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Properties

Label 20.0.88714761273...6272.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 71^{16}$
Root discriminant $352.70$
Ramified primes $2, 3, 71$
Class number $3579906550$ (GRH)
Class group $[5, 715981310]$ (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![26337021631, 15691933916, 31324467034, 22891449984, 16874705783, 10751869216, 5294165550, 1956241672, 594058091, 125920388, 14056380, -1537224, -1036720, -238944, -12750, 7188, 2039, -32, -46, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 46*x^18 - 32*x^17 + 2039*x^16 + 7188*x^15 - 12750*x^14 - 238944*x^13 - 1036720*x^12 - 1537224*x^11 + 14056380*x^10 + 125920388*x^9 + 594058091*x^8 + 1956241672*x^7 + 5294165550*x^6 + 10751869216*x^5 + 16874705783*x^4 + 22891449984*x^3 + 31324467034*x^2 + 15691933916*x + 26337021631)
 
gp: K = bnfinit(x^20 - 4*x^19 - 46*x^18 - 32*x^17 + 2039*x^16 + 7188*x^15 - 12750*x^14 - 238944*x^13 - 1036720*x^12 - 1537224*x^11 + 14056380*x^10 + 125920388*x^9 + 594058091*x^8 + 1956241672*x^7 + 5294165550*x^6 + 10751869216*x^5 + 16874705783*x^4 + 22891449984*x^3 + 31324467034*x^2 + 15691933916*x + 26337021631, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 46 x^{18} - 32 x^{17} + 2039 x^{16} + 7188 x^{15} - 12750 x^{14} - 238944 x^{13} - 1036720 x^{12} - 1537224 x^{11} + 14056380 x^{10} + 125920388 x^{9} + 594058091 x^{8} + 1956241672 x^{7} + 5294165550 x^{6} + 10751869216 x^{5} + 16874705783 x^{4} + 22891449984 x^{3} + 31324467034 x^{2} + 15691933916 x + 26337021631 \)

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:  \(887147612731669685863693358770856673695177241526272=2^{55}\cdot 3^{10}\cdot 71^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $352.70$
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:  $2, 3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3408=2^{4}\cdot 3\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{3408}(1,·)$, $\chi_{3408}(5,·)$, $\chi_{3408}(25,·)$, $\chi_{3408}(125,·)$, $\chi_{3408}(289,·)$, $\chi_{3408}(341,·)$, $\chi_{3408}(409,·)$, $\chi_{3408}(625,·)$, $\chi_{3408}(1421,·)$, $\chi_{3408}(1445,·)$, $\chi_{3408}(1705,·)$, $\chi_{3408}(1709,·)$, $\chi_{3408}(1729,·)$, $\chi_{3408}(1829,·)$, $\chi_{3408}(1993,·)$, $\chi_{3408}(2045,·)$, $\chi_{3408}(2113,·)$, $\chi_{3408}(2329,·)$, $\chi_{3408}(3125,·)$, $\chi_{3408}(3149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{69} a^{13} - \frac{4}{69} a^{12} + \frac{8}{69} a^{11} - \frac{4}{69} a^{10} - \frac{14}{69} a^{9} - \frac{34}{69} a^{8} + \frac{10}{23} a^{7} + \frac{14}{69} a^{6} - \frac{8}{23} a^{5} + \frac{5}{69} a^{4} - \frac{25}{69} a^{3} + \frac{10}{23} a^{2} - \frac{2}{23} a - \frac{1}{3}$, $\frac{1}{2829} a^{14} - \frac{5}{2829} a^{13} + \frac{449}{2829} a^{12} + \frac{157}{943} a^{11} - \frac{194}{2829} a^{10} - \frac{321}{943} a^{9} - \frac{925}{2829} a^{8} + \frac{102}{943} a^{7} + \frac{376}{2829} a^{6} + \frac{278}{943} a^{5} + \frac{404}{943} a^{4} + \frac{118}{943} a^{3} - \frac{334}{943} a^{2} - \frac{109}{2829} a - \frac{38}{123}$, $\frac{1}{2829} a^{15} + \frac{14}{2829} a^{13} - \frac{359}{2829} a^{12} - \frac{176}{2829} a^{11} - \frac{293}{2829} a^{10} + \frac{1}{3} a^{9} - \frac{752}{2829} a^{8} - \frac{964}{2829} a^{7} - \frac{197}{2829} a^{6} - \frac{809}{2829} a^{5} - \frac{117}{943} a^{4} + \frac{215}{943} a^{3} + \frac{166}{943} a^{2} - \frac{845}{2829} a + \frac{5}{41}$, $\frac{1}{2829} a^{16} - \frac{2}{2829} a^{13} - \frac{22}{943} a^{12} + \frac{124}{2829} a^{11} - \frac{106}{943} a^{10} - \frac{718}{2829} a^{9} + \frac{1285}{2829} a^{8} + \frac{119}{943} a^{7} + \frac{258}{943} a^{6} - \frac{55}{2829} a^{5} + \frac{381}{943} a^{4} - \frac{420}{943} a^{3} + \frac{349}{943} a^{2} - \frac{794}{2829} a - \frac{14}{41}$, $\frac{1}{2829} a^{17} + \frac{2}{943} a^{13} - \frac{83}{943} a^{12} + \frac{337}{2829} a^{11} + \frac{452}{2829} a^{10} - \frac{282}{943} a^{9} + \frac{459}{943} a^{8} + \frac{74}{2829} a^{7} - \frac{1}{69} a^{6} - \frac{1043}{2829} a^{5} - \frac{104}{943} a^{4} - \frac{1238}{2829} a^{3} - \frac{427}{943} a^{2} - \frac{733}{2829} a + \frac{2}{41}$, $\frac{1}{6251767876775063022239588523} a^{18} + \frac{444406354755399452421589}{6251767876775063022239588523} a^{17} - \frac{21554738302999781726150}{271815994642394044445199501} a^{16} + \frac{718030822600479843745592}{6251767876775063022239588523} a^{15} + \frac{285214366097165473235906}{2083922625591687674079862841} a^{14} + \frac{16562864839281063165894293}{6251767876775063022239588523} a^{13} - \frac{210368745934098862403561881}{2083922625591687674079862841} a^{12} + \frac{494156604801708562467375973}{6251767876775063022239588523} a^{11} - \frac{13062072001229606517992606}{90605331547464681481733167} a^{10} - \frac{715975464885250916905296934}{6251767876775063022239588523} a^{9} + \frac{551762378010435819934862981}{6251767876775063022239588523} a^{8} + \frac{260376261033560245893571515}{2083922625591687674079862841} a^{7} - \frac{3120776149212512820323875571}{6251767876775063022239588523} a^{6} - \frac{521322313537908007606597349}{6251767876775063022239588523} a^{5} + \frac{1136218635166832233145080451}{6251767876775063022239588523} a^{4} - \frac{53297618237038888372925}{6629658405912049864517061} a^{3} - \frac{2115705733490818274627160920}{6251767876775063022239588523} a^{2} + \frac{633832995461058053199261828}{2083922625591687674079862841} a + \frac{115224538785561678547287151}{271815994642394044445199501}$, $\frac{1}{13053932582128274441231920528508542272372159333992054157} a^{19} - \frac{667116392397969985118194052}{13053932582128274441231920528508542272372159333992054157} a^{18} + \frac{939525976359004840337473150251182949766359295141112}{13053932582128274441231920528508542272372159333992054157} a^{17} + \frac{233320370169163217947591577598265842352355710247379}{4351310860709424813743973509502847424124053111330684719} a^{16} + \frac{1499004210089873651377797606267896159898583899843218}{13053932582128274441231920528508542272372159333992054157} a^{15} - \frac{538415459084610470279319307969195072857349008401232}{4351310860709424813743973509502847424124053111330684719} a^{14} + \frac{4218519786346247031201599694175308378387308443705419}{4351310860709424813743973509502847424124053111330684719} a^{13} + \frac{30595889477771342987006366607401068404327967823503237}{189187428726496731032346674326210757570611004840464553} a^{12} - \frac{1613317684495330049599970908943058772987958626052547638}{13053932582128274441231920528508542272372159333992054157} a^{11} - \frac{1012640518834472411513227887356580123590795511952129100}{13053932582128274441231920528508542272372159333992054157} a^{10} + \frac{895875606622074711390061205436486382691929794179674233}{4351310860709424813743973509502847424124053111330684719} a^{9} + \frac{1783958187265232899431300917477411287392956628101848612}{13053932582128274441231920528508542272372159333992054157} a^{8} + \frac{19335492585558473859437456801649662496535939669529118}{189187428726496731032346674326210757570611004840464553} a^{7} - \frac{5194934360101187959809310131848288736337250638052790325}{13053932582128274441231920528508542272372159333992054157} a^{6} - \frac{886281458637106889992556167311941044257538829963847658}{4351310860709424813743973509502847424124053111330684719} a^{5} + \frac{993092466348488926316401597007348497715658404324913268}{13053932582128274441231920528508542272372159333992054157} a^{4} + \frac{592854924417156195592296814884852339636482960883338012}{13053932582128274441231920528508542272372159333992054157} a^{3} - \frac{1426896317429620877439605633280607543868418021341081711}{13053932582128274441231920528508542272372159333992054157} a^{2} + \frac{5830921845482231662101159800729816678531998437424778704}{13053932582128274441231920528508542272372159333992054157} a + \frac{195446500006566660136529316849397472476856986051673233}{567562286179490193097040022978632272711833014521393659}$

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

Class group and class number

$C_{5}\times C_{715981310}$, which has order $3579906550$ (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:  \( 116573225.49574807 \) (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

\(\Q(\sqrt{2}) \), 4.0.18432.2, 5.5.25411681.1, 10.10.21160051711861096448.1

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 R R $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
3Data not computed
$71$71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$