Normalized defining polynomial
\( x^{27} - 9 x^{26} - 189 x^{25} + 1680 x^{24} + 16011 x^{23} - 135216 x^{22} - 811224 x^{21} + 6143229 x^{20} + 27534006 x^{19} - 173103583 x^{18} - 657009639 x^{17} + 3125053791 x^{16} + 11114099142 x^{15} - 35932528620 x^{14} - 130592837952 x^{13} + 250836674670 x^{12} + 1021415604147 x^{11} - 930934482786 x^{10} - 4988671858122 x^{9} + 1049534716941 x^{8} + 13954608198684 x^{7} + 2985430435785 x^{6} - 20357236188306 x^{5} - 8830845112932 x^{4} + 13243359313719 x^{3} + 6479388550764 x^{2} - 2750267537172 x - 1151745452469 \)
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: | \(22874376238048151146999094376461057406235644645953493108395282961=3^{66}\cdot 67^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $241.92$ | 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: | $3, 67$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1809=3^{3}\cdot 67\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1809}(640,·)$, $\chi_{1809}(1,·)$, $\chi_{1809}(1609,·)$, $\chi_{1809}(967,·)$, $\chi_{1809}(841,·)$, $\chi_{1809}(202,·)$, $\chi_{1809}(1207,·)$, $\chi_{1809}(1168,·)$, $\chi_{1809}(1042,·)$, $\chi_{1809}(403,·)$, $\chi_{1809}(238,·)$, $\chi_{1809}(1369,·)$, $\chi_{1809}(1243,·)$, $\chi_{1809}(604,·)$, $\chi_{1809}(805,·)$, $\chi_{1809}(1570,·)$, $\chi_{1809}(163,·)$, $\chi_{1809}(1444,·)$, $\chi_{1809}(37,·)$, $\chi_{1809}(1771,·)$, $\chi_{1809}(364,·)$, $\chi_{1809}(1645,·)$, $\chi_{1809}(1006,·)$, $\chi_{1809}(1408,·)$, $\chi_{1809}(565,·)$, $\chi_{1809}(439,·)$, $\chi_{1809}(766,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{27} a^{18} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{2}{27} a^{9} + \frac{1}{3} a^{8} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{9}$, $\frac{1}{27} a^{19} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{2}{27} a^{10} + \frac{1}{3} a^{9} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{9} a$, $\frac{1}{27} a^{20} + \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{2}{27} a^{11} + \frac{1}{3} a^{10} + \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{9} a^{2}$, $\frac{1}{1431} a^{21} + \frac{11}{1431} a^{20} - \frac{23}{1431} a^{19} - \frac{23}{1431} a^{18} + \frac{43}{159} a^{17} - \frac{16}{159} a^{16} - \frac{37}{159} a^{15} - \frac{23}{53} a^{14} + \frac{64}{159} a^{13} + \frac{146}{1431} a^{12} + \frac{706}{1431} a^{11} - \frac{244}{1431} a^{10} - \frac{7}{1431} a^{9} - \frac{10}{477} a^{8} - \frac{1}{9} a^{7} + \frac{157}{477} a^{6} + \frac{41}{159} a^{5} + \frac{68}{159} a^{4} + \frac{23}{477} a^{3} - \frac{227}{477} a^{2} - \frac{202}{477} a + \frac{4}{9}$, $\frac{1}{1431} a^{22} + \frac{5}{477} a^{20} + \frac{2}{159} a^{19} + \frac{4}{1431} a^{18} - \frac{4}{53} a^{17} - \frac{20}{159} a^{16} - \frac{11}{53} a^{15} + \frac{28}{159} a^{14} + \frac{488}{1431} a^{13} + \frac{59}{159} a^{12} - \frac{179}{477} a^{11} - \frac{203}{477} a^{10} - \frac{271}{1431} a^{9} + \frac{24}{53} a^{8} + \frac{17}{159} a^{7} - \frac{14}{477} a^{6} - \frac{65}{159} a^{5} - \frac{154}{477} a^{4} - \frac{1}{159} a^{3} + \frac{76}{159} a^{2} - \frac{24}{53} a + \frac{4}{9}$, $\frac{1}{4293} a^{23} + \frac{4}{1431} a^{20} - \frac{25}{1431} a^{19} + \frac{26}{1431} a^{18} - \frac{29}{477} a^{17} + \frac{16}{159} a^{16} + \frac{1}{3} a^{15} + \frac{1217}{4293} a^{14} + \frac{2}{9} a^{13} + \frac{58}{159} a^{12} - \frac{32}{159} a^{11} - \frac{107}{1431} a^{10} - \frac{173}{1431} a^{9} + \frac{40}{159} a^{8} - \frac{40}{477} a^{7} + \frac{104}{477} a^{6} + \frac{386}{1431} a^{5} + \frac{13}{159} a^{4} + \frac{40}{159} a^{3} - \frac{103}{477} a^{2} + \frac{109}{477} a - \frac{1}{9}$, $\frac{1}{4293} a^{24} - \frac{16}{1431} a^{20} + \frac{4}{477} a^{19} + \frac{5}{1431} a^{18} + \frac{1}{53} a^{17} + \frac{11}{159} a^{16} - \frac{1942}{4293} a^{15} + \frac{139}{477} a^{14} + \frac{14}{159} a^{13} + \frac{559}{1431} a^{12} + \frac{37}{1431} a^{11} - \frac{121}{477} a^{10} - \frac{566}{1431} a^{9} + \frac{1}{9} a^{8} - \frac{12}{53} a^{7} + \frac{410}{1431} a^{6} - \frac{15}{53} a^{5} + \frac{11}{53} a^{4} - \frac{65}{159} a^{3} + \frac{10}{477} a^{2} - \frac{31}{159} a + \frac{2}{9}$, $\frac{1}{50069259} a^{25} + \frac{4531}{50069259} a^{24} - \frac{1250}{50069259} a^{23} + \frac{1354}{5563251} a^{22} - \frac{731}{5563251} a^{21} + \frac{120725}{16689753} a^{20} + \frac{94762}{5563251} a^{19} - \frac{128363}{16689753} a^{18} + \frac{2207671}{5563251} a^{17} - \frac{4488964}{50069259} a^{16} + \frac{18138653}{50069259} a^{15} - \frac{9541294}{50069259} a^{14} - \frac{5692256}{16689753} a^{13} + \frac{6227035}{16689753} a^{12} - \frac{264100}{16689753} a^{11} + \frac{73309}{5563251} a^{10} - \frac{1394101}{16689753} a^{9} - \frac{1605524}{5563251} a^{8} - \frac{3681055}{16689753} a^{7} - \frac{634330}{16689753} a^{6} - \frac{7933357}{16689753} a^{5} - \frac{598690}{1854417} a^{4} + \frac{305822}{1854417} a^{3} + \frac{615478}{5563251} a^{2} + \frac{271012}{1854417} a - \frac{161}{963}$, $\frac{1}{1783992706755650636630331571745071093099694943817158160245883574726726312849430904776551814650391754302437802230820269592604469531013} a^{26} + \frac{14759303943144727702387520480714356272231839451915477149226679577260339761029335235816679942880975798480317318242056402222219}{1783992706755650636630331571745071093099694943817158160245883574726726312849430904776551814650391754302437802230820269592604469531013} a^{25} - \frac{86458967917368593517795513846559299844772493469578958550095603936257505934086439098542719469651863300895139126006226259213906791}{1783992706755650636630331571745071093099694943817158160245883574726726312849430904776551814650391754302437802230820269592604469531013} a^{24} + \frac{16418945458432214797713250275647079141469450748190123518994642246824621029515965094764815224314118405139843533214182505094318941}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{23} - \frac{99434136594559419372450065994859167082083366236981352098829337006167689070246040811310339272327386859790372143406187677584283923}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{22} - \frac{85980279876301897570274234193924742299198223446829944994250242728199969444991966436129418065631227287114491961186719858596113371}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{21} + \frac{2986241796116256095456203579306207882718309213782826838223540065260627771065141159555910348743879538226105465188336398469963521099}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{20} + \frac{3190339753037126551088136313254606740014881273986975168221774612343301922879173727550110831599319366835248550815488260350646676571}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{19} - \frac{3880573322081367238046480728204768957941763863134866979026861298795017709225794307760852068615230613450086453235381033152543107985}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{18} - \frac{76111630244374486019270481011981322771878810990281697532272161750407778046207776544047286464912352917204825614206798493180124622407}{1783992706755650636630331571745071093099694943817158160245883574726726312849430904776551814650391754302437802230820269592604469531013} a^{17} + \frac{394036430950583177334065894619264835178678066671950460676635443945030760384061296406146813263849328873843069842315407721898254601034}{1783992706755650636630331571745071093099694943817158160245883574726726312849430904776551814650391754302437802230820269592604469531013} a^{16} - \frac{385556085470331251277680772132259838318432989672547381029521909759571116188483375022909146515787679363545986458436574453285321836735}{1783992706755650636630331571745071093099694943817158160245883574726726312849430904776551814650391754302437802230820269592604469531013} a^{15} + \frac{101575543876016513119958203460744629120173999261324691026239959831867749199733159510696382451701948990421907126013873976422940781}{22024601317970995513954710762284828309872777084162446422788686107737361887030011170080886600622120423486886447294077402377832957173} a^{14} - \frac{56638810417147867940788349096172874076668583354747008138414785504911856248871586420068960777819633749691400911044684522571763572110}{198221411861738959625592396860563454788854993757462017805098174969636256983270100530727979405599083811381978025646696621400496614557} a^{13} + \frac{32568121786349559500521825735090896287383453692021263905823460826185535726837586549203056633956403479124529600118272169186096394821}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{12} + \frac{28417257586287739629562415033837877233428902826656774628561447449408754325520228953048731029009506765973752036277414046416742443728}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{11} + \frac{45362164113518936949292467632849433372510505047068633355090353643489924839296737921226488259009148911696371650293728173486256094469}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{10} - \frac{121893530292772615459230407808605004035012084248656845220673709968220797464255826465968818296694770889980154674744349358148034002373}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{9} + \frac{198246980696474564059814465660768738185610187144742358319512556349636671653932379637228013594279065091543760242020494462467106557007}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{8} + \frac{121078048877755499185937808734827638358064939863597117647685051127527690112492634229013526393234729048720847277564057434464302819600}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{7} - \frac{291034440517588512061005629819872242006525803994047475712650484113731009985196243863855981985934769343380178204076144904135525417679}{594664235585216878876777190581690364366564981272386053415294524908908770949810301592183938216797251434145934076940089864201489843671} a^{6} + \frac{46921434014086814524354133582669310699140679045098557196396560891769343453508970152417568342786802009089051833982719520298148600595}{198221411861738959625592396860563454788854993757462017805098174969636256983270100530727979405599083811381978025646696621400496614557} a^{5} + \frac{68343398386156314825261798810794510100536244373126185032758735094971170743124823686771425905754049793699457918379829226418971986622}{198221411861738959625592396860563454788854993757462017805098174969636256983270100530727979405599083811381978025646696621400496614557} a^{4} + \frac{33059876126421762295000841439291625085206332634684400234287172268617866047395400041585068702200585570281722370435976663997929632428}{198221411861738959625592396860563454788854993757462017805098174969636256983270100530727979405599083811381978025646696621400496614557} a^{3} - \frac{13743259348892368901804145700457461544254834737035052711886552500753677690062695076683274342091934411493433635892528939929921572583}{198221411861738959625592396860563454788854993757462017805098174969636256983270100530727979405599083811381978025646696621400496614557} a^{2} - \frac{1072038815382303487098185926116045257550261084646270919699517544136894668881445915926184723357673552114010043848325477336120675136}{3740026638900735087275328242652140656393490448254000335945248584332759565722077368504301498218850637950603358974465973988688615369} a + \frac{91107500286226371744826869607491605612618728916983522979039433372578769297595951157508034382455977133614940857525822741388240}{647399452813005900515029988342070392313223203782932375964211283422669130296360977757365673917059137606128329405308286998215097}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 2233170119550491800000000 \) (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
| 3.3.363609.2, \(\Q(\zeta_{9})^+\), 3.3.363609.1, 3.3.4489.1, 9.9.48073293078275529.1, 9.9.2838679882979091711921.2, \(\Q(\zeta_{27})^+\), 9.9.2838679882979091711921.1 |
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 | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.9.22.8 | $x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 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 | $x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 67 | Data not computed | ||||||