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Properties

Label 20.8.13156433165...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 401^{11}$
Root discriminant $90.36$
Ramified primes $5, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![399411, 10036710, 18383031, -233541, -19686100, -75477676, 33741880, -4805649, 3413545, 1475347, -573908, 33476, -62529, 14837, -6100, 568, -7, -93, 32, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 32*x^18 - 93*x^17 - 7*x^16 + 568*x^15 - 6100*x^14 + 14837*x^13 - 62529*x^12 + 33476*x^11 - 573908*x^10 + 1475347*x^9 + 3413545*x^8 - 4805649*x^7 + 33741880*x^6 - 75477676*x^5 - 19686100*x^4 - 233541*x^3 + 18383031*x^2 + 10036710*x + 399411)
 
gp: K = bnfinit(x^20 - 4*x^19 + 32*x^18 - 93*x^17 - 7*x^16 + 568*x^15 - 6100*x^14 + 14837*x^13 - 62529*x^12 + 33476*x^11 - 573908*x^10 + 1475347*x^9 + 3413545*x^8 - 4805649*x^7 + 33741880*x^6 - 75477676*x^5 - 19686100*x^4 - 233541*x^3 + 18383031*x^2 + 10036710*x + 399411, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 32 x^{18} - 93 x^{17} - 7 x^{16} + 568 x^{15} - 6100 x^{14} + 14837 x^{13} - 62529 x^{12} + 33476 x^{11} - 573908 x^{10} + 1475347 x^{9} + 3413545 x^{8} - 4805649 x^{7} + 33741880 x^{6} - 75477676 x^{5} - 19686100 x^{4} - 233541 x^{3} + 18383031 x^{2} + 10036710 x + 399411 \)

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:  \(1315643316557894633135034313995361328125=5^{15}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.36$
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:  $5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{9} a^{8} + \frac{2}{9} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{15} - \frac{1}{27} a^{13} + \frac{1}{9} a^{11} - \frac{2}{27} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{2}{9} a^{6} - \frac{10}{27} a^{5} + \frac{2}{9} a^{4} + \frac{4}{27} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{14} + \frac{1}{9} a^{12} - \frac{2}{27} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{10}{27} a^{6} - \frac{1}{9} a^{5} + \frac{4}{27} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{17} + \frac{2}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{9} a^{10} + \frac{4}{27} a^{9} + \frac{1}{9} a^{8} - \frac{4}{27} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{27} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{18} - \frac{1}{27} a^{14} + \frac{4}{27} a^{12} - \frac{1}{9} a^{11} + \frac{4}{27} a^{10} + \frac{1}{9} a^{9} + \frac{2}{27} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{7}{27} a^{4} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{19} + \frac{5156478655660791096677634499816425782964895475978431072520955618284777218}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{18} + \frac{13069680208411742419486664501626487801829052926292140638969793195116668442}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{17} - \frac{8196033413920276320240800689113647841394115577368130614510108040955320803}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{16} - \frac{68727609491082631563259659029963613347589681131451515344517835616313608}{4776243199537832883740748331626054855652493902399494824623078493675623917} a^{15} + \frac{27218260953436476383533440260878848605910425523569017840332648258799765627}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{14} - \frac{15385774264596295610238285572083971269010546198987779086268965093397476214}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{13} + \frac{47450330776063370576379326440122669602338466688054014094266977752692201509}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{12} - \frac{42645220858675172583541588577465221184970945464035972869303715464713891249}{272245862373656474373222654902685126772192152436771205003515474139510563269} a^{11} - \frac{23071732131916657222350507491953975319187407175031105504206384401698647007}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{10} - \frac{117702596781990994808764885590385576095752691742703950578078334318351316902}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{9} - \frac{28557881303221547391766967606332511693967687256111948533706870703662688407}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{8} + \frac{52308189370343826146868944902213975450729889054611172079536769134668022626}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{7} - \frac{368281722563318501667183410837991810760689431352786804559022139146296131016}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{6} + \frac{47174421163079745515447094550098622688095234158187285288323427020787783147}{272245862373656474373222654902685126772192152436771205003515474139510563269} a^{5} - \frac{185793887273519939557955406341454037979504797289133102712484780325716555776}{816737587120969423119667964708055380316576457310313615010546422418531689807} a^{4} + \frac{16477541162554425252613958311050274449459511374022963574025103955790943055}{90748620791218824791074218300895042257397384145590401667838491379836854423} a^{3} - \frac{4086323827742034118314287905722000150716729975404236695135614998369097564}{272245862373656474373222654902685126772192152436771205003515474139510563269} a^{2} + \frac{4178040087148941810751408148093223841248711086737759622577135069783707720}{30249540263739608263691406100298347419132461381863467222612830459945618141} a - \frac{34904815666776008128094962573838044062916187651516697299568463707898387}{30249540263739608263691406100298347419132461381863467222612830459945618141}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1196517878250 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 28 conjugacy class representatives for t20n138
Character table for t20n138 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1

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 $20$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
401Data not computed