Normalized defining polynomial
\( x^{27} - 9 x^{26} - 171 x^{25} + 1662 x^{24} + 12105 x^{23} - 131490 x^{22} - 453414 x^{21} + 5848551 x^{20} + 9316386 x^{19} - 161570611 x^{18} - 87089931 x^{17} + 2896436709 x^{16} - 301799844 x^{15} - 34271005878 x^{14} + 17195762052 x^{13} + 267669198096 x^{12} - 196205674761 x^{11} - 1360829134422 x^{10} + 1139084249570 x^{9} + 4372192409787 x^{8} - 3630889722708 x^{7} - 8428513893225 x^{6} + 6048620755506 x^{5} + 8864208737262 x^{4} - 4498476763089 x^{3} - 4006776962952 x^{2} + 921282187860 x + 173820270061 \)
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: | \(4226100704227770327423556410760485273603323729873037153574412049=3^{66}\cdot 61^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $227.26$ | 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: | $3, 61$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1647=3^{3}\cdot 61\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1647}(1,·)$, $\chi_{1647}(1282,·)$, $\chi_{1647}(196,·)$, $\chi_{1647}(901,·)$, $\chi_{1647}(1099,·)$, $\chi_{1647}(13,·)$, $\chi_{1647}(1294,·)$, $\chi_{1647}(352,·)$, $\chi_{1647}(916,·)$, $\chi_{1647}(718,·)$, $\chi_{1647}(535,·)$, $\chi_{1647}(1111,·)$, $\chi_{1647}(733,·)$, $\chi_{1647}(1477,·)$, $\chi_{1647}(928,·)$, $\chi_{1647}(1633,·)$, $\chi_{1647}(550,·)$, $\chi_{1647}(169,·)$, $\chi_{1647}(1450,·)$, $\chi_{1647}(367,·)$, $\chi_{1647}(562,·)$, $\chi_{1647}(1267,·)$, $\chi_{1647}(745,·)$, $\chi_{1647}(184,·)$, $\chi_{1647}(1465,·)$, $\chi_{1647}(379,·)$, $\chi_{1647}(1084,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{27} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{2}{9} a^{15} - \frac{1}{3} a^{14} + \frac{2}{9} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{3} - \frac{1}{3} a - \frac{1}{27}$, $\frac{1}{27} a^{19} - \frac{1}{3} a^{17} - \frac{2}{9} a^{16} - \frac{1}{3} a^{15} + \frac{2}{9} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{2}{9} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{4} - \frac{1}{3} a^{2} - \frac{1}{27} a + \frac{1}{3}$, $\frac{1}{27} a^{20} - \frac{2}{9} a^{17} - \frac{1}{3} a^{16} + \frac{2}{9} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{2}{9} a^{8} - \frac{1}{3} a^{7} + \frac{4}{9} a^{5} - \frac{1}{3} a^{3} - \frac{1}{27} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{1431} a^{21} - \frac{4}{1431} a^{20} - \frac{13}{1431} a^{19} - \frac{4}{1431} a^{18} - \frac{175}{477} a^{17} + \frac{59}{477} a^{16} - \frac{62}{477} a^{15} - \frac{23}{477} a^{14} + \frac{31}{477} a^{13} - \frac{209}{477} a^{12} + \frac{46}{159} a^{11} - \frac{17}{53} a^{10} + \frac{202}{477} a^{9} + \frac{35}{477} a^{8} + \frac{71}{477} a^{7} + \frac{67}{159} a^{6} + \frac{203}{477} a^{5} - \frac{145}{477} a^{4} + \frac{32}{1431} a^{3} - \frac{86}{1431} a^{2} + \frac{490}{1431} a + \frac{2}{27}$, $\frac{1}{1431} a^{22} + \frac{8}{477} a^{20} - \frac{1}{477} a^{19} - \frac{11}{1431} a^{18} + \frac{23}{53} a^{17} + \frac{227}{477} a^{16} - \frac{59}{477} a^{15} - \frac{38}{159} a^{14} + \frac{7}{159} a^{13} + \frac{44}{477} a^{12} - \frac{26}{159} a^{11} + \frac{226}{477} a^{10} + \frac{16}{159} a^{9} - \frac{6}{53} a^{8} - \frac{98}{477} a^{7} - \frac{4}{9} a^{6} - \frac{26}{53} a^{5} + \frac{359}{1431} a^{4} + \frac{67}{477} a^{3} - \frac{128}{477} a^{2} + \frac{194}{477} a - \frac{2}{27}$, $\frac{1}{4293} a^{23} + \frac{1}{4293} a^{22} + \frac{1}{4293} a^{21} - \frac{46}{4293} a^{20} + \frac{20}{4293} a^{19} - \frac{40}{4293} a^{18} + \frac{7}{1431} a^{17} + \frac{613}{1431} a^{16} - \frac{284}{1431} a^{15} - \frac{41}{1431} a^{14} - \frac{701}{1431} a^{13} - \frac{527}{1431} a^{12} - \frac{482}{1431} a^{11} + \frac{454}{1431} a^{10} + \frac{118}{1431} a^{9} - \frac{107}{477} a^{8} + \frac{59}{477} a^{7} - \frac{15}{53} a^{6} + \frac{1868}{4293} a^{5} + \frac{230}{4293} a^{4} + \frac{194}{4293} a^{3} + \frac{1858}{4293} a^{2} + \frac{1873}{4293} a - \frac{25}{81}$, $\frac{1}{3603059091} a^{24} + \frac{2125}{400339899} a^{23} + \frac{6139}{44482211} a^{22} - \frac{740030}{3603059091} a^{21} + \frac{10011680}{1201019697} a^{20} + \frac{7249381}{400339899} a^{19} + \frac{718535}{67982247} a^{18} + \frac{156478544}{400339899} a^{17} - \frac{2263640}{7023507} a^{16} - \frac{15954233}{400339899} a^{15} + \frac{166366219}{400339899} a^{14} - \frac{58838200}{400339899} a^{13} + \frac{30176489}{133446633} a^{12} - \frac{964690}{44482211} a^{11} - \frac{107272549}{400339899} a^{10} + \frac{104924357}{1201019697} a^{9} + \frac{157554314}{400339899} a^{8} + \frac{19906504}{44482211} a^{7} - \frac{1336022266}{3603059091} a^{6} + \frac{72448735}{400339899} a^{5} - \frac{79730423}{400339899} a^{4} + \frac{15507776}{189634689} a^{3} - \frac{42278156}{1201019697} a^{2} - \frac{8649733}{133446633} a - \frac{171461}{67982247}$, $\frac{1}{3603059091} a^{25} - \frac{59507}{1201019697} a^{23} - \frac{857408}{3603059091} a^{22} - \frac{13}{21070521} a^{21} - \frac{4392148}{1201019697} a^{20} + \frac{776500}{189634689} a^{19} - \frac{18527759}{1201019697} a^{18} - \frac{70900481}{400339899} a^{17} - \frac{199019024}{400339899} a^{16} + \frac{552502}{21070521} a^{15} + \frac{44967154}{133446633} a^{14} - \frac{9731547}{44482211} a^{13} + \frac{13985195}{400339899} a^{12} + \frac{772060}{7023507} a^{11} + \frac{223101998}{1201019697} a^{10} - \frac{35440760}{133446633} a^{9} - \frac{35547289}{133446633} a^{8} + \frac{36943583}{3603059091} a^{7} - \frac{126889064}{400339899} a^{6} + \frac{420274745}{1201019697} a^{5} - \frac{943943632}{3603059091} a^{4} + \frac{99880865}{400339899} a^{3} + \frac{470844514}{1201019697} a^{2} - \frac{1100089240}{3603059091} a + \frac{1151488}{22660749}$, $\frac{1}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{26} + \frac{205344837499568879253836548401557983287878375625181503707640121863887631758168202331939475156758323838831199988319}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{25} + \frac{173765074356762281130496657807220434725175248462217801454251829452499121826309340027582908494099376229423662935233}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{24} + \frac{99099112181220492902600752116547843645020740250718109377632762131855009296327279255574424847899569543224267285735239631}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{23} - \frac{491482165431034669638152416886976860566052918893757164533526484535382664143289548931527007828426885569851520613786422640}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{22} - \frac{228413088306358380393492265839262864694265481631295631644402630049272233339972974097906219234501987313636275366067813925}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{21} - \frac{14212826905498005016311038856354675794726153224520128938682039199777048528629631745272566885904277549403799698985660352551}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{20} + \frac{22149954427927970502036352937356967027042700296034576318646525305341824958184876466104816397041357927354603254277917612647}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{19} - \frac{24699939117030257123836936588796865429975956470373474479196931188591662926526931245207743424497836668439809736444640549562}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{18} - \frac{13966150293795940743379196902005881281438525165806729043957973948340386187575978369391190409919613646732586160680314249459}{182533569852146471899203844841686331030190884332947139865924669144762739034677535674348401576202418577358506677990865916897} a^{17} - \frac{29992352608870741314029957089069529590081228297633350715812290396638644773952411976491402408629576513667051283568597513450}{182533569852146471899203844841686331030190884332947139865924669144762739034677535674348401576202418577358506677990865916897} a^{16} - \frac{62261354702269269387070393722284765113373526042054191127298974806516951840234517503508486144424017252438676845045251739593}{182533569852146471899203844841686331030190884332947139865924669144762739034677535674348401576202418577358506677990865916897} a^{15} + \frac{602470521254037392206727176190473685539357930808536502147958692049066959940311859907636577805905207584616874856263929529}{20281507761349607988800427204631814558910098259216348873991629904973637670519726186038711286244713175262056297554540657433} a^{14} - \frac{11304730339246925651498616730125144806483001738616392351858847939749598485627468770965044597103257853344824332533264237478}{182533569852146471899203844841686331030190884332947139865924669144762739034677535674348401576202418577358506677990865916897} a^{13} - \frac{38024233764362802127800381810987309019554610304013127456494691538800966370514163505409099431628826119094050501395499207592}{182533569852146471899203844841686331030190884332947139865924669144762739034677535674348401576202418577358506677990865916897} a^{12} + \frac{101149836710679216561309148558371998191350118062268905470717676541758852708960991970095841450556488817688480044308305772387}{547600709556439415697611534525058993090572652998841419597774007434288217104032607023045204728607255732075520033972597750691} a^{11} - \frac{272238526592991975088005342956731255318212042808413859127127868950258358546067903876875440268876503323655217490166851501314}{547600709556439415697611534525058993090572652998841419597774007434288217104032607023045204728607255732075520033972597750691} a^{10} - \frac{135559082162554198848173737902798019578698203812132904764500759374571333624989733703059904702816155972790475877900642972573}{547600709556439415697611534525058993090572652998841419597774007434288217104032607023045204728607255732075520033972597750691} a^{9} - \frac{290651329270258677378711759455083130329645937741927069658339510538929298285777935493195006697115362410452087699882088207321}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{8} - \frac{575495642816451050727190304367366139934886442655812880148088160638161297988771972416487726130209950248511759396558624439057}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{7} + \frac{805151719225472969260789361820064546923617096293247386870321781585176777492465032943858995341648600085786244441409812911790}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{6} + \frac{421606737071982408820032913801380060774834140520256775549414096790983156869066070098403086682829384296002047365292933685865}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{5} - \frac{708755448013841392304788828120639632881007856676854187043773105897696028185160700570287542668787546371178653285519794993106}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{4} - \frac{173621220213765352946788003966721470963974578250903989211561098903302999785035276761115062097418645906055514434727114867033}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{3} + \frac{468809669697516051735175249444684247274265919763374230605245099933648298222759361694020675626924821823330959631951553812326}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a^{2} - \frac{120356229768772953867139618667385280612741657933887767408638381779434312557651900733118992707642892323593126016774053276469}{1642802128669318247092834603575176979271717958996524258793322022302864651312097821069135614185821767196226560101917793252073} a - \frac{3472359244687855881565557404118881596598510417194266533182947792631789496965223947595274356676152875710756103769157298825}{30996266578666382020619520822173150552296565264085363373458906081186125496454675869228973852562674852758991700036184778341}$
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: | \( 665599783500463000000000 \) (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
| 3.3.301401.2, \(\Q(\zeta_{9})^+\), 3.3.3721.1, 3.3.301401.1, 9.9.27380039270784201.1, 9.9.1616763938900536284849.1, 9.9.1616763938900536284849.2, \(\Q(\zeta_{27})^+\) |
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 | $x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 61 | Data not computed | ||||||