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Properties

Label 20.8.13146741154...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{17}\cdot 11^{6}\cdot 9931^{4}$
Root discriminant $50.81$
Ramified primes $5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16189, 60617, -34699, -75655, 65866, -121, -21038, 13654, 38319, -2334, -3202, 11724, -1721, -2074, 622, -329, -39, 65, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 14*x^18 + 65*x^17 - 39*x^16 - 329*x^15 + 622*x^14 - 2074*x^13 - 1721*x^12 + 11724*x^11 - 3202*x^10 - 2334*x^9 + 38319*x^8 + 13654*x^7 - 21038*x^6 - 121*x^5 + 65866*x^4 - 75655*x^3 - 34699*x^2 + 60617*x - 16189)
 
gp: K = bnfinit(x^20 - 2*x^19 - 14*x^18 + 65*x^17 - 39*x^16 - 329*x^15 + 622*x^14 - 2074*x^13 - 1721*x^12 + 11724*x^11 - 3202*x^10 - 2334*x^9 + 38319*x^8 + 13654*x^7 - 21038*x^6 - 121*x^5 + 65866*x^4 - 75655*x^3 - 34699*x^2 + 60617*x - 16189, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 14 x^{18} + 65 x^{17} - 39 x^{16} - 329 x^{15} + 622 x^{14} - 2074 x^{13} - 1721 x^{12} + 11724 x^{11} - 3202 x^{10} - 2334 x^{9} + 38319 x^{8} + 13654 x^{7} - 21038 x^{6} - 121 x^{5} + 65866 x^{4} - 75655 x^{3} - 34699 x^{2} + 60617 x - 16189 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13146741154754246205048370361328125=5^{17}\cdot 11^{6}\cdot 9931^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.81$
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:  $5, 11, 9931$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{49410433741999192786412230683659036701243996477061129} a^{19} - \frac{21564103823413837258575626611890022095638601780210114}{49410433741999192786412230683659036701243996477061129} a^{18} + \frac{13565145291045249669672841262579631358853088196094394}{49410433741999192786412230683659036701243996477061129} a^{17} - \frac{15703277230935692996330630555774839191139930372058792}{49410433741999192786412230683659036701243996477061129} a^{16} - \frac{21817232377067676925549279558764456074086308235294159}{49410433741999192786412230683659036701243996477061129} a^{15} + \frac{7687861932578052606132537159100317007235765024835005}{49410433741999192786412230683659036701243996477061129} a^{14} - \frac{23379168883388707979446575384439275221377417319223101}{49410433741999192786412230683659036701243996477061129} a^{13} - \frac{5674075139362017122202880585844034038153555297128169}{49410433741999192786412230683659036701243996477061129} a^{12} + \frac{3174421861026357533168238381599773030707316759686731}{49410433741999192786412230683659036701243996477061129} a^{11} - \frac{7259376586593742362113699563251004648808858385495801}{49410433741999192786412230683659036701243996477061129} a^{10} - \frac{8859478050184493212160689327192825596322606543597637}{49410433741999192786412230683659036701243996477061129} a^{9} - \frac{4567017228911823219393348333100245249930167449878013}{49410433741999192786412230683659036701243996477061129} a^{8} + \frac{13170027054158985057697724975582199493788909458559208}{49410433741999192786412230683659036701243996477061129} a^{7} - \frac{22714041537464605887704812644729915531305817208595619}{49410433741999192786412230683659036701243996477061129} a^{6} + \frac{4779671264625892795588915657721761042358453956905585}{49410433741999192786412230683659036701243996477061129} a^{5} + \frac{24456932941434520736822858276511470076094576399588058}{49410433741999192786412230683659036701243996477061129} a^{4} - \frac{4035486446862243980046373088765398933522812112056544}{49410433741999192786412230683659036701243996477061129} a^{3} + \frac{18458880800098061224446609874949786506424893599090714}{49410433741999192786412230683659036701243996477061129} a^{2} + \frac{20698858257595965896192013637681158527419816202894725}{49410433741999192786412230683659036701243996477061129} a + \frac{15833975525315423807230718122635478598646566500457914}{49410433741999192786412230683659036701243996477061129}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 4350812050.15 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ $20$ R $20$ R $20$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
9931Data not computed