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Properties

Label 20.0.14339596913...7456.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 61^{18}$
Root discriminant $80.88$
Ramified primes $2, 61$
Class number $10065$ (GRH)
Class group $[10065]$ (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![169, 0, 16988, 0, 425738, 0, 850547, 0, 706127, 0, 314412, 0, 82081, 0, 12828, 0, 1163, 0, 55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 55*x^18 + 1163*x^16 + 12828*x^14 + 82081*x^12 + 314412*x^10 + 706127*x^8 + 850547*x^6 + 425738*x^4 + 16988*x^2 + 169)
 
gp: K = bnfinit(x^20 + 55*x^18 + 1163*x^16 + 12828*x^14 + 82081*x^12 + 314412*x^10 + 706127*x^8 + 850547*x^6 + 425738*x^4 + 16988*x^2 + 169, 1)
 

Normalized defining polynomial

\( x^{20} + 55 x^{18} + 1163 x^{16} + 12828 x^{14} + 82081 x^{12} + 314412 x^{10} + 706127 x^{8} + 850547 x^{6} + 425738 x^{4} + 16988 x^{2} + 169 \)

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:  \(143395969135330465829722243945339027456=2^{20}\cdot 61^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.88$
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:  $2, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(244=2^{2}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{244}(1,·)$, $\chi_{244}(3,·)$, $\chi_{244}(9,·)$, $\chi_{244}(203,·)$, $\chi_{244}(81,·)$, $\chi_{244}(131,·)$, $\chi_{244}(149,·)$, $\chi_{244}(217,·)$, $\chi_{244}(27,·)$, $\chi_{244}(95,·)$, $\chi_{244}(163,·)$, $\chi_{244}(241,·)$, $\chi_{244}(41,·)$, $\chi_{244}(235,·)$, $\chi_{244}(113,·)$, $\chi_{244}(243,·)$, $\chi_{244}(119,·)$, $\chi_{244}(121,·)$, $\chi_{244}(123,·)$, $\chi_{244}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{13} a^{9} - \frac{1}{13} a^{7} + \frac{1}{13} a^{3} - \frac{1}{13} a$, $\frac{1}{13} a^{10} - \frac{1}{13} a^{8} + \frac{1}{13} a^{4} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{11} - \frac{1}{13} a^{7} + \frac{1}{13} a^{5} - \frac{1}{13} a$, $\frac{1}{13} a^{12} - \frac{1}{13} a^{8} + \frac{1}{13} a^{6} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{7943} a^{16} + \frac{164}{7943} a^{14} + \frac{81}{7943} a^{12} - \frac{72}{7943} a^{10} - \frac{1075}{7943} a^{8} - \frac{725}{7943} a^{6} - \frac{2894}{7943} a^{4} - \frac{3085}{7943} a^{2} - \frac{11}{47}$, $\frac{1}{7943} a^{17} + \frac{164}{7943} a^{15} + \frac{81}{7943} a^{13} - \frac{72}{7943} a^{11} + \frac{147}{7943} a^{9} - \frac{1947}{7943} a^{7} - \frac{2894}{7943} a^{5} - \frac{1863}{7943} a^{3} - \frac{237}{611} a$, $\frac{1}{3438913907} a^{18} - \frac{160700}{3438913907} a^{16} + \frac{209725}{264531839} a^{14} + \frac{2827530}{264531839} a^{12} + \frac{1147086}{3438913907} a^{10} - \frac{1463389803}{3438913907} a^{8} - \frac{13727161}{312628537} a^{6} - \frac{241990282}{3438913907} a^{4} + \frac{12001115}{312628537} a^{2} - \frac{3068911}{20348603}$, $\frac{1}{3438913907} a^{19} - \frac{160700}{3438913907} a^{17} + \frac{209725}{264531839} a^{15} + \frac{2827530}{264531839} a^{13} + \frac{1147086}{3438913907} a^{11} + \frac{123801231}{3438913907} a^{9} + \frac{154611282}{312628537} a^{7} - \frac{241990282}{3438913907} a^{5} + \frac{156291209}{312628537} a^{3} + \frac{102544378}{264531839} a$

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

Class group and class number

$C_{10065}$, which has order $10065$ (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:  \( \frac{334}{312628537} a^{19} + \frac{444825}{312628537} a^{17} + \frac{22450756}{312628537} a^{15} + \frac{428108310}{312628537} a^{13} + \frac{4109912357}{312628537} a^{11} + \frac{21721592726}{312628537} a^{9} + \frac{63398148746}{312628537} a^{7} + \frac{94375064564}{312628537} a^{5} + \frac{55719178059}{312628537} a^{3} + \frac{255611388}{24048349} a \) (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:  \( 36549838.4715 \) (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

\(\Q(\sqrt{-61}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{61}) \), \(\Q(i, \sqrt{61})\), 5.5.13845841.1, 10.0.11974805599062160384.1, 10.0.196308288509215744.1, 10.10.11694146092834141.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$61$61.10.9.1$x^{10} - 61$$10$$1$$9$$C_{10}$$[\ ]_{10}$
61.10.9.1$x^{10} - 61$$10$$1$$9$$C_{10}$$[\ ]_{10}$