Normalized defining polynomial
\( x^{27} - 9 x^{26} - 153 x^{25} + 1380 x^{24} + 10011 x^{23} - 89964 x^{22} - 366086 x^{21} + 3289107 x^{20} + 8195640 x^{19} - 74863229 x^{18} - 115293333 x^{17} + 1111829427 x^{16} + 994569364 x^{15} - 10957304562 x^{14} - 4643357544 x^{13} + 71264766946 x^{12} + 4719112581 x^{11} - 297223405968 x^{10} + 61859763056 x^{9} + 747502977825 x^{8} - 308198445204 x^{7} - 1001646366449 x^{6} + 552392947014 x^{5} + 525970815906 x^{4} - 343918548357 x^{3} + 26874655680 x^{2} + 2448910590 x - 226688239 \)
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: | \(1197336751300349460100016353165918520894496565611371829528951009=3^{36}\cdot 7^{18}\cdot 19^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $216.89$ | 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: | $3, 7, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1197=3^{2}\cdot 7\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1197}(64,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(130,·)$, $\chi_{1197}(1156,·)$, $\chi_{1197}(709,·)$, $\chi_{1197}(967,·)$, $\chi_{1197}(841,·)$, $\chi_{1197}(484,·)$, $\chi_{1197}(142,·)$, $\chi_{1197}(655,·)$, $\chi_{1197}(403,·)$, $\chi_{1197}(919,·)$, $\chi_{1197}(856,·)$, $\chi_{1197}(25,·)$, $\chi_{1197}(1051,·)$, $\chi_{1197}(163,·)$, $\chi_{1197}(676,·)$, $\chi_{1197}(232,·)$, $\chi_{1197}(235,·)$, $\chi_{1197}(172,·)$, $\chi_{1197}(814,·)$, $\chi_{1197}(625,·)$, $\chi_{1197}(1138,·)$, $\chi_{1197}(499,·)$, $\chi_{1197}(1012,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(1087,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3} a^{21} + \frac{1}{3} a^{18} + \frac{1}{3} a^{15} + \frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{1137} a^{22} + \frac{73}{1137} a^{21} + \frac{5}{379} a^{20} + \frac{43}{1137} a^{19} + \frac{493}{1137} a^{18} - \frac{123}{379} a^{17} + \frac{154}{1137} a^{16} + \frac{289}{1137} a^{15} + \frac{75}{379} a^{14} + \frac{196}{1137} a^{13} + \frac{157}{1137} a^{12} + \frac{182}{379} a^{11} + \frac{436}{1137} a^{10} + \frac{163}{1137} a^{9} + \frac{159}{379} a^{8} - \frac{38}{1137} a^{7} - \frac{164}{1137} a^{6} + \frac{140}{379} a^{5} + \frac{131}{1137} a^{4} + \frac{557}{1137} a^{3} + \frac{37}{379} a^{2} - \frac{121}{1137} a - \frac{4}{1137}$, $\frac{1}{1137} a^{23} - \frac{8}{1137} a^{21} + \frac{85}{1137} a^{20} - \frac{124}{379} a^{19} - \frac{353}{1137} a^{18} - \frac{197}{1137} a^{17} + \frac{139}{379} a^{16} + \frac{352}{1137} a^{15} - \frac{311}{1137} a^{14} - \frac{169}{379} a^{13} + \frac{76}{1137} a^{12} + \frac{373}{1137} a^{11} + \frac{57}{379} a^{10} - \frac{431}{1137} a^{9} + \frac{388}{1137} a^{8} + \frac{112}{379} a^{7} - \frac{494}{1137} a^{6} + \frac{170}{1137} a^{5} + \frac{30}{379} a^{4} - \frac{376}{1137} a^{3} - \frac{265}{1137} a^{2} - \frac{89}{379} a - \frac{466}{1137}$, $\frac{1}{3322146861} a^{24} + \frac{450976}{1107382287} a^{23} + \frac{28115}{1107382287} a^{22} + \frac{67662032}{3322146861} a^{21} + \frac{547825127}{1107382287} a^{20} + \frac{205298309}{1107382287} a^{19} + \frac{52856689}{302013351} a^{18} + \frac{491156812}{1107382287} a^{17} - \frac{491475443}{1107382287} a^{16} + \frac{161537528}{3322146861} a^{15} + \frac{376081}{973951} a^{14} + \frac{132256928}{369127429} a^{13} + \frac{498219602}{3322146861} a^{12} + \frac{60922852}{369127429} a^{11} + \frac{13059986}{100671117} a^{10} + \frac{903168554}{3322146861} a^{9} + \frac{539151}{369127429} a^{8} + \frac{23732312}{1107382287} a^{7} + \frac{288976555}{1107382287} a^{6} + \frac{23004068}{369127429} a^{5} + \frac{3341942}{369127429} a^{4} - \frac{748699541}{3322146861} a^{3} + \frac{125161911}{369127429} a^{2} + \frac{55987227}{369127429} a - \frac{1409218969}{3322146861}$, $\frac{1}{3322146861} a^{25} + \frac{391994}{1107382287} a^{23} + \frac{716168}{3322146861} a^{22} + \frac{7776292}{369127429} a^{21} - \frac{447438109}{1107382287} a^{20} - \frac{169101895}{3322146861} a^{19} + \frac{396422579}{1107382287} a^{18} - \frac{248840615}{1107382287} a^{17} - \frac{1374359275}{3322146861} a^{16} + \frac{142923037}{1107382287} a^{15} - \frac{159924440}{369127429} a^{14} + \frac{41328829}{302013351} a^{13} - \frac{388717940}{1107382287} a^{12} + \frac{105693067}{1107382287} a^{11} - \frac{415539133}{3322146861} a^{10} + \frac{402638572}{1107382287} a^{9} - \frac{41581481}{100671117} a^{8} + \frac{108262973}{369127429} a^{7} - \frac{134968144}{369127429} a^{6} + \frac{12089147}{369127429} a^{5} - \frac{1377713255}{3322146861} a^{4} + \frac{394635179}{1107382287} a^{3} + \frac{181375636}{369127429} a^{2} + \frac{363289913}{3322146861} a + \frac{169873003}{1107382287}$, $\frac{1}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{26} - \frac{14355278589112162677058597478550902855836265762412252287822228604563033912244577994487980440128792954823}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{25} + \frac{54208060898912943644018729604547002888343370053637404270056128577877800681863861808874548296904172469101}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{24} - \frac{125280110285870241350519146531265760219867290145367122061337121548776769704304602579971323108218038553727249317}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{23} - \frac{75061248401271907889596755049044763993992435427065172713545531140678507999525670509011863968344835640245492418}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{22} - \frac{15575764660616787163793527590560013305180442148479007046314380782388172387599300500087327612108618290706805492472}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{21} - \frac{98810302804801599392963133407188147903824419286230544684482749127506777193316234364621308352299340325291942777692}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{20} + \frac{37047424710225159055885438100141898428675539778271477411246050629967001700222139869040257697552680555027814834215}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{19} + \frac{118202537966931687837281556908746135311075040349960416458099252199787175571085727359664555144680949216024504132137}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{18} + \frac{154979155789890281129524625357059134406988402145042743834292086187307390993608358449636697501355060757042567011897}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{17} - \frac{1683123119273525125705700319007304662271588859118124416179840257829742267315943229500725745039086956767259456872}{14222134129626214424864823540791020366079553187164987014500824780681895921559612230080617444543756906136667395303} a^{16} - \frac{9591361977841952144942755382215585372909747297309472427228080385276251823115951480447206841104837699398576698907}{47838087526924539429090770091751613958631224356827683594230046989566377190700513864816622313465364138823335784201} a^{15} + \frac{225378859756165153026221424846633761455050018521916531251672033557385960307267278707209583855353189506758377401453}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{14} - \frac{160095067937392384817228539304305806316117319561090120071301004162455956741521356500789034175890629761279578223665}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{13} - \frac{3790627414340613118773724620988311728971296069491367414820607665926033344534519096169361668094424664686030600890}{14222134129626214424864823540791020366079553187164987014500824780681895921559612230080617444543756906136667395303} a^{12} + \frac{191427855250576166450597621905578252690149017369237233694768321018217606508974386282515850074665218645402384451909}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{11} - \frac{167026485331055857954250343731476809128234135701614127734565464389454837848274800958156274027211145814666110916550}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{10} + \frac{4707624245954534912564241737493627563000981595610427405937318847881557481135067570593586837082658349780870704026}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{9} - \frac{86759134998620262659620120348010472136931457222864088774913733854836775764859721784786973476449739525716635883443}{175406320932056644573332823669755917848314489308368173178843505628410049699235217504327615149373001842352231208737} a^{8} + \frac{4662907632395446211818524702690154784095461649345670709671494629543766007990226557993626246889990662040468996372}{175406320932056644573332823669755917848314489308368173178843505628410049699235217504327615149373001842352231208737} a^{7} + \frac{19305508886526569580770792021830129997948133313974461212985919914970348210420766769199539479746256622270459603363}{58468773644018881524444274556585305949438163102789391059614501876136683233078405834775871716457667280784077069579} a^{6} - \frac{45311923002931534373178043239356290810598046746829511483870949183504423414060133069770281085726446434422217871136}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{5} + \frac{39793865363986842428365109485510405047872740626845017982682084922030341867049345966529919973768103322169835528696}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{4} + \frac{9895190554713672672255145901627715885115834197577182460176127124225284606112395934762386954943549687595434800942}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{3} + \frac{33668595701775579188197893142474032903801703738452657115808505544286190106689266352191768106719416038807592794032}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a^{2} - \frac{241815917358928166687225828953332995937706149238703475099329941299594184482804936106322004675321969472195642822034}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211} a - \frac{178688917383337026649358148805646445009963837745633983392164366099210812045288540336532853805257010422329103931379}{526218962796169933719998471009267753544943467925104519536530516885230149097705652512982845448119005527056693626211}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
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: | 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: | \( 19684261760036090000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times C_9$ (as 27T2):
| 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
| 3.3.17689.1, 3.3.361.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 9.9.5534900853769.1, 9.9.9025761726072081.1, 9.9.1061871841310654257569.5, 9.9.1061871841310654257569.6 |
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 | ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | 3.9.12.3 | $x^{9} + 6 x^{8} + 3 x^{6} + 9 x^{3} + 135$ | $3$ | $3$ | $12$ | $C_9$ | $[2]^{3}$ |
| 3.9.12.3 | $x^{9} + 6 x^{8} + 3 x^{6} + 9 x^{3} + 135$ | $3$ | $3$ | $12$ | $C_9$ | $[2]^{3}$ | |
| 3.9.12.3 | $x^{9} + 6 x^{8} + 3 x^{6} + 9 x^{3} + 135$ | $3$ | $3$ | $12$ | $C_9$ | $[2]^{3}$ | |
| $7$ | 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 19 | Data not computed | ||||||