times C_{2561680}$, which has order $5123360$ (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:  \( 1770719412.3197536 \) (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

\(\Q(\sqrt{-41}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{-5}, \sqrt{-41})\), 5.5.2825761.1, 10.0.335239100819416064.1, 10.0.25551760733187200000.2, 10.10.1023068544981128125.1

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$41$41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$