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Properties

Label 20.2.84117995262...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{20}\cdot 5^{10}\cdot 60662149^{5}$
Root discriminant $394.68$
Ramified primes $2, 5, 60662149$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-223230418947143342493949, 0, -9498393828889421896758, 0, -99376354587176198511, 0, 55091880668161990, 0, 7046810726338171, 0, 58487400397679, 0, 265312180916, 0, 775539485, 0, 1535084, 0, 1912, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 1912*x^18 + 1535084*x^16 + 775539485*x^14 + 265312180916*x^12 + 58487400397679*x^10 + 7046810726338171*x^8 + 55091880668161990*x^6 - 99376354587176198511*x^4 - 9498393828889421896758*x^2 - 223230418947143342493949)
 
gp: K = bnfinit(x^20 + 1912*x^18 + 1535084*x^16 + 775539485*x^14 + 265312180916*x^12 + 58487400397679*x^10 + 7046810726338171*x^8 + 55091880668161990*x^6 - 99376354587176198511*x^4 - 9498393828889421896758*x^2 - 223230418947143342493949, 1)
 

Normalized defining polynomial

\( x^{20} + 1912 x^{18} + 1535084 x^{16} + 775539485 x^{14} + 265312180916 x^{12} + 58487400397679 x^{10} + 7046810726338171 x^{8} + 55091880668161990 x^{6} - 99376354587176198511 x^{4} - 9498393828889421896758 x^{2} - 223230418947143342493949 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8411799526250997899915089732983470958069760000000000=-\,2^{20}\cdot 5^{10}\cdot 60662149^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $394.68$
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:  $2, 5, 60662149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{60662149} a^{16} + \frac{1912}{60662149} a^{14} + \frac{1535084}{60662149} a^{12} - \frac{13068452}{60662149} a^{10} - \frac{24058810}{60662149} a^{8} - \frac{10560671}{60662149} a^{6} + \frac{13170392}{60662149} a^{4} + \frac{14580742}{60662149} a^{2}$, $\frac{1}{60662149} a^{17} + \frac{1912}{60662149} a^{15} + \frac{1535084}{60662149} a^{13} - \frac{13068452}{60662149} a^{11} - \frac{24058810}{60662149} a^{9} - \frac{10560671}{60662149} a^{7} + \frac{13170392}{60662149} a^{5} + \frac{14580742}{60662149} a^{3}$, $\frac{1}{6785413178253979723568169200526596264158702786440195913095823574995610314140861469713036084798198478} a^{18} + \frac{13882029638914298793859067637454939445905673214107117808041808750240047064948286757513676345}{6785413178253979723568169200526596264158702786440195913095823574995610314140861469713036084798198478} a^{16} + \frac{1514679082406141352159360911319586642221887389163943075083350697905568829071259221462711944992675199}{6785413178253979723568169200526596264158702786440195913095823574995610314140861469713036084798198478} a^{14} - \frac{1231171771877285932436324533278865990108419423553283545818197135974611618338175194327118240187204720}{3392706589126989861784084600263298132079351393220097956547911787497805157070430734856518042399099239} a^{12} + \frac{2914511541816674522490486659012353599182244040221954592956488553770778778123823320220209981006285}{7728261023068314035954634624745553831615834608701817668674058741452859127723076844775667522549201} a^{10} - \frac{1469650047746822544777061515915530608171102083849332998482377550323361713350071876998742173254273375}{6785413178253979723568169200526596264158702786440195913095823574995610314140861469713036084798198478} a^{8} + \frac{72278427951237347314736852124932732038883684263779771945244993640652970353386752169784139516734895}{178563504690894203251793926329647270109439547011584102976205883552516060898443722887185160126268381} a^{6} - \frac{1578498630826242370568758410569684065937962017070908976658294713831798527754627030267783725279202820}{3392706589126989861784084600263298132079351393220097956547911787497805157070430734856518042399099239} a^{4} - \frac{6538930522591692913432481712236549970216798873804175690040080648507225161474081527408333731}{111855799540731399469019292417856747280065907101975498973764077744684091461066792568015288822} a^{2} + \frac{61806527729175025093803160284498214483350989076897170685353099997942897946215993521}{1843914226328041881091606108742648521734136176118249601967844524345553459721098778878}$, $\frac{1}{6785413178253979723568169200526596264158702786440195913095823574995610314140861469713036084798198478} a^{19} + \frac{13882029638914298793859067637454939445905673214107117808041808750240047064948286757513676345}{6785413178253979723568169200526596264158702786440195913095823574995610314140861469713036084798198478} a^{17} + \frac{1514679082406141352159360911319586642221887389163943075083350697905568829071259221462711944992675199}{6785413178253979723568169200526596264158702786440195913095823574995610314140861469713036084798198478} a^{15} - \frac{1231171771877285932436324533278865990108419423553283545818197135974611618338175194327118240187204720}{3392706589126989861784084600263298132079351393220097956547911787497805157070430734856518042399099239} a^{13} + \frac{2914511541816674522490486659012353599182244040221954592956488553770778778123823320220209981006285}{7728261023068314035954634624745553831615834608701817668674058741452859127723076844775667522549201} a^{11} - \frac{1469650047746822544777061515915530608171102083849332998482377550323361713350071876998742173254273375}{6785413178253979723568169200526596264158702786440195913095823574995610314140861469713036084798198478} a^{9} + \frac{72278427951237347314736852124932732038883684263779771945244993640652970353386752169784139516734895}{178563504690894203251793926329647270109439547011584102976205883552516060898443722887185160126268381} a^{7} - \frac{1578498630826242370568758410569684065937962017070908976658294713831798527754627030267783725279202820}{3392706589126989861784084600263298132079351393220097956547911787497805157070430734856518042399099239} a^{5} - \frac{6538930522591692913432481712236549970216798873804175690040080648507225161474081527408333731}{111855799540731399469019292417856747280065907101975498973764077744684091461066792568015288822} a^{3} + \frac{61806527729175025093803160284498214483350989076897170685353099997942897946215993521}{1843914226328041881091606108742648521734136176118249601967844524345553459721098778878} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $10$
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:  \( 238962828323000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.24264859600.2, 10.6.189569215625.1

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 24 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 R $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
60662149Data not computed