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Properties

Label 20.0.46077954434...5936.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 41^{19}$
Root discriminant $68.10$
Ramified primes $2, 41$
Class number $964$ (GRH)
Class group $[2, 482]$ (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![41, 0, 1517, 0, 13120, 0, 45141, 0, 71545, 0, 55801, 0, 23165, 0, 5289, 0, 656, 0, 41, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41)
 
gp: K = bnfinit(x^20 + 41*x^18 + 656*x^16 + 5289*x^14 + 23165*x^12 + 55801*x^10 + 71545*x^8 + 45141*x^6 + 13120*x^4 + 1517*x^2 + 41, 1)
 

Normalized defining polynomial

\( x^{20} + 41 x^{18} + 656 x^{16} + 5289 x^{14} + 23165 x^{12} + 55801 x^{10} + 71545 x^{8} + 45141 x^{6} + 13120 x^{4} + 1517 x^{2} + 41 \)

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:  \(4607795443446634940843146923591335936=2^{20}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.10$
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:  $2, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(164=2^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{164}(1,·)$, $\chi_{164}(131,·)$, $\chi_{164}(133,·)$, $\chi_{164}(81,·)$, $\chi_{164}(141,·)$, $\chi_{164}(143,·)$, $\chi_{164}(43,·)$, $\chi_{164}(87,·)$, $\chi_{164}(25,·)$, $\chi_{164}(91,·)$, $\chi_{164}(159,·)$, $\chi_{164}(155,·)$, $\chi_{164}(37,·)$, $\chi_{164}(39,·)$, $\chi_{164}(105,·)$, $\chi_{164}(103,·)$, $\chi_{164}(45,·)$, $\chi_{164}(113,·)$, $\chi_{164}(115,·)$, $\chi_{164}(57,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{7}$, $\frac{1}{657} a^{16} + \frac{2}{73} a^{14} + \frac{35}{219} a^{12} - \frac{10}{219} a^{10} + \frac{53}{657} a^{8} - \frac{34}{219} a^{4} - \frac{85}{219} a^{2} - \frac{110}{657}$, $\frac{1}{657} a^{17} + \frac{2}{73} a^{15} + \frac{35}{219} a^{13} - \frac{10}{219} a^{11} + \frac{53}{657} a^{9} - \frac{34}{219} a^{5} - \frac{85}{219} a^{3} - \frac{110}{657} a$, $\frac{1}{3858449967} a^{18} + \frac{1762651}{3858449967} a^{16} + \frac{93209570}{1286149989} a^{14} - \frac{9177052}{428716663} a^{12} - \frac{3830878}{52855479} a^{10} + \frac{338193440}{3858449967} a^{8} - \frac{562305310}{1286149989} a^{6} + \frac{75059731}{428716663} a^{4} - \frac{1443614141}{3858449967} a^{2} + \frac{653515540}{3858449967}$, $\frac{1}{3858449967} a^{19} + \frac{1762651}{3858449967} a^{17} + \frac{93209570}{1286149989} a^{15} - \frac{9177052}{428716663} a^{13} - \frac{3830878}{52855479} a^{11} + \frac{338193440}{3858449967} a^{9} - \frac{562305310}{1286149989} a^{7} + \frac{75059731}{428716663} a^{5} - \frac{1443614141}{3858449967} a^{3} + \frac{653515540}{3858449967} a$

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

Class group and class number

$C_{2}\times C_{482}$, which has order $964$ (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:  \( 5104264.63655 \) (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

\(\Q(\sqrt{41}) \), 4.0.1102736.1, 5.5.2825761.1, 10.10.327381934393961.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
41Data not computed