Normalized defining polynomial
\( x^{27} - x^{26} - 234 x^{25} + 451 x^{24} + 22410 x^{23} - 59336 x^{22} - 1146120 x^{21} + 3693810 x^{20} + 34438763 x^{19} - 128044212 x^{18} - 633592894 x^{17} + 2668309905 x^{16} + 7191540555 x^{15} - 34553534532 x^{14} - 49158187704 x^{13} + 278963008001 x^{12} + 190196186526 x^{11} - 1372095146729 x^{10} - 381761929893 x^{9} + 3905637763092 x^{8} + 497849168333 x^{7} - 5908214090044 x^{6} - 1034156707955 x^{5} + 4070335505780 x^{4} + 1245264135357 x^{3} - 878470562528 x^{2} - 387476751728 x - 27950290933 \)
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: | \(7511955241024825880501672185173495875424957194130773062050731864857009=487^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $387.25$ | 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: | $487$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(487\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{487}(1,·)$, $\chi_{487}(259,·)$, $\chi_{487}(392,·)$, $\chi_{487}(138,·)$, $\chi_{487}(143,·)$, $\chi_{487}(144,·)$, $\chi_{487}(19,·)$, $\chi_{487}(220,·)$, $\chi_{487}(25,·)$, $\chi_{487}(282,·)$, $\chi_{487}(475,·)$, $\chi_{487}(284,·)$, $\chi_{487}(482,·)$, $\chi_{487}(187,·)$, $\chi_{487}(292,·)$, $\chi_{487}(166,·)$, $\chi_{487}(39,·)$, $\chi_{487}(232,·)$, $\chi_{487}(41,·)$, $\chi_{487}(362,·)$, $\chi_{487}(301,·)$, $\chi_{487}(51,·)$, $\chi_{487}(361,·)$, $\chi_{487}(443,·)$, $\chi_{487}(60,·)$, $\chi_{487}(254,·)$, $\chi_{487}(191,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{41} a^{20} + \frac{9}{41} a^{19} - \frac{7}{41} a^{18} - \frac{12}{41} a^{17} + \frac{9}{41} a^{16} + \frac{18}{41} a^{15} + \frac{6}{41} a^{14} - \frac{9}{41} a^{13} - \frac{9}{41} a^{12} + \frac{19}{41} a^{11} + \frac{1}{41} a^{10} - \frac{3}{41} a^{9} - \frac{5}{41} a^{8} - \frac{10}{41} a^{7} - \frac{2}{41} a^{6} - \frac{13}{41} a^{5} + \frac{9}{41} a^{3} - \frac{18}{41} a^{2} - \frac{20}{41} a$, $\frac{1}{41} a^{21} - \frac{6}{41} a^{19} + \frac{10}{41} a^{18} - \frac{6}{41} a^{17} + \frac{19}{41} a^{16} + \frac{8}{41} a^{15} + \frac{19}{41} a^{14} - \frac{10}{41} a^{13} + \frac{18}{41} a^{12} - \frac{6}{41} a^{11} - \frac{12}{41} a^{10} - \frac{19}{41} a^{9} - \frac{6}{41} a^{8} + \frac{6}{41} a^{7} + \frac{5}{41} a^{6} - \frac{6}{41} a^{5} + \frac{9}{41} a^{4} - \frac{17}{41} a^{3} + \frac{19}{41} a^{2} + \frac{16}{41} a$, $\frac{1}{41} a^{22} - \frac{18}{41} a^{19} - \frac{7}{41} a^{18} - \frac{12}{41} a^{17} - \frac{20}{41} a^{16} + \frac{4}{41} a^{15} - \frac{15}{41} a^{14} + \frac{5}{41} a^{13} - \frac{19}{41} a^{12} + \frac{20}{41} a^{11} - \frac{13}{41} a^{10} + \frac{17}{41} a^{9} + \frac{17}{41} a^{8} - \frac{14}{41} a^{7} - \frac{18}{41} a^{6} + \frac{13}{41} a^{5} - \frac{17}{41} a^{4} - \frac{9}{41} a^{3} - \frac{10}{41} a^{2} + \frac{3}{41} a$, $\frac{1}{41} a^{23} - \frac{9}{41} a^{19} - \frac{15}{41} a^{18} + \frac{10}{41} a^{17} + \frac{2}{41} a^{16} - \frac{19}{41} a^{15} - \frac{10}{41} a^{14} - \frac{17}{41} a^{13} - \frac{19}{41} a^{12} + \frac{1}{41} a^{11} - \frac{6}{41} a^{10} + \frac{4}{41} a^{9} + \frac{19}{41} a^{8} + \frac{7}{41} a^{7} + \frac{18}{41} a^{6} - \frac{5}{41} a^{5} - \frac{9}{41} a^{4} - \frac{12}{41} a^{3} + \frac{7}{41} a^{2} + \frac{9}{41} a$, $\frac{1}{205} a^{24} + \frac{2}{205} a^{23} + \frac{1}{205} a^{22} - \frac{1}{205} a^{21} + \frac{77}{205} a^{19} + \frac{64}{205} a^{18} - \frac{2}{41} a^{17} + \frac{68}{205} a^{16} - \frac{19}{41} a^{15} - \frac{17}{205} a^{14} + \frac{9}{41} a^{13} - \frac{73}{205} a^{12} + \frac{14}{41} a^{11} - \frac{1}{5} a^{10} + \frac{77}{205} a^{9} + \frac{23}{205} a^{8} + \frac{86}{205} a^{7} + \frac{31}{205} a^{6} - \frac{7}{41} a^{5} - \frac{56}{205} a^{4} - \frac{2}{41} a^{3} - \frac{4}{205} a^{2} + \frac{71}{205} a - \frac{1}{5}$, $\frac{1}{12852749495} a^{25} - \frac{5062689}{2570549899} a^{24} + \frac{91629907}{12852749495} a^{23} - \frac{106974868}{12852749495} a^{22} - \frac{98505403}{12852749495} a^{21} - \frac{84814928}{12852749495} a^{20} - \frac{562777113}{2570549899} a^{19} - \frac{2317707433}{12852749495} a^{18} + \frac{5604875383}{12852749495} a^{17} - \frac{3309351291}{12852749495} a^{16} + \frac{1041586193}{12852749495} a^{15} + \frac{1956983519}{12852749495} a^{14} - \frac{1932107473}{12852749495} a^{13} + \frac{5431030656}{12852749495} a^{12} + \frac{2461893994}{12852749495} a^{11} + \frac{3968511389}{12852749495} a^{10} - \frac{3362693541}{12852749495} a^{9} - \frac{23189104}{2570549899} a^{8} + \frac{5467247529}{12852749495} a^{7} + \frac{3142583423}{12852749495} a^{6} - \frac{4271601301}{12852749495} a^{5} + \frac{2157566867}{12852749495} a^{4} + \frac{5262824681}{12852749495} a^{3} - \frac{4932699301}{12852749495} a^{2} + \frac{760811217}{12852749495} a - \frac{140804333}{313481695}$, $\frac{1}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{26} - \frac{240772964008434430786578997813755799324644305669024709149711991633478892181871609111469236742542347012523162132237628136}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{25} + \frac{111560978048206849265664667985378768886825565932070291246224956866971238564557572580890451943258833333677447569711264677831450}{81720447061654600730885420122893802975013737408373957344966493019325430476615155693552347962762877603351174010117230913424735021} a^{24} - \frac{1013827173165472210956483392849918023353680822227714127357045022115249930119550254898341274510607062220313952246448790755613462609}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{23} - \frac{515866625456504582830466367862590014089703814462972327703577418773996055609959784959272421996691885019999944975106906086529280332}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{22} - \frac{930839294402927046981541371179495476952151287351763060946471800152213596179319411451128632925473268414546956020353156600772282923}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{21} - \frac{557526928986613140018669943363279588488626041753034205963463769419483047351685419331075271902673032061168585381399773539370314077}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{20} + \frac{18796306018753125322853065969485587037757284400176545597175807585589058414115480823135911091739381338324827524822173769909193936673}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{19} + \frac{26793958716319514678268402271734432585113854784505638140320902092360389840791272812583168766439480552687737174617981431864408272978}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{18} + \frac{34657077597083381145112389845612081972654488668377339818662627771079241387465099596993042364594473831476353398038056833685630575811}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{17} + \frac{4626973829384523525945819421704473413986038266503421141410030542879372619938139320426157421963323081832605572183984743205155885063}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{16} - \frac{46141507132470130816108558039113838050986048446421562611907263361993616479532341220722009155846758721792085734205115122989871979319}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{15} - \frac{538810017751717652074071779334260024850898306312359296848122099660961440546785225635081156436247033285391479366377165494144344348}{2322056605532380728084914986418811718680268392213552690411852789451564061103820887389963057966311034339124822482599366198532104865} a^{14} - \frac{37001731870070317631850424453370775741747468613202711168643170204922037247496033009965141148930602860288411248127739931705344932241}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{13} - \frac{24724082275556116409886231343041643610830149981584389067665374892525159584903316646278992117453992312932726875656702688711317708526}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{12} - \frac{6259847982274661855650949473406193594365941329730381506781919011004308788372622275431513912211004091861676592150058785215516425894}{19040864165365521970296302888634256093178200816151132061377192873502825301051331276597697075323750481580823544357314802827963259893} a^{11} + \frac{40684794409378589295486014781368672015109671475297552635997557358565987237132313065702903399661191452793174969837229625710825000457}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{10} + \frac{43823523204106350862952419619580852056040098028408212441439855724622847190343287166103793719770595354822235131036020504944208021277}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{9} - \frac{45636017848818991730073236174467035592128776475165001773095482773454080407524402348801678322094814384947226158570187543364937883357}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{8} + \frac{12884458276658799253115324649650038514327028725098096578663624264811009778292041187145920549737447260265121391200591925471044750302}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{7} + \frac{7194011367702359960282522469783464567003579332662796110800882788992771448347886259853220272300158267370737478990000501098532838019}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{6} + \frac{6845808221630241352736036629467722048845196263584145022105399123531716704028231898030164328195383448155292642722286699274584321723}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{5} - \frac{37503822395629486940211179126680901778055180200049341174021173642852527116350166651922002621504000573830894449318449183511368192529}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{4} - \frac{38907539014134593874133251310040772051211090403093769114489784492969352138769463831633192449677324165790280941055670659879724166637}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{3} - \frac{31238864462522494537962874761193501580505546412398024534126127621148552312544007413101694671526192554724159767954924445445336352384}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a^{2} - \frac{42915063801702322157803157958489892787855136216073689293404655693999128765005305906270884441346593808920284146780925138406740959247}{95204320826827609851481514443171280465891004080755660306885964367514126505256656382988485376618752407904117721786574014139816299465} a + \frac{129882155766585099055436006509455275596818775550415520037779305789750323349438411738655996246482575502143938402741973098355824945}{464411321106476145616982997283762343736053678442710538082370557890312812220764177477992611593262206867824964496519873239706420973}$
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: | \( 265538399404326520000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 27 |
| The 27 conjugacy class representatives for $C_{27}$ |
| Character table for $C_{27}$ is not computed |
Intermediate fields
| 3.3.237169.1, 9.9.3163965138861484662721.1 |
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 | $27$ | $27$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | $27$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | $27$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{27}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | $27$ | $27$ | $27$ |
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 |
|---|---|---|---|---|---|---|---|
| 487 | Data not computed | ||||||