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Properties

Label 27.27.2904636282...3089.3
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 109^{26}$
Root discriminant $1343.58$
Ramified primes $3, 109$
Class number Not computed
Class group Not computed
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1699412318998420661, 68809894376068855206, -56974306727633418045, -71561295489954712743, 43373094180308101017, 33977534604641591400, -8650195833611503449, -7923812360425415109, 207644338670160276, 865257605899112586, 77344277257641249, -47356437284334108, -8109166737504153, 1267991081780472, 356369675039019, -10743842518713, -8313083080686, -232610440878, 108416615105, 6903675666, -777428766, -73225110, 2805660, 382590, -3924, -981, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 981*x^25 - 3924*x^24 + 382590*x^23 + 2805660*x^22 - 73225110*x^21 - 777428766*x^20 + 6903675666*x^19 + 108416615105*x^18 - 232610440878*x^17 - 8313083080686*x^16 - 10743842518713*x^15 + 356369675039019*x^14 + 1267991081780472*x^13 - 8109166737504153*x^12 - 47356437284334108*x^11 + 77344277257641249*x^10 + 865257605899112586*x^9 + 207644338670160276*x^8 - 7923812360425415109*x^7 - 8650195833611503449*x^6 + 33977534604641591400*x^5 + 43373094180308101017*x^4 - 71561295489954712743*x^3 - 56974306727633418045*x^2 + 68809894376068855206*x - 1699412318998420661)
 
gp: K = bnfinit(x^27 - 981*x^25 - 3924*x^24 + 382590*x^23 + 2805660*x^22 - 73225110*x^21 - 777428766*x^20 + 6903675666*x^19 + 108416615105*x^18 - 232610440878*x^17 - 8313083080686*x^16 - 10743842518713*x^15 + 356369675039019*x^14 + 1267991081780472*x^13 - 8109166737504153*x^12 - 47356437284334108*x^11 + 77344277257641249*x^10 + 865257605899112586*x^9 + 207644338670160276*x^8 - 7923812360425415109*x^7 - 8650195833611503449*x^6 + 33977534604641591400*x^5 + 43373094180308101017*x^4 - 71561295489954712743*x^3 - 56974306727633418045*x^2 + 68809894376068855206*x - 1699412318998420661, 1)
 

Normalized defining polynomial

\( x^{27} - 981 x^{25} - 3924 x^{24} + 382590 x^{23} + 2805660 x^{22} - 73225110 x^{21} - 777428766 x^{20} + 6903675666 x^{19} + 108416615105 x^{18} - 232610440878 x^{17} - 8313083080686 x^{16} - 10743842518713 x^{15} + 356369675039019 x^{14} + 1267991081780472 x^{13} - 8109166737504153 x^{12} - 47356437284334108 x^{11} + 77344277257641249 x^{10} + 865257605899112586 x^{9} + 207644338670160276 x^{8} - 7923812360425415109 x^{7} - 8650195833611503449 x^{6} + 33977534604641591400 x^{5} + 43373094180308101017 x^{4} - 71561295489954712743 x^{3} - 56974306727633418045 x^{2} + 68809894376068855206 x - 1699412318998420661 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2904636282635430627257935704503025621208529312293717639293800815805291320812817603089=3^{66}\cdot 109^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1343.58$
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:  $3, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2943=3^{3}\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{2943}(1,·)$, $\chi_{2943}(517,·)$, $\chi_{2943}(2689,·)$, $\chi_{2943}(1357,·)$, $\chi_{2943}(910,·)$, $\chi_{2943}(661,·)$, $\chi_{2943}(2134,·)$, $\chi_{2943}(1753,·)$, $\chi_{2943}(2713,·)$, $\chi_{2943}(2074,·)$, $\chi_{2943}(2587,·)$, $\chi_{2943}(349,·)$, $\chi_{2943}(2596,·)$, $\chi_{2943}(2533,·)$, $\chi_{2943}(2791,·)$, $\chi_{2943}(2503,·)$, $\chi_{2943}(2305,·)$, $\chi_{2943}(877,·)$, $\chi_{2943}(1006,·)$, $\chi_{2943}(1117,·)$, $\chi_{2943}(2800,·)$, $\chi_{2943}(1138,·)$, $\chi_{2943}(2419,·)$, $\chi_{2943}(2869,·)$, $\chi_{2943}(187,·)$, $\chi_{2943}(124,·)$, $\chi_{2943}(1135,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{26} + \frac{8205428139204239703883314759948090861267547489045367138243327982213967169215014025273952357348439316011675785308242213737337769267281999424663554562626813738482301123491936581485682344273728378076384617805427941927523367870546146898741033887632462}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{25} - \frac{59160300016755342526044340524467397726802502579366643881004422289104517858391495244752908207910263515859819402434235634150072446115388182889495656791861952857920992086404029457183695574624678311282696381886205040646793299154205773496035325935689493}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{24} + \frac{60537539141468958110545304679174973653603130463618796292319612254356668693614789897358708792610875080364273092779593768097572576209966265600261963713825582640337653680398845913144845455957285173959905921800854056739741893975969998945732321388621445}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{23} + \frac{10538676973964954381798671308337945156876748248943124730706530839072992069365412640824033853891124763641063329450803868695494434629497368601163981389447820668249450588041192887200673451582341434228538487637638614423109122111459347458249503635016132}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{22} - \frac{1780531138523538564087450138726727248261828807912224209607098515217366499708747984641886233143376534408895106644708766869778003765131193383958083119987081395123431203387971714330434814095795441860088957157054131898402557903520526738633797454745364}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{21} + \frac{21595914344306033299149492492881826731741274483651765019779999195159800966569075171257175176224650871747258717906930372781151941044678985217122752168422115895929820284978263421947231145160258856249210080569271633119946187758762837346816983529980537}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{20} - \frac{20710997160577569311868063671481863843556641852755054390620946950594664413794580448138781535286345062135828627253394788986081274209393620746210654782842221940840893020653301017483059475050672191488941388877193354416154362012318373135992323003925866}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{19} + \frac{5058914786868945895998310271934625162014667539087973364330091686740306136567387156854668275243300031017300026621035736659424323849536938168470827055186463017601961796698941615658533134091063122906588030855219268989069882237302399611415607487422200}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{18} - \frac{42317259892482376855963410896449520185299000472851701445485222248676600327244848145085903179900819447004499286188543841413758152469231938021126355779793322275751859265091509701051078825953866160052760039680617543112064604788774905701693614682968772}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{17} + \frac{69519313517429494020819353128293668082064429853747296477922561811162291128290405981535739163053983612784586332881241049680010805065677980618911717023676304071605138354932192445493483242141694788545325728245718540836206917758965266557659564475965017}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{16} + \frac{66998285579718342622979706774849735810475647814711811363850271518120436576315310703609454736410101600471594862458485809279511550790115318954864160966803420984947674524912464895835258937070932950220355216959836388891765866060877115047424424063178811}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{15} - \frac{2234999975713422878470881776880631569847551155794965058548108707387865750294378597952032924891165790460682440219925764821316053349073254608797816689137201419900352976361297442352766511177065870008197566009804907106953679523781279240069998430936487}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{14} - \frac{11690849306521036757928414546799230622981918856854667176526427244391459969168489264566440567739657766754398894690991621740814525943672488773646941595490926397650467738539144708894592884458313120279358628758343620830835484581790245878536673434138143}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{13} + \frac{59166419931530655331836136272848147214023409709280672795669778573242316539327166307375691665955649204602537810260622091808209865135857575666729830196117639006457786994031834708317807940090270063295811265623256715938266058748216223540785748660080719}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{12} + \frac{64158907027941717641616083515185891355561572003639508913761545669571974196920594861842339197442851187039892641301529939809914314632636773685328564760669996831254745506957057820428147556825625915813375174806125983091669900086622001248330028516466977}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{11} + \frac{26540374525222220512160634662298400664149909863337828282188044309951083411232289173350515725459854883257268809615368175378508179586755851698834708460087831074559049504117709810002769884069561092966227716970492690924106547861864834463798087798851433}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{10} + \frac{65645175608022767922717751965942091097416651498629905505537041392305340703488788608411639229638537032235177487693893246796375111404547714900892569846310331759062044004533018156954715303428789457840098570531242347961990899558189946471949769828281306}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{9} + \frac{50375071873867329887966752666216044498936055098181692129452163906537935954277680020972329862205079622621845493523074307082640470825536081272837207923436558349321346271626842728069361739486955528659706813983655879049841718997137584462688167892052532}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{8} - \frac{11756909094427829398102718624053470323650031086249339872214020489743628047427437621289937842951753493847681675749993655775312292697118288118141954910341292124739379493084134949425485118116615598749479161998122597758036107303109466977457930441579181}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{7} - \frac{7438244375676416363352775139643977051629732290550423257250146036276686372777862810122122623246098419887584371089530951050606674779438624925301883194186245182117806809106233144200554790063223748657399395825146135319862161459457178944041534755204810}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{6} - \frac{50663016719757223138215547987183651239770300378403078776636836090032028425376528311895683690633304313519879007157572989945055958384866945936995682841256596752299580797616445558750497656221980930108093535760330427725928281064341623114777907282074565}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{5} + \frac{54646783244465009558886740720522979632586471515706441549563641519752833422075245743820605854305962045139132026919379205353694334407981387470388951806056574407333701236806808898496432339921292379287989307361052123763116656583036757316262741082368714}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{4} + \frac{40761557532025369803984399094264918066534651262019658775912486715595630698649600023220414599663433840266792882514307696525405326212667825142558315886337933502379520818668812172333000939130541521651938685632606002730293619446574685149495123422213655}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{3} + \frac{52269724075118862247676712693230142091793601090824760466655246606967907123196599105041284626216117847226182064594280124927740186559917769580708358484138285139834217223268051231653929521201720340815485481745568822986734719678950935044408973686556436}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a^{2} - \frac{31355327874685735012695436170479796628964215189665569465691483868551974773484644499445425971180538927074948875197333877684869743769011409447872847943232706573557020497314874596067410715257536723668255740180166608345673536260675541661796972691722123}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751} a - \frac{10972693468553517871256819685516189632069105490446162130178796476519879140228872416622945266386481409735775388955953649387808455324674861049267177780942404549689887224477798530507111218955563342373749901134833410798093073442396911618989551789557906}{148943807876537254139364382917158809995583867399407140271720942263517514764824734758305166695619565860482430731851919515375004519289882268119433959660683805580870525106252338291099538078625938607278954629834264726051689484587372868179527979549370751}$

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:  $26$
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:  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

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

3.3.11881.1, 9.9.10589294828624773798161.4

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $27$ $27$ $27$

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
109Data not computed