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Properties

Label 27.27.3063651216...3609.5
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 37^{24}$
Root discriminant $1135.13$
Ramified primes $3, 37$
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![5039013988182624813037, -4425702387444901252881, -5904517674133907030826, 5197683939054223670559, 1063877058402505771320, -1306717860991648955517, -73800480627921571524, 148485095274118612266, 2564495465835112524, -9477982179287626548, -50057719303087782, 377261368227838926, 573962383491588, -9934810805722716, -3835507463190, 179005600103112, 13821648516, -2241127346283, -20753226, 19481903595, 0, -115336881, 0, 443556, 0, -999, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 20753226*x^18 - 2241127346283*x^17 + 13821648516*x^16 + 179005600103112*x^15 - 3835507463190*x^14 - 9934810805722716*x^13 + 573962383491588*x^12 + 377261368227838926*x^11 - 50057719303087782*x^10 - 9477982179287626548*x^9 + 2564495465835112524*x^8 + 148485095274118612266*x^7 - 73800480627921571524*x^6 - 1306717860991648955517*x^5 + 1063877058402505771320*x^4 + 5197683939054223670559*x^3 - 5904517674133907030826*x^2 - 4425702387444901252881*x + 5039013988182624813037)
 
gp: K = bnfinit(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 20753226*x^18 - 2241127346283*x^17 + 13821648516*x^16 + 179005600103112*x^15 - 3835507463190*x^14 - 9934810805722716*x^13 + 573962383491588*x^12 + 377261368227838926*x^11 - 50057719303087782*x^10 - 9477982179287626548*x^9 + 2564495465835112524*x^8 + 148485095274118612266*x^7 - 73800480627921571524*x^6 - 1306717860991648955517*x^5 + 1063877058402505771320*x^4 + 5197683939054223670559*x^3 - 5904517674133907030826*x^2 - 4425702387444901252881*x + 5039013988182624813037, 1)
 

Normalized defining polynomial

\( x^{27} - 999 x^{25} + 443556 x^{23} - 115336881 x^{21} + 19481903595 x^{19} - 20753226 x^{18} - 2241127346283 x^{17} + 13821648516 x^{16} + 179005600103112 x^{15} - 3835507463190 x^{14} - 9934810805722716 x^{13} + 573962383491588 x^{12} + 377261368227838926 x^{11} - 50057719303087782 x^{10} - 9477982179287626548 x^{9} + 2564495465835112524 x^{8} + 148485095274118612266 x^{7} - 73800480627921571524 x^{6} - 1306717860991648955517 x^{5} + 1063877058402505771320 x^{4} + 5197683939054223670559 x^{3} - 5904517674133907030826 x^{2} - 4425702387444901252881 x + 5039013988182624813037 \)

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:  \(30636512167696967158640831949436414926745966008852301931537630975865004722034443609=3^{94}\cdot 37^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1135.13$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2997=3^{4}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2997}(1,·)$, $\chi_{2997}(1999,·)$, $\chi_{2997}(2824,·)$, $\chi_{2997}(268,·)$, $\chi_{2997}(10,·)$, $\chi_{2997}(1099,·)$, $\chi_{2997}(2956,·)$, $\chi_{2997}(589,·)$, $\chi_{2997}(2893,·)$, $\chi_{2997}(1681,·)$, $\chi_{2997}(2008,·)$, $\chi_{2997}(2266,·)$, $\chi_{2997}(2587,·)$, $\chi_{2997}(1825,·)$, $\chi_{2997}(100,·)$, $\chi_{2997}(1957,·)$, $\chi_{2997}(1894,·)$, $\chi_{2997}(1000,·)$, $\chi_{2997}(682,·)$, $\chi_{2997}(1009,·)$, $\chi_{2997}(2098,·)$, $\chi_{2997}(1267,·)$, $\chi_{2997}(1588,·)$, $\chi_{2997}(2680,·)$, $\chi_{2997}(826,·)$, $\chi_{2997}(958,·)$, $\chi_{2997}(895,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{37} a^{9}$, $\frac{1}{37} a^{10}$, $\frac{1}{37} a^{11}$, $\frac{1}{37} a^{12}$, $\frac{1}{37} a^{13}$, $\frac{1}{1369} a^{14} - \frac{15}{37} a^{5}$, $\frac{1}{1369} a^{15} - \frac{15}{37} a^{6}$, $\frac{1}{1369} a^{16} - \frac{15}{37} a^{7}$, $\frac{1}{1369} a^{17} - \frac{15}{37} a^{8}$, $\frac{1}{28550355687075206393} a^{18} - \frac{18}{771631234785816389} a^{16} + \frac{135}{20854898237454497} a^{14} - \frac{546}{563645898309581} a^{12} + \frac{47619}{563645898309581} a^{10} + \frac{2258258994184435}{771631234785816389} a^{9} - \frac{2439558}{563645898309581} a^{8} + \frac{530567289794582}{20854898237454497} a^{7} + \frac{70205058}{563645898309581} a^{6} + \frac{99235825544997}{563645898309581} a^{5} - \frac{1012046940}{563645898309581} a^{4} - \frac{134173762016143}{563645898309581} a^{3} + \frac{5616860517}{563645898309581} a^{2} + \frac{192943192267151}{563645898309581} a - \frac{276662758441731}{563645898309581}$, $\frac{1}{28550355687075206393} a^{19} - \frac{18}{771631234785816389} a^{17} + \frac{135}{20854898237454497} a^{15} - \frac{546}{563645898309581} a^{13} + \frac{47619}{563645898309581} a^{11} + \frac{2258258994184435}{771631234785816389} a^{10} - \frac{2439558}{563645898309581} a^{9} + \frac{530567289794582}{20854898237454497} a^{8} + \frac{70205058}{563645898309581} a^{7} + \frac{99235825544997}{563645898309581} a^{6} - \frac{1012046940}{563645898309581} a^{5} - \frac{134173762016143}{563645898309581} a^{4} + \frac{5616860517}{563645898309581} a^{3} + \frac{192943192267151}{563645898309581} a^{2} - \frac{276662758441731}{563645898309581} a$, $\frac{1}{28550355687075206393} a^{20} - \frac{189}{20854898237454497} a^{16} + \frac{1884}{563645898309581} a^{14} - \frac{316017}{563645898309581} a^{12} + \frac{2258258994184435}{771631234785816389} a^{11} + \frac{29274696}{563645898309581} a^{10} + \frac{33078608514999}{20854898237454497} a^{9} - \frac{1554540570}{563645898309581} a^{8} + \frac{67466770584596}{563645898309581} a^{7} + \frac{45744521688}{563645898309581} a^{6} + \frac{10315948730882}{563645898309581} a^{5} - \frac{668406401523}{563645898309581} a^{4} - \frac{110730377570289}{563645898309581} a^{3} - \frac{272921929337409}{563645898309581} a^{2} - \frac{11098764661902}{563645898309581} a + \frac{54811625040141}{563645898309581}$, $\frac{1}{1056363160421782636541} a^{21} - \frac{189}{771631234785816389} a^{17} + \frac{1884}{20854898237454497} a^{15} - \frac{8541}{563645898309581} a^{13} + \frac{335936630793456387}{28550355687075206393} a^{12} + \frac{791208}{563645898309581} a^{11} - \frac{530567289794582}{771631234785816389} a^{10} - \frac{42014610}{563645898309581} a^{9} + \frac{7958509346918730}{20854898237454497} a^{8} + \frac{1236338424}{563645898309581} a^{7} - \frac{14954863502127}{563645898309581} a^{6} - \frac{18065037879}{563645898309581} a^{5} - \frac{63927404616449}{563645898309581} a^{4} + \frac{5363537053758401}{20854898237454497} a^{3} + \frac{91102070951232}{563645898309581} a^{2} + \frac{92883432835071}{563645898309581} a$, $\frac{1}{1056363160421782636541} a^{22} - \frac{1518}{20854898237454497} a^{16} + \frac{16974}{563645898309581} a^{14} + \frac{335936630793456387}{28550355687075206393} a^{13} - \frac{3026970}{563645898309581} a^{12} - \frac{530567289794582}{771631234785816389} a^{11} + \frac{290985057}{563645898309581} a^{10} + \frac{198471651089994}{20854898237454497} a^{9} - \frac{15823490670}{563645898309581} a^{8} - \frac{66706991431547}{563645898309581} a^{7} + \frac{472878932715}{563645898309581} a^{6} + \frac{347242617743}{4438156679603} a^{5} + \frac{5101679016455861}{20854898237454497} a^{4} - \frac{279240920793983}{563645898309581} a^{3} + \frac{132162138430452}{563645898309581} a^{2} - \frac{116537028949971}{563645898309581} a - \frac{269946784542891}{563645898309581}$, $\frac{1}{281591835210805062245778430082047} a^{23} + \frac{2401022295}{7610590140832569249885903515731} a^{22} - \frac{23}{7610590140832569249885903515731} a^{21} - \frac{64617469}{5559233119673169649295765899} a^{20} + \frac{230}{205691625427907277023943338263} a^{19} + \frac{1055844228}{205691625427907277023943338263} a^{18} - \frac{1311}{5559233119673169649295765899} a^{17} - \frac{118155634870545}{5559233119673169649295765899} a^{16} + \frac{4692}{150249543774950531062047727} a^{15} + \frac{155335224622521682208508763}{7610590140832569249885903515731} a^{14} + \frac{420359920870556210437456845}{205691625427907277023943338263} a^{13} - \frac{2174692523357262037260283271}{205691625427907277023943338263} a^{12} + \frac{58192318544877354650183042}{5559233119673169649295765899} a^{11} - \frac{59155577237506119950055384}{5559233119673169649295765899} a^{10} - \frac{9172348190298730353747864}{5559233119673169649295765899} a^{9} - \frac{16233171066244606560896459}{150249543774950531062047727} a^{8} - \frac{46257614301783488023269820}{150249543774950531062047727} a^{7} + \frac{797378173090665802797529}{4060798480404068407082371} a^{6} + \frac{283048058988276175257240310}{5559233119673169649295765899} a^{5} - \frac{12320999708158215576138956}{150249543774950531062047727} a^{4} - \frac{37164781880641313771821442}{150249543774950531062047727} a^{3} - \frac{31395649236465737811323}{4060798480404068407082371} a^{2} - \frac{1973470208691646694805909}{4060798480404068407082371} a + \frac{585214532562378806181326}{4060798480404068407082371}$, $\frac{1}{281591835210805062245778430082047} a^{24} - \frac{24}{7610590140832569249885903515731} a^{22} + \frac{2383577823}{7610590140832569249885903515731} a^{21} + \frac{252}{205691625427907277023943338263} a^{20} + \frac{10175942}{5559233119673169649295765899} a^{19} - \frac{1520}{5559233119673169649295765899} a^{18} - \frac{457273385919}{5559233119673169649295765899} a^{17} + \frac{5814}{150249543774950531062047727} a^{16} + \frac{155335156905884898144887431}{7610590140832569249885903515731} a^{15} - \frac{14688}{4060798480404068407082371} a^{14} - \frac{2330027351179286229037892106}{205691625427907277023943338263} a^{13} - \frac{2371365174314535346738593841}{205691625427907277023943338263} a^{12} + \frac{6956532361985519735643129}{5559233119673169649295765899} a^{11} + \frac{22856488497712742752298005}{5559233119673169649295765899} a^{10} - \frac{1628874226287106452619120}{150249543774950531062047727} a^{9} - \frac{14350735977405816602646059}{150249543774950531062047727} a^{8} - \frac{1764534572116922041122660}{4060798480404068407082371} a^{7} + \frac{1809011542909805478353401805}{5559233119673169649295765899} a^{6} + \frac{1564107409484698012999391}{4060798480404068407082371} a^{5} - \frac{69177466836220389656606329}{150249543774950531062047727} a^{4} + \frac{19453906967859986816523574}{150249543774950531062047727} a^{3} + \frac{413322360479948443499887}{4060798480404068407082371} a^{2} - \frac{856244098932025658278139}{4060798480404068407082371} a + \frac{568146269983606312531673}{4060798480404068407082371}$, $\frac{1}{10418897902799787303093801913035739} a^{25} + \frac{2389392647}{7610590140832569249885903515731} a^{22} - \frac{300}{7610590140832569249885903515731} a^{21} + \frac{266028198}{205691625427907277023943338263} a^{20} + \frac{4000}{205691625427907277023943338263} a^{19} + \frac{338111013}{205691625427907277023943338263} a^{18} - \frac{25650}{5559233119673169649295765899} a^{17} - \frac{83233161645314005424989939070}{281591835210805062245778430082047} a^{16} + \frac{97920}{150249543774950531062047727} a^{15} + \frac{1398016487035619604875349810}{7610590140832569249885903515731} a^{14} + \frac{2265365270529274895720547287}{205691625427907277023943338263} a^{13} + \frac{702182924832744828281236441}{205691625427907277023943338263} a^{12} - \frac{67588477255626628753436463}{5559233119673169649295765899} a^{11} + \frac{36444235553001878184073789}{5559233119673169649295765899} a^{10} - \frac{36481968268313153717436006}{5559233119673169649295765899} a^{9} - \frac{54285344235136826548826083}{150249543774950531062047727} a^{8} - \frac{24813620200833479026935161341}{205691625427907277023943338263} a^{7} + \frac{1266402639439405725927874}{4060798480404068407082371} a^{6} + \frac{872011732840090728053280756}{5559233119673169649295765899} a^{5} + \frac{62865736849615946159676316}{150249543774950531062047727} a^{4} + \frac{1815047805858530096178097}{4060798480404068407082371} a^{3} + \frac{112894496766087363976539}{4060798480404068407082371} a^{2} + \frac{374623703595850897932095}{4060798480404068407082371} a + \frac{1547180213355832442465317}{4060798480404068407082371}$, $\frac{1}{10418897902799787303093801913035739} a^{26} - \frac{325}{7610590140832569249885903515731} a^{22} - \frac{2867661921}{7610590140832569249885903515731} a^{21} + \frac{4550}{205691625427907277023943338263} a^{20} - \frac{2236207155}{205691625427907277023943338263} a^{19} - \frac{30875}{5559233119673169649295765899} a^{18} - \frac{83233161638392207712139627893}{281591835210805062245778430082047} a^{17} + \frac{125970}{150249543774950531062047727} a^{16} + \frac{1398016409793650266933526868}{7610590140832569249885903515731} a^{15} - \frac{331500}{4060798480404068407082371} a^{14} + \frac{2378037499533881930897921800}{205691625427907277023943338263} a^{13} - \frac{49359101483965278811783106}{205691625427907277023943338263} a^{12} + \frac{36522481048624558784131499}{5559233119673169649295765899} a^{11} + \frac{27761051756702158829453595}{5559233119673169649295765899} a^{10} - \frac{1889601413477706177984077}{150249543774950531062047727} a^{9} + \frac{66022062911866429455383267173}{205691625427907277023943338263} a^{8} - \frac{1845006435565039129747338}{4060798480404068407082371} a^{7} + \frac{2267929090374162828619612886}{5559233119673169649295765899} a^{6} + \frac{263883019399070925009866}{4060798480404068407082371} a^{5} - \frac{54940436386637842180389554}{150249543774950531062047727} a^{4} - \frac{29048204302641336779268277}{150249543774950531062047727} a^{3} - \frac{1704557784605416863727188}{4060798480404068407082371} a^{2} + \frac{1848020108941680831013851}{4060798480404068407082371} a + \frac{2021261308377952592044712}{4060798480404068407082371}$

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

\(\Q(\zeta_{9})^+\), 9.9.80515213381214514081.2

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 $27$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ $27$ R $27$ $27$ $27$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ $27$

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
$37$37.9.8.5$x^{9} - 592$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.5$x^{9} - 592$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.5$x^{9} - 592$$9$$1$$8$$C_9$$[\ ]_{9}$