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Properties

Label 20.0.83419362232...0741.1
Degree $20$
Signature $[0, 10]$
Discriminant $61^{19}$
Root discriminant $49.67$
Ramified prime $61$
Class number $41$ (GRH)
Class group $[41]$ (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![611, -522, 2983, -439, -261, 1820, -1330, -541, 3278, 691, -745, 1778, 414, -267, 413, 51, -29, 36, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 2*x^18 + 36*x^17 - 29*x^16 + 51*x^15 + 413*x^14 - 267*x^13 + 414*x^12 + 1778*x^11 - 745*x^10 + 691*x^9 + 3278*x^8 - 541*x^7 - 1330*x^6 + 1820*x^5 - 261*x^4 - 439*x^3 + 2983*x^2 - 522*x + 611)
 
gp: K = bnfinit(x^20 - x^19 + 2*x^18 + 36*x^17 - 29*x^16 + 51*x^15 + 413*x^14 - 267*x^13 + 414*x^12 + 1778*x^11 - 745*x^10 + 691*x^9 + 3278*x^8 - 541*x^7 - 1330*x^6 + 1820*x^5 - 261*x^4 - 439*x^3 + 2983*x^2 - 522*x + 611, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 2 x^{18} + 36 x^{17} - 29 x^{16} + 51 x^{15} + 413 x^{14} - 267 x^{13} + 414 x^{12} + 1778 x^{11} - 745 x^{10} + 691 x^{9} + 3278 x^{8} - 541 x^{7} - 1330 x^{6} + 1820 x^{5} - 261 x^{4} - 439 x^{3} + 2983 x^{2} - 522 x + 611 \)

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:  \(8341936223273428359616333847680741=61^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.67$
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:  $61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(61\)
Dirichlet character group:    $\lbrace$$\chi_{61}(1,·)$, $\chi_{61}(3,·)$, $\chi_{61}(8,·)$, $\chi_{61}(9,·)$, $\chi_{61}(11,·)$, $\chi_{61}(20,·)$, $\chi_{61}(23,·)$, $\chi_{61}(24,·)$, $\chi_{61}(27,·)$, $\chi_{61}(28,·)$, $\chi_{61}(33,·)$, $\chi_{61}(34,·)$, $\chi_{61}(37,·)$, $\chi_{61}(38,·)$, $\chi_{61}(41,·)$, $\chi_{61}(50,·)$, $\chi_{61}(52,·)$, $\chi_{61}(53,·)$, $\chi_{61}(58,·)$, $\chi_{61}(60,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} + \frac{3}{13} a^{10} - \frac{1}{13} a^{8} - \frac{3}{13} a^{7} + \frac{1}{13} a^{5} + \frac{3}{13} a^{4} - \frac{1}{13} a^{2} - \frac{3}{13} a$, $\frac{1}{13} a^{12} + \frac{4}{13} a^{10} - \frac{1}{13} a^{9} - \frac{4}{13} a^{7} + \frac{1}{13} a^{6} + \frac{4}{13} a^{4} - \frac{1}{13} a^{3} - \frac{4}{13} a$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{611} a^{16} + \frac{17}{611} a^{15} + \frac{1}{47} a^{14} - \frac{1}{611} a^{13} + \frac{4}{611} a^{12} + \frac{4}{611} a^{11} - \frac{271}{611} a^{10} + \frac{113}{611} a^{9} + \frac{87}{611} a^{8} - \frac{223}{611} a^{7} + \frac{277}{611} a^{6} + \frac{69}{611} a^{5} - \frac{168}{611} a^{4} - \frac{255}{611} a^{3} + \frac{295}{611} a^{2} + \frac{38}{611} a$, $\frac{1}{611} a^{17} + \frac{6}{611} a^{15} + \frac{1}{47} a^{14} + \frac{21}{611} a^{13} - \frac{17}{611} a^{12} - \frac{10}{611} a^{11} - \frac{215}{611} a^{10} - \frac{48}{611} a^{9} - \frac{198}{611} a^{8} - \frac{162}{611} a^{7} + \frac{295}{611} a^{6} + \frac{210}{611} a^{5} + \frac{110}{611} a^{4} + \frac{24}{611} a^{3} - \frac{42}{611} a^{2} + \frac{12}{611} a$, $\frac{1}{7943} a^{18} + \frac{6}{7943} a^{17} - \frac{4}{7943} a^{16} - \frac{215}{7943} a^{15} - \frac{172}{7943} a^{14} + \frac{119}{7943} a^{13} - \frac{105}{7943} a^{12} + \frac{202}{7943} a^{11} + \frac{3722}{7943} a^{10} - \frac{1052}{7943} a^{9} - \frac{2737}{7943} a^{8} + \frac{1036}{7943} a^{7} - \frac{132}{7943} a^{6} - \frac{3691}{7943} a^{5} - \frac{2007}{7943} a^{4} - \frac{2189}{7943} a^{3} + \frac{1322}{7943} a^{2} - \frac{3880}{7943} a + \frac{3}{13}$, $\frac{1}{1459187418855718731023} a^{19} + \frac{71588855505119702}{1459187418855718731023} a^{18} - \frac{9866776487534390}{31046540826717419809} a^{17} - \frac{808552113296549277}{1459187418855718731023} a^{16} - \frac{26568493924809314463}{1459187418855718731023} a^{15} + \frac{29067909814587736332}{1459187418855718731023} a^{14} - \frac{22205036471015043602}{1459187418855718731023} a^{13} - \frac{53057262527011422338}{1459187418855718731023} a^{12} + \frac{50661271734256702284}{1459187418855718731023} a^{11} - \frac{5286136387776578952}{112245186065824517771} a^{10} - \frac{606396326625827563232}{1459187418855718731023} a^{9} + \frac{609579628493693634544}{1459187418855718731023} a^{8} - \frac{321144900268723759463}{1459187418855718731023} a^{7} - \frac{289926439569863927836}{1459187418855718731023} a^{6} + \frac{42823027067232406473}{1459187418855718731023} a^{5} - \frac{85030557258461786654}{1459187418855718731023} a^{4} - \frac{723497774961558747364}{1459187418855718731023} a^{3} + \frac{630816575343483879501}{1459187418855718731023} a^{2} + \frac{544587768651235071472}{1459187418855718731023} a - \frac{597908067529066610}{2388195448209032293}$

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

Class group and class number

$C_{41}$, which has order $41$ (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:  \( 36549838.4715 \) (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

\(\Q(\sqrt{61}) \), 4.0.226981.1, 5.5.13845841.1, 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{20}$ $20$ $20$

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
61Data not computed