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Properties

Label 27.27.5659106580...6209.6
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 7^{18}\cdot 13^{18}$
Root discriminant $296.70$
Ramified primes $3, 7, 13$
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![5057089505859, 15086119963302, -126122964539436, -199876274766459, 188746081240014, 450767216583810, 84350717576493, -274189790401506, -189487015010469, -1312384302218, 37741694932692, 9081521720235, -2946031839456, -1330149204990, 82399099806, 96103309782, 2545292439, -4239416871, -285174031, 123276078, 10339083, -2430132, -199422, 32067, 2058, -261, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 261*x^25 + 2058*x^24 + 32067*x^23 - 199422*x^22 - 2430132*x^21 + 10339083*x^20 + 123276078*x^19 - 285174031*x^18 - 4239416871*x^17 + 2545292439*x^16 + 96103309782*x^15 + 82399099806*x^14 - 1330149204990*x^13 - 2946031839456*x^12 + 9081521720235*x^11 + 37741694932692*x^10 - 1312384302218*x^9 - 189487015010469*x^8 - 274189790401506*x^7 + 84350717576493*x^6 + 450767216583810*x^5 + 188746081240014*x^4 - 199876274766459*x^3 - 126122964539436*x^2 + 15086119963302*x + 5057089505859)
 
gp: K = bnfinit(x^27 - 9*x^26 - 261*x^25 + 2058*x^24 + 32067*x^23 - 199422*x^22 - 2430132*x^21 + 10339083*x^20 + 123276078*x^19 - 285174031*x^18 - 4239416871*x^17 + 2545292439*x^16 + 96103309782*x^15 + 82399099806*x^14 - 1330149204990*x^13 - 2946031839456*x^12 + 9081521720235*x^11 + 37741694932692*x^10 - 1312384302218*x^9 - 189487015010469*x^8 - 274189790401506*x^7 + 84350717576493*x^6 + 450767216583810*x^5 + 188746081240014*x^4 - 199876274766459*x^3 - 126122964539436*x^2 + 15086119963302*x + 5057089505859, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 261 x^{25} + 2058 x^{24} + 32067 x^{23} - 199422 x^{22} - 2430132 x^{21} + 10339083 x^{20} + 123276078 x^{19} - 285174031 x^{18} - 4239416871 x^{17} + 2545292439 x^{16} + 96103309782 x^{15} + 82399099806 x^{14} - 1330149204990 x^{13} - 2946031839456 x^{12} + 9081521720235 x^{11} + 37741694932692 x^{10} - 1312384302218 x^{9} - 189487015010469 x^{8} - 274189790401506 x^{7} + 84350717576493 x^{6} + 450767216583810 x^{5} + 188746081240014 x^{4} - 199876274766459 x^{3} - 126122964539436 x^{2} + 15086119963302 x + 5057089505859 \)

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:  \(5659106580522258442740055894009895600932750567734671570320232266209=3^{66}\cdot 7^{18}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $296.70$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2457=3^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2457}(1,·)$, $\chi_{2457}(1093,·)$, $\chi_{2457}(646,·)$, $\chi_{2457}(1537,·)$, $\chi_{2457}(2185,·)$, $\chi_{2457}(1738,·)$, $\chi_{2457}(718,·)$, $\chi_{2457}(274,·)$, $\chi_{2457}(547,·)$, $\chi_{2457}(1366,·)$, $\chi_{2457}(919,·)$, $\chi_{2457}(2011,·)$, $\chi_{2457}(991,·)$, $\chi_{2457}(1465,·)$, $\chi_{2457}(2083,·)$, $\chi_{2457}(100,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(1192,·)$, $\chi_{2457}(172,·)$, $\chi_{2457}(1810,·)$, $\chi_{2457}(1264,·)$, $\chi_{2457}(2356,·)$, $\chi_{2457}(2284,·)$, $\chi_{2457}(1912,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(445,·)$, $\chi_{2457}(373,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{27} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{2}{9} a^{15} - \frac{1}{3} a^{14} + \frac{2}{9} a^{12} + \frac{1}{3} a^{10} + \frac{13}{27} a^{9} - \frac{1}{3} a^{8} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9}$, $\frac{1}{27} a^{19} - \frac{1}{3} a^{17} + \frac{2}{9} a^{16} - \frac{1}{3} a^{15} + \frac{2}{9} a^{13} + \frac{1}{3} a^{11} + \frac{13}{27} a^{10} + \frac{2}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{9} a$, $\frac{1}{27} a^{20} + \frac{2}{9} a^{17} - \frac{1}{3} a^{16} + \frac{2}{9} a^{14} + \frac{1}{3} a^{12} + \frac{13}{27} a^{11} + \frac{1}{3} a^{9} + \frac{2}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{21} - \frac{1}{3} a^{17} - \frac{1}{9} a^{15} + \frac{1}{3} a^{13} + \frac{4}{27} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{9} a^{3} + \frac{1}{3}$, $\frac{1}{27} a^{22} - \frac{1}{9} a^{16} + \frac{1}{3} a^{14} + \frac{4}{27} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{9} a^{4} + \frac{1}{3} a$, $\frac{1}{81} a^{23} - \frac{1}{81} a^{22} + \frac{1}{81} a^{21} + \frac{1}{81} a^{20} - \frac{1}{81} a^{19} + \frac{1}{81} a^{18} - \frac{11}{27} a^{17} + \frac{2}{27} a^{16} - \frac{2}{27} a^{15} - \frac{35}{81} a^{14} + \frac{26}{81} a^{13} + \frac{1}{81} a^{12} + \frac{4}{81} a^{11} + \frac{23}{81} a^{10} - \frac{23}{81} a^{9} + \frac{2}{27} a^{8} + \frac{7}{27} a^{7} + \frac{11}{27} a^{6} - \frac{2}{27} a^{5} - \frac{10}{27} a^{4} - \frac{8}{27} a^{3} + \frac{4}{27} a^{2} - \frac{13}{27} a + \frac{13}{27}$, $\frac{1}{81} a^{24} - \frac{1}{81} a^{21} + \frac{1}{81} a^{18} + \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{4}{81} a^{15} + \frac{2}{9} a^{14} + \frac{29}{81} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{20}{81} a^{9} - \frac{1}{3} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{7}{27} a^{3} - \frac{1}{3} a^{2} + \frac{10}{27}$, $\frac{1}{34911} a^{25} + \frac{14}{3879} a^{24} + \frac{22}{34911} a^{23} - \frac{587}{34911} a^{22} - \frac{119}{34911} a^{21} + \frac{151}{34911} a^{20} + \frac{43}{11637} a^{19} + \frac{313}{34911} a^{18} + \frac{2581}{11637} a^{17} - \frac{15875}{34911} a^{16} + \frac{2968}{11637} a^{15} - \frac{14414}{34911} a^{14} - \frac{4697}{34911} a^{13} + \frac{2797}{34911} a^{12} + \frac{7948}{34911} a^{11} - \frac{5479}{11637} a^{10} - \frac{44}{34911} a^{9} - \frac{3613}{11637} a^{8} - \frac{3962}{11637} a^{7} + \frac{3998}{11637} a^{6} + \frac{2494}{11637} a^{5} + \frac{1531}{11637} a^{4} + \frac{646}{11637} a^{3} + \frac{2458}{11637} a^{2} - \frac{229}{1293} a + \frac{2692}{11637}$, $\frac{1}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{26} + \frac{83565151375938792449556201285146713168480183339995616130902137715979663828530558130903176255741960573715731612688178710236314606996671691453357092}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{25} - \frac{11880798210496801464013793632517041207750038855796592415225462213669344742439213745814708485189405523877764791877673125866155675928303637480334990613}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{24} + \frac{21187382979290509442543999883180048479549360550117820363300839048432349096288823148360420486555245525594654718198611691248060573591886127753171387954}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{23} + \frac{8311552387449638203433104356843124672124603245431082789782954609800706223704265268434989687111969180685674758429929798575014323218561162233091473936}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{22} + \frac{106375683728559958434719842336809967345587682115953813126655243209832765463377585246783071825562753437305078904044603275935755985182437057895918995179}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{21} + \frac{43069408745333388645414716240460582975606304935033919646142275128953254793718894793102938957665948630291746061722359050331296186329253056296789671746}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{20} - \frac{51518653968814836895363897850877503697045832212216130246882965300076560536672757099384283181747914082091563387635789902679076337078021811673977833808}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{19} - \frac{58818235044620648959374170741699087363008139677083695551931667652294602226307583257776477982794021769187652100479039810464785444487217522605450289138}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{18} - \frac{1478263036113277107582483642991688428158725276956548723924932580959562265102546288212391049204781898252351214691672607492235116368593571004039426252448}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{17} + \frac{2043151231902075997138118456436044079053963143965441596696993055020789881915944680314302332505830863265530604560247526927835896551481583612718955367394}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{16} - \frac{3250325576416135208996985150467738678818964384598803022120991124414302114163888599211283219900196358714124883065894928894251723742287300269277592568702}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{15} - \frac{898858099720374054458640482209958476361517368941021673667361695194371463040597645761799157233759884453838557688717048440317396233062919332595763307981}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{14} + \frac{267384793262157027955887647501248578960156172454637323149465634551812415953308166179447836750831680628810029622584039338023576099839214375421080575299}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{13} + \frac{799147339084313455996581449472155586689231967073899772108282510573389992946582796852691028066423885247590108518220608681802591753366274211008553489401}{9535432028531463683811932296065200302733455314382315635108800871209913549253610723525543251177541158409797590285661892066081711948599062403854209096151} a^{12} - \frac{167718933035081159168926229534909165000209011874404338611340740519867026069038847083381827329515394777601326989527423938962746713387305065666176197767}{353164149204869025326367862817229640841979826458604282781807439674441242564948545315760861154723746607770281121691181187632655998096261570513118855413} a^{11} + \frac{1190425324195142204929451658772942877889769276354662566738424355157128321526599016769958731618370338563002939137325897428573631350857488279584055145743}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{10} + \frac{463244951378125248508257170783013598669155538076582891405981798483923025058887173118640650183322332485538310363652292875925394949995307977703605393133}{1059492447614607075979103588451688922525939479375812848345422319023323727694845635947282583464171239823310843365073543562897967994288784711539356566239} a^{9} + \frac{57073834029947256069606355636976903840446888192835231616307206439965205290949045185731296615943276814847750896818144529432334573163236317179517136387}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{8} + \frac{643096272300420514193759612600556748672605382186955549046415617048106094602758190307464890973325715487856080150735184435675133372581687010247702394333}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{7} + \frac{235401032416112797240755296898622734359042664086821073654217087015309092262008868940703344186702839490752987961877636154311274291326266541794058583049}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{6} - \frac{1171636363889485058476020098739233026365683044866625348789479335599506893450713377434761541385070514484772967919740167686405421230811858489669974093551}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{5} + \frac{760138417931720633370377150921909874585498877326865208148065880158297459423073088148355982069169627407147303153300723010570763641866017210434495842970}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{4} + \frac{781651285075916181994081128165688435794025729442271758053929598577943577631049758707773414761568293852479965474052108241658731050445939940945652261690}{3178477342843821227937310765355066767577818438127438545036266957069971183084536907841847750392513719469932530095220630688693903982866354134618069698717} a^{3} + \frac{246150523010788721203060169170483071505875070265703323898176626969008229128479008801860915675788793788536080241958813709094708470652944848158832952244}{1059492447614607075979103588451688922525939479375812848345422319023323727694845635947282583464171239823310843365073543562897967994288784711539356566239} a^{2} + \frac{160615218746127877615694783015124232534509277579931374899479912451646611172125699901915547836065435699014142165392739263942910754833572445332854585033}{353164149204869025326367862817229640841979826458604282781807439674441242564948545315760861154723746607770281121691181187632655998096261570513118855413} a - \frac{194778126613964163101222350280944630075588364065272440813017591795286718435649845430159921928648881585371535574697644456051386779295549230178664191550}{1059492447614607075979103588451688922525939479375812848345422319023323727694845635947282583464171239823310843365073543562897967994288784711539356566239}$

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

3.3.8281.2, \(\Q(\zeta_{9})^+\), 3.3.670761.2, 3.3.670761.4, 9.9.301789003173921081.10, 9.9.17820338848416865911969.4, \(\Q(\zeta_{27})^+\), 9.9.17820338848416865911969.3

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

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
$3$3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
7Data not computed
13Data not computed