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Properties

Label 22.0.80175413461...1947.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 11^{40}$
Root discriminant $135.52$
Ramified primes $3, 11$
Class number $91136$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 178]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, -2670327, 120982653, 3952476, 140760675, 15287250, 109177024, 11785983, 46593316, 2608749, 14038057, 317505, 2823634, -67485, 407473, -15048, 36520, -2211, 2200, -66, 55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 55*x^20 - 66*x^19 + 2200*x^18 - 2211*x^17 + 36520*x^16 - 15048*x^15 + 407473*x^14 - 67485*x^13 + 2823634*x^12 + 317505*x^11 + 14038057*x^10 + 2608749*x^9 + 46593316*x^8 + 11785983*x^7 + 109177024*x^6 + 15287250*x^5 + 140760675*x^4 + 3952476*x^3 + 120982653*x^2 - 2670327*x + 59049)
 
gp: K = bnfinit(x^22 + 55*x^20 - 66*x^19 + 2200*x^18 - 2211*x^17 + 36520*x^16 - 15048*x^15 + 407473*x^14 - 67485*x^13 + 2823634*x^12 + 317505*x^11 + 14038057*x^10 + 2608749*x^9 + 46593316*x^8 + 11785983*x^7 + 109177024*x^6 + 15287250*x^5 + 140760675*x^4 + 3952476*x^3 + 120982653*x^2 - 2670327*x + 59049, 1)
 

Normalized defining polynomial

\( x^{22} + 55 x^{20} - 66 x^{19} + 2200 x^{18} - 2211 x^{17} + 36520 x^{16} - 15048 x^{15} + 407473 x^{14} - 67485 x^{13} + 2823634 x^{12} + 317505 x^{11} + 14038057 x^{10} + 2608749 x^{9} + 46593316 x^{8} + 11785983 x^{7} + 109177024 x^{6} + 15287250 x^{5} + 140760675 x^{4} + 3952476 x^{3} + 120982653 x^{2} - 2670327 x + 59049 \)

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:  \(-80175413461356653345578817535138847697621761947=-\,3^{11}\cdot 11^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.52$
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:  $3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(363=3\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{363}(320,·)$, $\chi_{363}(1,·)$, $\chi_{363}(67,·)$, $\chi_{363}(133,·)$, $\chi_{363}(199,·)$, $\chi_{363}(265,·)$, $\chi_{363}(331,·)$, $\chi_{363}(23,·)$, $\chi_{363}(89,·)$, $\chi_{363}(155,·)$, $\chi_{363}(221,·)$, $\chi_{363}(287,·)$, $\chi_{363}(353,·)$, $\chi_{363}(34,·)$, $\chi_{363}(100,·)$, $\chi_{363}(166,·)$, $\chi_{363}(232,·)$, $\chi_{363}(298,·)$, $\chi_{363}(56,·)$, $\chi_{363}(122,·)$, $\chi_{363}(188,·)$, $\chi_{363}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{3}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{9} + \frac{1}{9} a^{7} + \frac{2}{27} a^{5} + \frac{4}{27} a^{3}$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{10} + \frac{2}{27} a^{6} + \frac{4}{27} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{9} - \frac{4}{27} a^{7} - \frac{1}{9} a^{5} - \frac{2}{27} a^{3}$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{10} - \frac{1}{27} a^{8} - \frac{1}{9} a^{6} - \frac{2}{27} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{15} + \frac{1}{27} a^{10} - \frac{2}{81} a^{9} - \frac{1}{27} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{27} a^{4} + \frac{1}{81} a^{3} + \frac{4}{27} a^{2} + \frac{1}{9} a$, $\frac{1}{243} a^{16} + \frac{1}{81} a^{14} - \frac{1}{81} a^{12} - \frac{11}{243} a^{10} + \frac{2}{81} a^{8} + \frac{1}{9} a^{7} + \frac{4}{81} a^{6} + \frac{19}{243} a^{4} - \frac{1}{9} a$, $\frac{1}{729} a^{17} + \frac{1}{729} a^{16} - \frac{1}{243} a^{15} - \frac{2}{243} a^{14} + \frac{2}{243} a^{13} - \frac{1}{243} a^{12} - \frac{2}{729} a^{11} + \frac{34}{729} a^{10} + \frac{2}{243} a^{8} + \frac{1}{243} a^{7} - \frac{5}{243} a^{6} - \frac{71}{729} a^{5} - \frac{296}{729} a^{4} - \frac{104}{243} a^{3} + \frac{25}{81} a^{2} + \frac{2}{9} a$, $\frac{1}{729} a^{18} - \frac{1}{729} a^{16} - \frac{1}{243} a^{15} - \frac{2}{243} a^{14} - \frac{1}{81} a^{13} - \frac{8}{729} a^{12} + \frac{1}{81} a^{11} - \frac{40}{729} a^{10} + \frac{11}{243} a^{9} - \frac{13}{243} a^{8} - \frac{2}{81} a^{7} + \frac{61}{729} a^{6} - \frac{4}{81} a^{5} + \frac{95}{729} a^{4} - \frac{19}{243} a^{3} + \frac{11}{81} a^{2}$, $\frac{1}{2187} a^{19} + \frac{1}{2187} a^{18} - \frac{1}{2187} a^{17} - \frac{1}{2187} a^{16} + \frac{1}{243} a^{15} - \frac{11}{729} a^{14} + \frac{37}{2187} a^{13} + \frac{19}{2187} a^{12} - \frac{31}{2187} a^{11} + \frac{95}{2187} a^{10} - \frac{32}{729} a^{9} + \frac{32}{729} a^{8} - \frac{254}{2187} a^{7} + \frac{358}{2187} a^{6} - \frac{103}{2187} a^{5} + \frac{284}{2187} a^{4} - \frac{16}{729} a^{3} - \frac{46}{243} a^{2} + \frac{10}{27} a - \frac{1}{3}$, $\frac{1}{8995131} a^{20} - \frac{40}{999459} a^{19} + \frac{4168}{8995131} a^{18} - \frac{391}{2998377} a^{17} + \frac{4783}{8995131} a^{16} + \frac{5305}{2998377} a^{15} - \frac{130448}{8995131} a^{14} - \frac{17870}{999459} a^{13} - \frac{1676}{8995131} a^{12} - \frac{30226}{2998377} a^{11} - \frac{78875}{8995131} a^{10} - \frac{15347}{2998377} a^{9} - \frac{143504}{8995131} a^{8} + \frac{1658}{111051} a^{7} - \frac{466352}{8995131} a^{6} - \frac{226639}{2998377} a^{5} + \frac{1808896}{8995131} a^{4} + \frac{126763}{2998377} a^{3} - \frac{149081}{999459} a^{2} - \frac{10114}{111051} a + \frac{3658}{12339}$, $\frac{1}{9310989035692488804807978190045401039369184103502518484423} a^{21} + \frac{26089848928550188898843095710169297064990444454176}{1034554337299165422756442021116155671041020455944724276047} a^{20} - \frac{1158120101988574146370603766483546454686700103324540166}{9310989035692488804807978190045401039369184103502518484423} a^{19} - \frac{1898634934581775919704509680336245794665194691552341039}{3103663011897496268269326063348467013123061367834172828141} a^{18} + \frac{633644881722378930361853551973408852439203666377392145}{9310989035692488804807978190045401039369184103502518484423} a^{17} - \frac{2525233674250297017663319784784089554025886921665462447}{3103663011897496268269326063348467013123061367834172828141} a^{16} + \frac{56208191545435075642457165418204062304117185199677431477}{9310989035692488804807978190045401039369184103502518484423} a^{15} + \frac{17522699560294902321015965991616703883040497084273882670}{1034554337299165422756442021116155671041020455944724276047} a^{14} + \frac{26857720251932399777685855829802735528136060134568216550}{9310989035692488804807978190045401039369184103502518484423} a^{13} - \frac{8252712823299221298792322321478568210152024843031154}{915264822146120987398798603169704220915087398358647251} a^{12} + \frac{115607458735917483932889734925629058082345370248080437110}{9310989035692488804807978190045401039369184103502518484423} a^{11} - \frac{90742657979855520894752984442843795027731316176794385762}{3103663011897496268269326063348467013123061367834172828141} a^{10} + \frac{43875687812843723473515266763961962271724852979522207160}{9310989035692488804807978190045401039369184103502518484423} a^{9} + \frac{1951387730142623664390488199994334338796021673371269675}{114950481922129491417382446790683963449002272882747141783} a^{8} - \frac{382413422396761279302342327642021340531484294564577691781}{9310989035692488804807978190045401039369184103502518484423} a^{7} - \frac{378184635319505875955686284811720765873727528786905184917}{3103663011897496268269326063348467013123061367834172828141} a^{6} - \frac{715652861125620242506416935878986144405078095742947379548}{9310989035692488804807978190045401039369184103502518484423} a^{5} - \frac{292236478938218278517178989006073239135764282903069409681}{3103663011897496268269326063348467013123061367834172828141} a^{4} + \frac{373063851714546678415398708019090404612140769440458640869}{1034554337299165422756442021116155671041020455944724276047} a^{3} - \frac{10001910017002813651510020945804161300021633719754633663}{38316827307376497139127482263561321149667424294249047261} a^{2} - \frac{229994975711949143306633781350263610144359900964576194}{473047250708351816532438052636559520366264497459864781} a + \frac{63087520987997673194755686595614787646341880966548867}{1419141752125055449597314157909678561098793492379594343}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{178}$, which has order $91136$ (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:  \( \frac{282465819824296943767440657491483692552736754338}{754598349598224232499228315912586193319489756341884957} a^{21} - \frac{507267438179264276305773293281025978226255632}{83844261066469359166580923990287354813276639593542773} a^{20} + \frac{15534055138882840716779260766492805852613891547111}{754598349598224232499228315912586193319489756341884957} a^{19} - \frac{6293902307991215600064435799188886183084054354490}{251532783199408077499742771970862064439829918780628319} a^{18} + \frac{621643961362143706078739661323832977838789431477297}{754598349598224232499228315912586193319489756341884957} a^{17} - \frac{211274633156835180769842764880835974602628794104003}{251532783199408077499742771970862064439829918780628319} a^{16} + \frac{10321703377400076008062142158240219179447438409754288}{754598349598224232499228315912586193319489756341884957} a^{15} - \frac{487594774403013885928945007185504990618160703523288}{83844261066469359166580923990287354813276639593542773} a^{14} + \frac{115091051906078583503552378232957216658896569370591725}{754598349598224232499228315912586193319489756341884957} a^{13} - \frac{2015327816349408013898473806658168162846680047900}{74176580123682712326671416092852274974883491235809} a^{12} + \frac{797220850739617556044659698700057352352742822781446566}{754598349598224232499228315912586193319489756341884957} a^{11} + \frac{27079704506434305298465788294478524076639138179602114}{251532783199408077499742771970862064439829918780628319} a^{10} + \frac{3959953123889595832686048550562091607313027230173371528}{754598349598224232499228315912586193319489756341884957} a^{9} + \frac{8647109759633411101554347227939340660156081848958543}{9316029007385484351842324887809706090364071065949197} a^{8} + \frac{13138690491232699763604795492916275165472417020764085743}{754598349598224232499228315912586193319489756341884957} a^{7} + \frac{1083062755343391773229805786495776356443179324544645555}{251532783199408077499742771970862064439829918780628319} a^{6} + \frac{30766749870638688848167183051599500923418802713313967517}{754598349598224232499228315912586193319489756341884957} a^{5} + \frac{1403963358548796922739799169823163480836862922618117280}{251532783199408077499742771970862064439829918780628319} a^{4} + \frac{4412382941374127795443438806157623032023278322733966975}{83844261066469359166580923990287354813276639593542773} a^{3} + \frac{6113179756111248465402207259289960351820358972979780}{3105343002461828117280774962603235363454690355316399} a^{2} + \frac{15608533668705764146170378391416370838102031203918539}{345038111384647568586752773622581707050521150590711} a + \frac{175598094021745168675036203510536322303751947008}{115012703794882522862250924540860569016840383530237} \) (order $6$)
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:  \( 285114946276.13544 \) (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

\(\Q(\sqrt{-3}) \), 11.11.672749994932560009201.1

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 $22$ R $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ $22$ $22$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
11Data not computed