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Properties

Label 22.22.1991232159...0993.1
Degree $22$
Signature $[22, 0]$
Discriminant $3^{21}\cdot 11^{33}\cdot 31^{10}$
Root discriminant $495.92$
Ramified primes $3, 11, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T11

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![122440026897, 648146044458, 841782926853, -577327236750, -1357714398150, 211943243358, 712727063631, -34101623160, -180180897930, 2424565242, 25255546767, -62767890, -2070767160, 0, 100074348, 0, -2784870, 0, 42669, 0, -330, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 330*x^20 + 42669*x^18 - 2784870*x^16 + 100074348*x^14 - 2070767160*x^12 - 62767890*x^11 + 25255546767*x^10 + 2424565242*x^9 - 180180897930*x^8 - 34101623160*x^7 + 712727063631*x^6 + 211943243358*x^5 - 1357714398150*x^4 - 577327236750*x^3 + 841782926853*x^2 + 648146044458*x + 122440026897)
 
gp: K = bnfinit(x^22 - 330*x^20 + 42669*x^18 - 2784870*x^16 + 100074348*x^14 - 2070767160*x^12 - 62767890*x^11 + 25255546767*x^10 + 2424565242*x^9 - 180180897930*x^8 - 34101623160*x^7 + 712727063631*x^6 + 211943243358*x^5 - 1357714398150*x^4 - 577327236750*x^3 + 841782926853*x^2 + 648146044458*x + 122440026897, 1)
 

Normalized defining polynomial

\( x^{22} - 330 x^{20} + 42669 x^{18} - 2784870 x^{16} + 100074348 x^{14} - 2070767160 x^{12} - 62767890 x^{11} + 25255546767 x^{10} + 2424565242 x^{9} - 180180897930 x^{8} - 34101623160 x^{7} + 712727063631 x^{6} + 211943243358 x^{5} - 1357714398150 x^{4} - 577327236750 x^{3} + 841782926853 x^{2} + 648146044458 x + 122440026897 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(199123215915132850596583390027485698365466594244243490410993=3^{21}\cdot 11^{33}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $495.92$
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:  $3, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{9} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{10} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{1116} a^{15} + \frac{7}{186} a^{13} - \frac{1}{62} a^{11} + \frac{3}{31} a^{9} - \frac{9}{62} a^{7} + \frac{15}{124} a^{5} + \frac{15}{124} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{1116} a^{16} + \frac{7}{186} a^{14} - \frac{1}{62} a^{12} + \frac{3}{31} a^{10} - \frac{9}{62} a^{8} + \frac{15}{124} a^{6} + \frac{15}{124} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{71889372} a^{17} - \frac{4903}{71889372} a^{16} - \frac{5407}{71889372} a^{15} - \frac{65793}{1996927} a^{14} + \frac{72235}{3993854} a^{13} - \frac{208507}{23963124} a^{12} + \frac{113350}{5990781} a^{11} + \frac{3286657}{23963124} a^{10} - \frac{677251}{11981562} a^{9} - \frac{514610}{5990781} a^{8} - \frac{2933145}{7987708} a^{7} + \frac{846953}{7987708} a^{6} - \frac{527295}{3993854} a^{5} - \frac{3870741}{7987708} a^{4} + \frac{15040}{64417} a^{3} - \frac{38509}{128834} a^{2} - \frac{89283}{257668} a - \frac{16605}{257668}$, $\frac{1}{71889372} a^{18} - \frac{17275}{71889372} a^{16} - \frac{6751}{23963124} a^{15} + \frac{191413}{11981562} a^{14} + \frac{714565}{23963124} a^{13} + \frac{183701}{23963124} a^{12} + \frac{104335}{7987708} a^{11} + \frac{242891}{23963124} a^{10} - \frac{73934}{1996927} a^{9} + \frac{3635201}{23963124} a^{8} + \frac{311901}{1996927} a^{7} - \frac{271484}{1996927} a^{6} - \frac{991543}{1996927} a^{5} + \frac{3962053}{7987708} a^{4} + \frac{50853}{257668} a^{3} - \frac{15833}{128834} a^{2} + \frac{71195}{257668} a - \frac{120061}{257668}$, $\frac{1}{2228570532} a^{19} + \frac{11}{2228570532} a^{17} + \frac{5183}{35944686} a^{16} + \frac{3283}{60231636} a^{15} - \frac{86911}{23963124} a^{14} - \frac{232309}{742856844} a^{13} - \frac{119492}{5990781} a^{12} - \frac{50023}{10038606} a^{11} + \frac{2377667}{23963124} a^{10} - \frac{45633977}{371428422} a^{9} + \frac{38243653}{371428422} a^{8} - \frac{836678}{1996927} a^{7} - \frac{745836}{1996927} a^{6} - \frac{1620583}{3993854} a^{5} - \frac{1169337}{7987708} a^{4} + \frac{49001}{128834} a^{3} + \frac{22175}{64417} a^{2} - \frac{109173}{257668} a - \frac{32143}{64417}$, $\frac{1}{2228570532} a^{20} + \frac{11}{2228570532} a^{18} + \frac{52753}{1114285266} a^{16} + \frac{3107}{71889372} a^{15} + \frac{1994731}{742856844} a^{14} - \frac{313529}{23963124} a^{13} + \frac{96981}{247618948} a^{12} + \frac{214747}{11981562} a^{11} + \frac{19203739}{742856844} a^{10} - \frac{12943965}{123809474} a^{9} + \frac{335387}{5990781} a^{8} + \frac{458993}{7987708} a^{7} + \frac{544207}{1996927} a^{6} + \frac{3912561}{7987708} a^{5} + \frac{823713}{7987708} a^{4} + \frac{6675}{64417} a^{3} + \frac{17092}{64417} a^{2} + \frac{46609}{128834} a - \frac{30907}{64417}$, $\frac{1}{177035103608400898686222413216243525551912649498071802707415640000593292} a^{21} + \frac{201343915502435842235887405357096126622917710650460562175167}{3052329372558636184245214020969715957791597405139169012196821379320574} a^{20} + \frac{17623225893115460460008316912577974151392149093129960690335835}{177035103608400898686222413216243525551912649498071802707415640000593292} a^{19} - \frac{41117730518316770578480381974521730098583342777382443966011057}{6104658745117272368490428041939431915583194810278338024393642758641148} a^{18} - \frac{2958663389002707493908625054464884324742976109408415959811721}{2240950678587353147926865990079031969011552525292048135536906835450548} a^{17} + \frac{778441105227734759828091039704328234402675200170086077033150665}{77274161330598384411271241037207998931432845699725797777134718463812} a^{16} - \frac{571336073663819238080459955091695024945772644973319820292101115985}{3052329372558636184245214020969715957791597405139169012196821379320574} a^{15} - \frac{12443107242283026027339368526191300057259883744235392216052437416361}{2034886248372424122830142680646477305194398270092779341464547586213716} a^{14} - \frac{272826698380819485684144841659259919913996343762787817308418714935137}{19670567067600099854024712579582613950212516610896866967490626666732588} a^{13} + \frac{1228568744526051501574702955535361181878577851269658045179992578263}{169573854031035343569178556720539775432866522507731611788712298851143} a^{12} + \frac{1824799240955954944011022568503918425699032785371234723293732060915539}{59011701202800299562074137738747841850637549832690600902471880000197764} a^{11} - \frac{23612441853132159803521932775748372653225933183400071223723106537487}{169573854031035343569178556720539775432866522507731611788712298851143} a^{10} - \frac{7604756761886788185300769839510055615492886998592914404998425798874}{508721562093106030707535670161619326298599567523194835366136896553429} a^{9} - \frac{76220270374532202834028215520283544670616216548698572851240112217187}{508721562093106030707535670161619326298599567523194835366136896553429} a^{8} + \frac{61364700135704949309642919032378516391005065299765884449816301140924}{158633605383871773016328327254698499598488037184652152963634086022037} a^{7} + \frac{530899055940458828059928606272508276935199028379826576687461755717}{5470124323581785276425114732920637917189242661539729412539106414553} a^{6} + \frac{3906955004572553842304249706063069991799104106868284946679184615611}{634534421535487092065313309018793998393952148738608611854536344088148} a^{5} + \frac{4425069461667927682500756862171761415805456750667630464981687700177}{21880497294327141105700458931682551668756970646158917650156425658212} a^{4} + \frac{4147113375553450729873736912267847358166545763532169537596796550457}{10234426153798178904279246919657967716031486269977558255718328130454} a^{3} - \frac{64841053239126748133254770451365368469644677404460962734298344099}{176455623341347912142745636545827029586749763275475142339971174663} a^{2} + \frac{7859013991258847421092830526048217865603020407437239412202652646637}{20468852307596357808558493839315935432062972539955116511436656260908} a - \frac{1298645063466101658898412344832336167197947637681010280845580963}{705822493365391648570982546183308118346999053101900569359884698652}$

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:  $21$
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:  \( 307607921484000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T11:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1210
The 25 conjugacy class representatives for t22n11
Character table for t22n11 is not computed

Intermediate fields

\(\Q(\sqrt{33}) \)

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

Sibling fields

Degree 22 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
11Data not computed
31Data not computed