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Properties

Label 22.0.38086571779...5952.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{48}\cdot 3^{23}\cdot 337^{8}\cdot 310501^{8}$
Root discriminant $11{,}799.22$
Ramified primes $2, 3, 337, 310501$
Class number Not computed
Class group Not computed
Galois group 22T27

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9655790329507504656039075, 0, 20751898885985795263838019, 0, 2110663170149138275578021, 0, 43869578581905971661981, 0, 411884307283617185094, 0, 2145714219228954438, 0, 6763660792069842, 0, 13399040743938, 0, 16760085111, 0, 12830103, 0, 5481, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 5481*x^20 + 12830103*x^18 + 16760085111*x^16 + 13399040743938*x^14 + 6763660792069842*x^12 + 2145714219228954438*x^10 + 411884307283617185094*x^8 + 43869578581905971661981*x^6 + 2110663170149138275578021*x^4 + 20751898885985795263838019*x^2 + 9655790329507504656039075)
 
gp: K = bnfinit(x^22 + 5481*x^20 + 12830103*x^18 + 16760085111*x^16 + 13399040743938*x^14 + 6763660792069842*x^12 + 2145714219228954438*x^10 + 411884307283617185094*x^8 + 43869578581905971661981*x^6 + 2110663170149138275578021*x^4 + 20751898885985795263838019*x^2 + 9655790329507504656039075, 1)
 

Normalized defining polynomial

\( x^{22} + 5481 x^{20} + 12830103 x^{18} + 16760085111 x^{16} + 13399040743938 x^{14} + 6763660792069842 x^{12} + 2145714219228954438 x^{10} + 411884307283617185094 x^{8} + 43869578581905971661981 x^{6} + 2110663170149138275578021 x^{4} + 20751898885985795263838019 x^{2} + 9655790329507504656039075 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-380865717798637777174570181499852006314367491547703593645250892679521572925208115336445952=-\,2^{48}\cdot 3^{23}\cdot 337^{8}\cdot 310501^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}799.22$
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:  $2, 3, 337, 310501$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{6} a^{4} - \frac{1}{2}$, $\frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{72} a^{6} - \frac{1}{24} a^{4} - \frac{3}{8} a^{2} - \frac{3}{8}$, $\frac{1}{72} a^{7} - \frac{1}{24} a^{5} - \frac{1}{24} a^{3} - \frac{3}{8} a$, $\frac{1}{216} a^{8} - \frac{1}{2} a^{2} + \frac{1}{8}$, $\frac{1}{432} a^{9} - \frac{1}{432} a^{8} + \frac{1}{12} a^{3} + \frac{1}{4} a^{2} - \frac{7}{16} a + \frac{7}{16}$, $\frac{1}{864} a^{10} - \frac{1}{864} a^{8} - \frac{1}{144} a^{6} - \frac{1}{48} a^{4} + \frac{3}{32} a^{2} - \frac{11}{32}$, $\frac{1}{5184} a^{11} - \frac{1}{1728} a^{10} + \frac{1}{1728} a^{9} - \frac{1}{576} a^{8} - \frac{1}{288} a^{7} - \frac{1}{288} a^{6} + \frac{1}{32} a^{5} - \frac{5}{96} a^{4} - \frac{1}{192} a^{3} + \frac{25}{64} a^{2} + \frac{7}{64} a + \frac{3}{64}$, $\frac{1}{8843904} a^{12} + \frac{487}{1473984} a^{10} - \frac{3959}{2947968} a^{8} + \frac{911}{245664} a^{6} + \frac{4859}{109184} a^{4} - \frac{1}{6} a^{3} + \frac{3545}{54592} a^{2} - \frac{1}{2} a - \frac{17647}{109184}$, $\frac{1}{26531712} a^{13} - \frac{61}{736992} a^{11} - \frac{1}{1728} a^{10} - \frac{751}{2947968} a^{9} + \frac{1}{1728} a^{8} - \frac{559}{122832} a^{7} - \frac{1}{288} a^{6} - \frac{5377}{327552} a^{5} + \frac{1}{32} a^{4} + \frac{733}{27296} a^{3} + \frac{9}{64} a^{2} + \frac{20845}{109184} a + \frac{23}{64}$, $\frac{1}{53063424} a^{14} - \frac{1}{17687808} a^{12} - \frac{53}{1965312} a^{10} - \frac{4121}{1965312} a^{8} + \frac{10157}{1965312} a^{6} - \frac{3855}{218368} a^{4} - \frac{1}{6} a^{3} + \frac{19101}{218368} a^{2} - \frac{45247}{218368}$, $\frac{1}{53063424} a^{15} - \frac{1}{53063424} a^{13} + \frac{1471}{17687808} a^{11} + \frac{355}{655104} a^{9} - \frac{5611}{1965312} a^{7} - \frac{1847}{655104} a^{5} - \frac{1}{12} a^{4} - \frac{92293}{655104} a^{3} - \frac{75209}{218368} a - \frac{1}{4}$, $\frac{1}{5415334672896} a^{16} - \frac{293}{75212981568} a^{14} - \frac{14977}{300851926272} a^{12} + \frac{7856123}{16713995904} a^{10} + \frac{4462357}{2089249488} a^{8} + \frac{15122071}{4178498976} a^{6} - \frac{849938665}{11142663936} a^{4} - \frac{1}{6} a^{3} + \frac{372676663}{1857110656} a^{2} - \frac{1}{2} a - \frac{1163353489}{7428442624}$, $\frac{1}{10830669345792} a^{17} - \frac{1}{10830669345792} a^{16} - \frac{293}{150425963136} a^{15} + \frac{293}{150425963136} a^{14} - \frac{10913}{1805111557632} a^{13} - \frac{6347}{200567950848} a^{12} - \frac{3199727}{33427991808} a^{11} - \frac{40135135}{100283975424} a^{10} - \frac{12088741}{11142663936} a^{9} - \frac{39757937}{100283975424} a^{8} + \frac{550445}{116069416} a^{7} - \frac{15308635}{4178498976} a^{6} + \frac{708187789}{22285327872} a^{5} - \frac{501017459}{7428442624} a^{4} + \frac{1818970879}{11142663936} a^{3} + \frac{1363840183}{3714221312} a^{2} - \frac{789767813}{14856885248} a + \frac{6078206093}{14856885248}$, $\frac{1}{32492008037376} a^{18} - \frac{1}{10830669345792} a^{16} - \frac{181}{66855983616} a^{14} - \frac{9929}{200567950848} a^{12} - \frac{15340405}{50141987712} a^{10} - \frac{160590077}{100283975424} a^{8} + \frac{317566613}{66855983616} a^{6} - \frac{489972271}{22285327872} a^{4} - \frac{1}{6} a^{3} + \frac{5844007957}{14856885248} a^{2} - \frac{1}{2} a - \frac{1272429915}{14856885248}$, $\frac{1}{25408750285228032} a^{19} - \frac{1}{64984016074752} a^{18} - \frac{281}{8469583428409344} a^{17} + \frac{1}{21661338691584} a^{16} - \frac{1442405}{470532412689408} a^{15} + \frac{181}{133711967232} a^{14} + \frac{25214051}{1411597238068224} a^{13} - \frac{38249}{1203407705088} a^{12} - \frac{1835265137}{39211034390784} a^{11} + \frac{56808347}{100283975424} a^{10} + \frac{25486185863}{26140689593856} a^{9} - \frac{285079741}{200567950848} a^{8} + \frac{161071038829}{52281379187712} a^{7} + \frac{275781065}{44570655744} a^{6} + \frac{532645608589}{17427126395904} a^{5} - \frac{2823176821}{44570655744} a^{4} - \frac{4314651515}{683416721408} a^{3} + \frac{5726738491}{29713770496} a^{2} - \frac{439859538107}{11618084263936} a - \frac{14433136397}{29713770496}$, $\frac{1}{2289080061020400131049369650529139436180077853106595020190476702554112} a^{20} + \frac{1885091781733974440774562454718347719550598075356037341}{381513343503400021841561608421523239363346308851099170031746117092352} a^{18} - \frac{12406693279498483214042690278226741884645933594814720717}{254342229002266681227707738947682159575564205900732780021164078061568} a^{16} - \frac{40504831731218378971803655824394862059331532712873577827949}{10597592875094445051154489122820089982315175245863865834215169919232} a^{14} - \frac{704768827404508721833862677818802334914639399669020712670557}{14130123833459260068205985497093453309753566994485154445620226558976} a^{12} - \frac{165818078145350501462343249913982438221183306778234469767984965}{785006879636625559344776972060747406097420388582508580312234808832} a^{10} + \frac{830863731161395915802362112292217421467262828605788764779592099}{523337919757750372896517981373831604064946925721672386874823205888} a^{8} - \frac{1}{144} a^{7} + \frac{50786121607133346632610134872107230308861315175824317998949943}{43611493313145864408043165114485967005412243810139365572901933824} a^{6} + \frac{1}{48} a^{5} - \frac{1171533678659911905559700572725169867344683144915099913992525527}{20523055676774524427314430642111043296664585322418524975483262976} a^{4} - \frac{7}{48} a^{3} - \frac{64636541082480872836771995340736232167437772391514264412610660343}{523337919757750372896517981373831604064946925721672386874823205888} a^{2} - \frac{5}{16} a + \frac{3497300910871820011321050977177935432799915462462072598297029}{52488633444436123855024119289286555745945231003627941113767936}$, $\frac{1}{129667904415179586841461252787920538597990363190042847959240699483507481815040} a^{21} + \frac{51686047269452479511953978179577055805479193010954131446101}{21611317402529931140243542131320089766331727198340474659873449913917913635840} a^{19} - \frac{175926505795167651162794882877337298156987751104864217349790783}{14407544935019954093495694754213393177554484798893649773248966609278609090560} a^{17} - \frac{1}{10830669345792} a^{16} + \frac{327379188405993220910737475942663945818975775706826183506874676537}{75039296536562260903623410178194756133096274994237759235671701089992755680} a^{15} + \frac{293}{150425963136} a^{14} + \frac{9192184100792642852779946692230878189947506140665232098431222967229}{800419163056664116305316375234077398753026933271869431847164811626589393920} a^{13} - \frac{6347}{200567950848} a^{12} + \frac{3275890298969840725132982780142768947762772635316157667266573803653027}{44467731280925784239184243068559855486279274070659412880398045090366077440} a^{11} + \frac{17899573}{100283975424} a^{10} + \frac{24193547080371670885880300610335597126068093715072253347870669461259267}{29645154187283856159456162045706570324186182713772941920265363393577384960} a^{9} + \frac{134346187}{100283975424} a^{8} - \frac{294117508805322777006456522754691693881336165547573753251785103411831}{1235214757803494006644006751904440430174424279740539246677723474732391040} a^{7} + \frac{7054349}{1044624744} a^{6} + \frac{42945337551693046339489343070344124712554370412710488691552102647046373}{1162555066167994359194359295910061581340634616226389879226092682101073920} a^{5} + \frac{1050474775}{22285327872} a^{4} + \frac{1393319215531417639099845563719279718162343041386459888099365943355910809}{29645154187283856159456162045706570324186182713772941920265363393577384960} a^{3} + \frac{1073666643}{3714221312} a^{2} + \frac{20154506892986839761954685938297754788227002647679029469966292030348581}{68385592127529079952609370347650681255331448013317051719181922476533760} a + \frac{6310344925}{14856885248}$

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

Class group and class number

Not computed

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:  \( -\frac{31818886217381155485372444787633773079}{741928348175010208383090783784881371934159109054118467397445120} a^{21} - \frac{125297646224066048133914623579657173989801}{494618898783340138922060522523254247956106072702745644931630080} a^{19} - \frac{1532345135976662573753278532033968694157619}{2389463279146570719430244070160648540850753974409399250877440} a^{17} - \frac{24957481229440419348705226789005285995692285367}{27478827710185563273447806806847458219783670705708091385090560} a^{15} - \frac{7217464082969853330426906116061682363998371557727}{9159609236728521091149268935615819406594556901902697128363520} a^{13} - \frac{658087523839187059916436073131513908333494105502063}{1526601539454753515191544822602636567765759483650449521393920} a^{11} - \frac{9794228123454643392422148344792372388756694193231419}{66373979976293631095284557504462459468076499289149979191040} a^{9} - \frac{30825788606705594702239302906393325438834918761194614739}{1017734359636502343461029881735091045177172989100299680929280} a^{7} - \frac{33837456035317058868818737489045426459942648089432915043}{9977787839573552386872841977795010246835029304904898832640} a^{5} - \frac{112081595455862220556663242337135805819615804309326331327939}{678489573091001562307353254490060696784781992733533120619520} a^{3} - \frac{1031398728818815565784924817772516831875236254854591627773}{782571595260670775441007213944706686026276808227835202560} a + \frac{1}{2} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T27:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15840
The 20 conjugacy class representatives for t22n27
Character table for t22n27

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.11.118769262421915560193703211428553337469927424.1

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

Sibling fields

Degree 22 sibling: data not computed
Degree 24 sibling: data not computed
Degree 44 sibling: 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ $22$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
3Data not computed
337Data not computed
310501Data not computed