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Properties

Label 27.27.4439054271...0569.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 79^{18}$
Root discriminant $270.01$
Ramified primes $3, 79$
Class number Not computed
Class group Not computed
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2072482797673, -31044631374180, -178295033152692, -533618215210377, -944724683324952, -1034723340656082, -677653689800451, -202686294270924, 45728358013593, 62786902579694, 16864294008150, -3198616042653, -2730350841870, -275778124056, 169288875360, 42769856514, -4572350253, -2481341895, -6907183, 83303766, 4127985, -1789356, -122580, 25263, 1644, -225, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 225*x^25 + 1644*x^24 + 25263*x^23 - 122580*x^22 - 1789356*x^21 + 4127985*x^20 + 83303766*x^19 - 6907183*x^18 - 2481341895*x^17 - 4572350253*x^16 + 42769856514*x^15 + 169288875360*x^14 - 275778124056*x^13 - 2730350841870*x^12 - 3198616042653*x^11 + 16864294008150*x^10 + 62786902579694*x^9 + 45728358013593*x^8 - 202686294270924*x^7 - 677653689800451*x^6 - 1034723340656082*x^5 - 944724683324952*x^4 - 533618215210377*x^3 - 178295033152692*x^2 - 31044631374180*x - 2072482797673)
 
gp: K = bnfinit(x^27 - 9*x^26 - 225*x^25 + 1644*x^24 + 25263*x^23 - 122580*x^22 - 1789356*x^21 + 4127985*x^20 + 83303766*x^19 - 6907183*x^18 - 2481341895*x^17 - 4572350253*x^16 + 42769856514*x^15 + 169288875360*x^14 - 275778124056*x^13 - 2730350841870*x^12 - 3198616042653*x^11 + 16864294008150*x^10 + 62786902579694*x^9 + 45728358013593*x^8 - 202686294270924*x^7 - 677653689800451*x^6 - 1034723340656082*x^5 - 944724683324952*x^4 - 533618215210377*x^3 - 178295033152692*x^2 - 31044631374180*x - 2072482797673, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 225 x^{25} + 1644 x^{24} + 25263 x^{23} - 122580 x^{22} - 1789356 x^{21} + 4127985 x^{20} + 83303766 x^{19} - 6907183 x^{18} - 2481341895 x^{17} - 4572350253 x^{16} + 42769856514 x^{15} + 169288875360 x^{14} - 275778124056 x^{13} - 2730350841870 x^{12} - 3198616042653 x^{11} + 16864294008150 x^{10} + 62786902579694 x^{9} + 45728358013593 x^{8} - 202686294270924 x^{7} - 677653689800451 x^{6} - 1034723340656082 x^{5} - 944724683324952 x^{4} - 533618215210377 x^{3} - 178295033152692 x^{2} - 31044631374180 x - 2072482797673 \)

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:  \(443905427170893592618091312247244421283763327307281442575957150569=3^{66}\cdot 79^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $270.01$
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:  $3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2133=3^{3}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{2133}(1,·)$, $\chi_{2133}(1603,·)$, $\chi_{2133}(1477,·)$, $\chi_{2133}(712,·)$, $\chi_{2133}(1186,·)$, $\chi_{2133}(892,·)$, $\chi_{2133}(655,·)$, $\chi_{2133}(529,·)$, $\chi_{2133}(1129,·)$, $\chi_{2133}(1366,·)$, $\chi_{2133}(1240,·)$, $\chi_{2133}(475,·)$, $\chi_{2133}(2077,·)$, $\chi_{2133}(1951,·)$, $\chi_{2133}(418,·)$, $\chi_{2133}(292,·)$, $\chi_{2133}(1897,·)$, $\chi_{2133}(1003,·)$, $\chi_{2133}(238,·)$, $\chi_{2133}(1840,·)$, $\chi_{2133}(1714,·)$, $\chi_{2133}(1423,·)$, $\chi_{2133}(181,·)$, $\chi_{2133}(55,·)$, $\chi_{2133}(1660,·)$, $\chi_{2133}(766,·)$, $\chi_{2133}(949,·)$$\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}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{26} - \frac{21503064870164220096669626011398008820435275653423427462138278253393231578814302517612829099688062524022776586178510829541080586410719060664015}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{25} + \frac{83888453418793688686779752370169667629797959401504557001683343467827722301295809425818403099191073034306005567894314069230120006603869496793298}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{24} + \frac{76965036598380015190280899265808489215713491726773255201945036277920791981355209351337595415108794666067536265041109011492104463692658478352207}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{23} - \frac{67605198666676587760058983768441821407819127349268487330361945387390098315645744784947281491498205567974582733943200619703357634685819439186697}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{22} + \frac{63548650717760179371062515189381976509766664501068101555148851013846273410199429074736091753551059704678592808673425662942120752004925963340594}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{21} + \frac{45475951254310811814591399964919877188641039737306840184320136786863615488221701313443880067243115093783116445869502800778248683066650605517005}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{20} + \frac{47784608050600833269036237110859795516829889764921370430413036216109446763719201247642055131799953758157979969247134531192354251435098451782082}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{19} + \frac{52371594588002478421688254753175599124833054203087750941728083856102123800896768713810451804085164019124589722021212198058064050012462032389653}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{18} - \frac{50733451195830568083351344819314045191699498188351860057278664688299159752505077721587545515246249648561017695330291478618261578958481718593852}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{17} + \frac{63065232103806276349306812114050077971249764494297973574839724285878311629271207448413659327692817298778017980767322506124580276143243734149954}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{16} + \frac{66703073289824067061354116164142163664222061794890897785785625896139055327859574965261303101311759861387247665262378822587423223898701364104225}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{15} + \frac{77507815088883899674328924410248396045117445019877524069867646848704204360581244627609784114413607038411122225999424981124043655889359794825889}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{14} - \frac{30680532204518712040051347041950796791908088382448114520617312410300742929434400896239102279285742381547922883831803234262844586762265597055548}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{13} - \frac{37186302472523113684850239432122850149563638020662626472900102875030535837406123189411923956588964166788082873652534081882857391317375830376062}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{12} - \frac{67067041862498254075890794237093220320388411683879656415161011061741003848819078308346085479185426196095590591428231920078832951542478064547753}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{11} - \frac{1264530677615963113176526046881316637110654804207980353572316628910503992541845108660071183067159544643564430471670345156960588530574817040196}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{10} - \frac{77862124945825422476549058706929068086528206183029143645202000270873462453716068848437774937090239552424845606712109221386253031899123156941689}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{9} + \frac{70700747297411312334958911255744543354215944676095608398234491302827180026845796777408806971332143927811716164665408058684422503357109784932942}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{8} + \frac{64079018535265724578948937763478985928255626312328048033053241959282439545760854370732443988292111997493594895005234607161057471335729621560413}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{7} - \frac{64988520457467759183390467512428522431361466010720064470486886615303830594463039885154348761011284827403912316123348054685242568162939392429789}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{6} - \frac{60368200786779934351036285989829682517662559192158707466261720412904174501320057847482366946316807359791288723966628115622774746936860205169605}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{5} + \frac{58941918058999665871817929812304787319434952357101477887986246512358185082742363910033796332704214766549226797887700397918974030334085489966096}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{4} - \frac{72548482471078843784527154902572545479411955712275697585043687632048677512439552929754732360968029253669331332103217837822034454310539064573461}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{3} + \frac{10694703308475378380404062644792414081319384248985769775805119139062063714915556188902823303071503142863287565651381920973301451898366825096092}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a^{2} - \frac{28478610059278438509109112106423830684042614930038304972937191850013135025206269476168156203453859784456721458677132638980623297546366857227811}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351} a - \frac{38923210360964028794689316063786051634215445298153091351879465738777514796327497891492169360456245469870702428167901456045093273702479194423462}{168475247183302034580418808796394904119665892513968704591746642917725438760955597455286704104321261316078833626276990478152865132249097515939351}$

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

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.6241.1, \(\Q(\zeta_{9})^+\), 3.3.505521.2, 3.3.505521.1, 9.9.129186640449535761.1, 9.9.7628341931904637151289.2, \(\Q(\zeta_{27})^+\), 9.9.7628341931904637151289.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
79Data not computed