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Properties

Label 20.8.31976199190...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{13}\cdot 419^{6}\cdot 695771^{2}$
Root discriminant $66.87$
Ramified primes $5, 419, 695771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11257231, 45658124, -49913465, 19084901, -8610893, -24683413, 21372876, -5483095, 2844358, -1198093, 214863, 84798, 13881, 22634, -6543, -224, -497, -71, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 - 71*x^17 - 497*x^16 - 224*x^15 - 6543*x^14 + 22634*x^13 + 13881*x^12 + 84798*x^11 + 214863*x^10 - 1198093*x^9 + 2844358*x^8 - 5483095*x^7 + 21372876*x^6 - 24683413*x^5 - 8610893*x^4 + 19084901*x^3 - 49913465*x^2 + 45658124*x - 11257231)
 
gp: K = bnfinit(x^20 + 20*x^18 - 71*x^17 - 497*x^16 - 224*x^15 - 6543*x^14 + 22634*x^13 + 13881*x^12 + 84798*x^11 + 214863*x^10 - 1198093*x^9 + 2844358*x^8 - 5483095*x^7 + 21372876*x^6 - 24683413*x^5 - 8610893*x^4 + 19084901*x^3 - 49913465*x^2 + 45658124*x - 11257231, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} - 71 x^{17} - 497 x^{16} - 224 x^{15} - 6543 x^{14} + 22634 x^{13} + 13881 x^{12} + 84798 x^{11} + 214863 x^{10} - 1198093 x^{9} + 2844358 x^{8} - 5483095 x^{7} + 21372876 x^{6} - 24683413 x^{5} - 8610893 x^{4} + 19084901 x^{3} - 49913465 x^{2} + 45658124 x - 11257231 \)

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:  \(3197619919037143226174828028564453125=5^{13}\cdot 419^{6}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.87$
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:  $5, 419, 695771$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{115} a^{18} + \frac{4}{115} a^{17} + \frac{47}{115} a^{16} + \frac{2}{5} a^{15} - \frac{26}{115} a^{14} - \frac{52}{115} a^{13} - \frac{22}{115} a^{12} + \frac{9}{115} a^{11} - \frac{2}{23} a^{10} - \frac{13}{115} a^{9} - \frac{4}{115} a^{8} + \frac{48}{115} a^{7} - \frac{19}{115} a^{6} - \frac{18}{115} a^{5} - \frac{8}{23} a^{4} - \frac{56}{115} a^{3} - \frac{12}{115} a^{2} - \frac{13}{115} a + \frac{31}{115}$, $\frac{1}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{19} + \frac{24150682973218010423111141358986238121756245131616808375276011072587524132333334}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{18} - \frac{665091263970590528613874832214916380044605851890648061399490592748322841999783999}{1484196342291219190304034715365012774386508155691451124094700243250702665568030755} a^{17} + \frac{490229975255295559599842615284864549149480232637805428649995194458030688856856506}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{16} - \frac{863391956368052756041009838426013073288913884012417551016142703528439631200819561}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{15} - \frac{1480591088303889788025395724257911220191968431719957880547709186014741352703183782}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{14} + \frac{2157283689350678489913232475065003870539146056552868845081330072351040181179180708}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{13} + \frac{1184971116229012906969165593098653152790221999184661344023615646051801503467276809}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{12} + \frac{18707091466803627200608984788049360242907733364836867958064628879152523403058616}{56158780519127212606098610851649132003813822107244096587367036231107668426898461} a^{11} - \frac{716319174988892133106973028619386392257499424576118156547103183793981612428712693}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{10} - \frac{2501373326193919080669337003120723115329702214465170536004124986719675825141845719}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{9} - \frac{1646862838035627872252657936347460143958339821366992454863433834759136767157145942}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{8} - \frac{462254075205687127678504438619417306112409833746506142542428652918154097147859152}{1484196342291219190304034715365012774386508155691451124094700243250702665568030755} a^{7} - \frac{2921469409209013949001042922322229759391033853886428947145221541913255669818991053}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{6} + \frac{168265157312812083724563779104701200367208152705652103623973044565620660180869388}{2077874879207706866425648601511017884141111417968031573732580340550983731795243057} a^{5} + \frac{4644947081673662397472397891179485104436657587613304063083557718575079284140770609}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{4} + \frac{2796187879596561177265655461820453609318406827351964828366003579221589757713385538}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{3} - \frac{769812945404502061326472919857681130931375936783220208111748856997480749766784648}{10389374396038534332128243007555089420705557089840157868662901702754918658976215285} a^{2} - \frac{201896756768844549605011540618818901556126838951238742271538775139120934675248003}{451711930262544970962097522067612583508937264775659037767952247945866028651139795} a - \frac{42301940927724721681215397459714702286002749874359304074889093507249282030578796}{2077874879207706866425648601511017884141111417968031573732580340550983731795243057}$

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:  \( 37597322292.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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

Sibling fields

Degree 20 siblings: data not computed
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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
419Data not computed
695771Data not computed