↑

Properties

Label 22.0.24689351133...3179.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,11^{11}\cdot 89^{21}$
Root discriminant $240.70$
Ramified primes $11, 89$
Class number $82867144$ (GRH)
Class group $[82867144]$ (GRH)
Galois group $C_{22}$ (as 22T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![364599064657, -383812617252, 728343357757, -675845066158, 549328483842, -333264099002, 199796166276, -72371663593, 35724961604, -12395573094, 3941677367, -1208195986, 333560957, -63602020, 20490513, -1847925, 812810, -28841, 19126, -230, 225, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 225*x^20 - 230*x^19 + 19126*x^18 - 28841*x^17 + 812810*x^16 - 1847925*x^15 + 20490513*x^14 - 63602020*x^13 + 333560957*x^12 - 1208195986*x^11 + 3941677367*x^10 - 12395573094*x^9 + 35724961604*x^8 - 72371663593*x^7 + 199796166276*x^6 - 333264099002*x^5 + 549328483842*x^4 - 675845066158*x^3 + 728343357757*x^2 - 383812617252*x + 364599064657)
 
gp: K = bnfinit(x^22 - x^21 + 225*x^20 - 230*x^19 + 19126*x^18 - 28841*x^17 + 812810*x^16 - 1847925*x^15 + 20490513*x^14 - 63602020*x^13 + 333560957*x^12 - 1208195986*x^11 + 3941677367*x^10 - 12395573094*x^9 + 35724961604*x^8 - 72371663593*x^7 + 199796166276*x^6 - 333264099002*x^5 + 549328483842*x^4 - 675845066158*x^3 + 728343357757*x^2 - 383812617252*x + 364599064657, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 225 x^{20} - 230 x^{19} + 19126 x^{18} - 28841 x^{17} + 812810 x^{16} - 1847925 x^{15} + 20490513 x^{14} - 63602020 x^{13} + 333560957 x^{12} - 1208195986 x^{11} + 3941677367 x^{10} - 12395573094 x^{9} + 35724961604 x^{8} - 72371663593 x^{7} + 199796166276 x^{6} - 333264099002 x^{5} + 549328483842 x^{4} - 675845066158 x^{3} + 728343357757 x^{2} - 383812617252 x + 364599064657 \)

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:  \(-24689351133194187911921679520154637032130685006493179=-\,11^{11}\cdot 89^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.70$
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:  $11, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(979=11\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{979}(978,·)$, $\chi_{979}(1,·)$, $\chi_{979}(67,·)$, $\chi_{979}(901,·)$, $\chi_{979}(769,·)$, $\chi_{979}(648,·)$, $\chi_{979}(331,·)$, $\chi_{979}(716,·)$, $\chi_{979}(78,·)$, $\chi_{979}(912,·)$, $\chi_{979}(210,·)$, $\chi_{979}(340,·)$, $\chi_{979}(406,·)$, $\chi_{979}(87,·)$, $\chi_{979}(934,·)$, $\chi_{979}(615,·)$, $\chi_{979}(263,·)$, $\chi_{979}(364,·)$, $\chi_{979}(45,·)$, $\chi_{979}(892,·)$, $\chi_{979}(573,·)$, $\chi_{979}(639,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{21} + \frac{435181412632840434463223211280951836151755003082777437586383504523418189807250927341571846493114702973580398841221451214601788171428800}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{20} - \frac{38183455914123103019948088473091418026272512071589437325622885460903385682741807155018545841604804790640061010338065337393574591607531}{83541433582583439081693412041284794531986585613274068030439474135478434875241303072002050971856239931909311684282909025856161613848323} a^{19} - \frac{291538590377950483995195873347120286665918435724375239578121658919344363274830814922183929238253412510825222621612192357436528771536432}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{18} + \frac{1336932460822908292261438524417807098025773165990155230111479808790428270540168556547272934192074626536781773720672277372796142731293779}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{17} + \frac{1255597000442757593727620546080292792397779242631603583609371161488362445132955433758985227164686784989942596391588284597268428728121586}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{16} + \frac{846944064383434771023615288988828560379833028983856869809187061328466234884268962459166717208214690335440210142508483308039604990371036}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{15} - \frac{233220047735191012544849749760312237786875085380393874533029375946158282682910664241875037168608316533054292755166165862044832337625440}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{14} + \frac{1321719561205352353568080867502964783203553725450567220534671914849731008480005756961686212141923407670509603048986951831563123743468498}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{13} - \frac{416704702445180947924011478102176309497745099692618261667905131446006293223109432018324136656799971986960928311749603683181404874482508}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{12} - \frac{859261124912010784308810360401542017798898479883608848032009066642276943841196293861881486144516939641056612139640485336458640036958923}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{11} - \frac{781659299589663412170050309686297957732073595237860034290758030305051614272323173149994480654742990082409398240118178970821597710813774}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{10} + \frac{806133360622394209607005969216139355582101293637200151380699103285139409060536805162761560295904290541166172522408680525904179663927714}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{9} - \frac{1231535268836186322140189952423112580950488994630775193341710283896906566331780734424789027483926291420301104845270478967961385354496299}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{8} + \frac{19695151307094783559758096578769875304825713272793981887428773076024614902631264419805144268487795273126487110308066723473223884163436}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{7} + \frac{1214369711478003163378921601805773572686206831991058257925006345128421620462221788304978496809836200554891593852620479265258156699311330}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{6} - \frac{1209975476639883372018450840043763042744858385330464431774731482015354086035747965173715463859264288147596780382772460448160746201407502}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{5} - \frac{1298913045763765784056212346604470453067987282020772962048216045084572741173335548858938543801245589186161474816502753404303503987716622}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{4} - \frac{164270691682155598884486960592905613590978240057137901481294032187909149800557699824905686572215105087759572949128412056520957862620394}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{3} - \frac{455010113279915886873738460710601967530505030169762964496353855353006034119188267712188431445247923226681009058538770446434954324128431}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a^{2} - \frac{1008419851710360741107189971951086282152149541632810091512673904009391339190896563680584000028776677379624085664531216019693085024287622}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951} a + \frac{1274818166258691114835942030863253084208471301431242513866093403230396551545540092053048994036864262401732722798841941130704675841122799}{3091033042555587246022656245527537397683503667691140517126260543012702090383928213664075885958680877480644532318467633956677979712387951}$

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

Class group and class number

$C_{82867144}$, which has order $82867144$ (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:  \( 866679281.3791491 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-979}) \), 11.11.31181719929966183601.1

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 $22$ $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
11Data not computed
89Data not computed