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Properties

Label 36.0.11564874589...6837.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{88}\cdot 13^{27}$
Root discriminant $100.40$
Ramified primes $3, 13$
Class number $11069217$ (GRH)
Class group $[9, 1229913]$ (GRH)
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![510583503, -261336159, 1258558695, -2003355207, 666088029, -179083917, 4681659129, -4539381993, -421255737, 2255635343, -836274987, 371741373, 239017128, -418326174, 189154899, -33922617, -5674194, 46453752, -20060275, 7764570, 3302046, -2780889, 2148039, -310914, -168765, 147294, -58959, 32299, -234, -9936, 4254, 297, -594, 114, 18, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 + 18*x^34 + 114*x^33 - 594*x^32 + 297*x^31 + 4254*x^30 - 9936*x^29 - 234*x^28 + 32299*x^27 - 58959*x^26 + 147294*x^25 - 168765*x^24 - 310914*x^23 + 2148039*x^22 - 2780889*x^21 + 3302046*x^20 + 7764570*x^19 - 20060275*x^18 + 46453752*x^17 - 5674194*x^16 - 33922617*x^15 + 189154899*x^14 - 418326174*x^13 + 239017128*x^12 + 371741373*x^11 - 836274987*x^10 + 2255635343*x^9 - 421255737*x^8 - 4539381993*x^7 + 4681659129*x^6 - 179083917*x^5 + 666088029*x^4 - 2003355207*x^3 + 1258558695*x^2 - 261336159*x + 510583503)
 
gp: K = bnfinit(x^36 - 9*x^35 + 18*x^34 + 114*x^33 - 594*x^32 + 297*x^31 + 4254*x^30 - 9936*x^29 - 234*x^28 + 32299*x^27 - 58959*x^26 + 147294*x^25 - 168765*x^24 - 310914*x^23 + 2148039*x^22 - 2780889*x^21 + 3302046*x^20 + 7764570*x^19 - 20060275*x^18 + 46453752*x^17 - 5674194*x^16 - 33922617*x^15 + 189154899*x^14 - 418326174*x^13 + 239017128*x^12 + 371741373*x^11 - 836274987*x^10 + 2255635343*x^9 - 421255737*x^8 - 4539381993*x^7 + 4681659129*x^6 - 179083917*x^5 + 666088029*x^4 - 2003355207*x^3 + 1258558695*x^2 - 261336159*x + 510583503, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} + 18 x^{34} + 114 x^{33} - 594 x^{32} + 297 x^{31} + 4254 x^{30} - 9936 x^{29} - 234 x^{28} + 32299 x^{27} - 58959 x^{26} + 147294 x^{25} - 168765 x^{24} - 310914 x^{23} + 2148039 x^{22} - 2780889 x^{21} + 3302046 x^{20} + 7764570 x^{19} - 20060275 x^{18} + 46453752 x^{17} - 5674194 x^{16} - 33922617 x^{15} + 189154899 x^{14} - 418326174 x^{13} + 239017128 x^{12} + 371741373 x^{11} - 836274987 x^{10} + 2255635343 x^{9} - 421255737 x^{8} - 4539381993 x^{7} + 4681659129 x^{6} - 179083917 x^{5} + 666088029 x^{4} - 2003355207 x^{3} + 1258558695 x^{2} - 261336159 x + 510583503 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1156487458975635277051336992772878401742742331832969956377584210296826837=3^{88}\cdot 13^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.40$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(351=3^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{351}(1,·)$, $\chi_{351}(259,·)$, $\chi_{351}(265,·)$, $\chi_{351}(268,·)$, $\chi_{351}(142,·)$, $\chi_{351}(274,·)$, $\chi_{351}(148,·)$, $\chi_{351}(151,·)$, $\chi_{351}(25,·)$, $\chi_{351}(157,·)$, $\chi_{351}(31,·)$, $\chi_{351}(34,·)$, $\chi_{351}(40,·)$, $\chi_{351}(298,·)$, $\chi_{351}(304,·)$, $\chi_{351}(307,·)$, $\chi_{351}(181,·)$, $\chi_{351}(313,·)$, $\chi_{351}(187,·)$, $\chi_{351}(190,·)$, $\chi_{351}(64,·)$, $\chi_{351}(196,·)$, $\chi_{351}(70,·)$, $\chi_{351}(73,·)$, $\chi_{351}(79,·)$, $\chi_{351}(337,·)$, $\chi_{351}(343,·)$, $\chi_{351}(346,·)$, $\chi_{351}(220,·)$, $\chi_{351}(226,·)$, $\chi_{351}(229,·)$, $\chi_{351}(103,·)$, $\chi_{351}(235,·)$, $\chi_{351}(109,·)$, $\chi_{351}(112,·)$, $\chi_{351}(118,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{27} a^{27} - \frac{1}{3} a^{25} + \frac{1}{3} a^{23} + \frac{2}{9} a^{21} + \frac{1}{3} a^{19} - \frac{1}{9} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{9} a^{15} + \frac{1}{3} a^{14} + \frac{2}{9} a^{12} + \frac{1}{3} a^{10} - \frac{10}{27} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9}$, $\frac{1}{1431} a^{28} + \frac{22}{1431} a^{27} + \frac{8}{159} a^{26} + \frac{11}{159} a^{25} + \frac{40}{159} a^{24} - \frac{74}{159} a^{23} + \frac{137}{477} a^{22} - \frac{82}{477} a^{21} - \frac{14}{159} a^{20} - \frac{178}{477} a^{19} + \frac{233}{477} a^{18} - \frac{18}{53} a^{17} - \frac{173}{477} a^{16} - \frac{110}{477} a^{15} - \frac{77}{159} a^{14} + \frac{11}{477} a^{13} - \frac{190}{477} a^{12} + \frac{4}{159} a^{11} - \frac{487}{1431} a^{10} - \frac{418}{1431} a^{9} + \frac{5}{159} a^{8} - \frac{181}{477} a^{7} - \frac{199}{477} a^{6} - \frac{59}{159} a^{5} + \frac{25}{53} a^{4} + \frac{14}{53} a^{3} - \frac{10}{53} a^{2} + \frac{199}{477} a - \frac{4}{9}$, $\frac{1}{1431} a^{29} + \frac{4}{477} a^{27} - \frac{2}{53} a^{26} + \frac{10}{159} a^{25} + \frac{92}{477} a^{23} - \frac{26}{53} a^{22} + \frac{25}{53} a^{21} - \frac{208}{477} a^{20} + \frac{58}{159} a^{19} + \frac{4}{159} a^{18} - \frac{107}{477} a^{17} + \frac{13}{159} a^{16} + \frac{76}{159} a^{15} + \frac{164}{477} a^{14} + \frac{5}{53} a^{13} - \frac{23}{53} a^{12} + \frac{152}{1431} a^{11} - \frac{22}{159} a^{10} + \frac{236}{477} a^{9} + \frac{125}{477} a^{8} - \frac{11}{159} a^{7} - \frac{13}{159} a^{6} + \frac{16}{53} a^{5} - \frac{6}{53} a^{4} - \frac{206}{477} a^{2} + \frac{20}{53} a - \frac{1}{3}$, $\frac{1}{1431} a^{30} + \frac{73}{159} a^{26} + \frac{9}{53} a^{25} + \frac{83}{477} a^{24} + \frac{5}{53} a^{23} + \frac{4}{159} a^{22} - \frac{19}{477} a^{21} + \frac{67}{159} a^{20} - \frac{79}{159} a^{19} + \frac{118}{477} a^{18} + \frac{25}{159} a^{17} - \frac{9}{53} a^{16} - \frac{2}{9} a^{15} - \frac{5}{53} a^{14} + \frac{46}{159} a^{13} + \frac{314}{1431} a^{12} - \frac{70}{159} a^{11} - \frac{67}{159} a^{10} - \frac{217}{477} a^{9} - \frac{71}{159} a^{8} + \frac{25}{53} a^{7} - \frac{19}{53} a^{6} + \frac{18}{53} a^{5} + \frac{18}{53} a^{4} + \frac{190}{477} a^{3} - \frac{19}{53} a^{2} - \frac{18}{53} a$, $\frac{1}{1431} a^{31} + \frac{7}{477} a^{27} + \frac{9}{53} a^{26} + \frac{83}{477} a^{25} + \frac{5}{53} a^{24} + \frac{4}{159} a^{23} - \frac{19}{477} a^{22} - \frac{13}{53} a^{21} - \frac{79}{159} a^{20} + \frac{118}{477} a^{19} + \frac{26}{53} a^{18} - \frac{9}{53} a^{17} - \frac{2}{9} a^{16} - \frac{68}{159} a^{15} + \frac{46}{159} a^{14} + \frac{314}{1431} a^{13} - \frac{17}{159} a^{12} - \frac{67}{159} a^{11} - \frac{217}{477} a^{10} - \frac{1}{477} a^{9} + \frac{25}{53} a^{8} - \frac{19}{53} a^{7} - \frac{52}{159} a^{6} + \frac{18}{53} a^{5} + \frac{190}{477} a^{4} - \frac{19}{53} a^{3} - \frac{18}{53} a^{2} - \frac{1}{3}$, $\frac{1}{4293} a^{32} - \frac{1}{4293} a^{31} + \frac{1}{4293} a^{30} + \frac{1}{4293} a^{29} - \frac{1}{4293} a^{28} - \frac{38}{4293} a^{27} + \frac{152}{1431} a^{26} - \frac{335}{1431} a^{25} - \frac{682}{1431} a^{24} - \frac{575}{1431} a^{23} + \frac{5}{1431} a^{22} - \frac{548}{1431} a^{21} - \frac{1}{9} a^{20} - \frac{214}{477} a^{19} + \frac{67}{477} a^{18} - \frac{209}{477} a^{17} - \frac{103}{477} a^{16} + \frac{79}{159} a^{15} + \frac{1670}{4293} a^{14} + \frac{787}{4293} a^{13} + \frac{176}{4293} a^{12} - \frac{1318}{4293} a^{11} - \frac{410}{4293} a^{10} - \frac{775}{4293} a^{9} - \frac{496}{1431} a^{8} - \frac{581}{1431} a^{7} + \frac{458}{1431} a^{6} - \frac{383}{1431} a^{5} - \frac{433}{1431} a^{4} + \frac{289}{1431} a^{3} - \frac{143}{1431} a^{2} - \frac{226}{1431} a - \frac{7}{27}$, $\frac{1}{459351} a^{33} - \frac{19}{459351} a^{32} - \frac{74}{459351} a^{31} - \frac{1}{4293} a^{30} - \frac{49}{459351} a^{29} - \frac{1}{8667} a^{28} - \frac{1004}{153117} a^{27} - \frac{53462}{153117} a^{26} + \frac{66101}{153117} a^{25} - \frac{26810}{153117} a^{24} + \frac{32174}{153117} a^{23} - \frac{44675}{153117} a^{22} - \frac{13738}{51039} a^{21} - \frac{2510}{5671} a^{20} + \frac{6476}{51039} a^{19} + \frac{5990}{51039} a^{18} - \frac{21053}{51039} a^{17} - \frac{1969}{51039} a^{16} + \frac{151754}{459351} a^{15} - \frac{194909}{459351} a^{14} + \frac{47492}{459351} a^{13} - \frac{90787}{459351} a^{12} + \frac{96001}{459351} a^{11} + \frac{133412}{459351} a^{10} - \frac{2159}{51039} a^{9} - \frac{22934}{153117} a^{8} - \frac{30829}{153117} a^{7} - \frac{70049}{153117} a^{6} + \frac{72305}{153117} a^{5} + \frac{48733}{153117} a^{4} - \frac{50768}{153117} a^{3} + \frac{21920}{153117} a^{2} + \frac{33499}{153117} a + \frac{454}{963}$, $\frac{1}{93539022183} a^{34} - \frac{19544}{93539022183} a^{33} - \frac{6135448}{93539022183} a^{32} + \frac{3771296}{31179674061} a^{31} + \frac{21700300}{93539022183} a^{30} - \frac{14420833}{93539022183} a^{29} - \frac{31780165}{93539022183} a^{28} - \frac{399472099}{31179674061} a^{27} + \frac{147553181}{588295737} a^{26} + \frac{4842433982}{31179674061} a^{25} - \frac{3494987621}{31179674061} a^{24} + \frac{11758652351}{31179674061} a^{23} + \frac{12746262491}{31179674061} a^{22} - \frac{4072874975}{10393224687} a^{21} + \frac{2797661774}{10393224687} a^{20} + \frac{308873201}{3464408229} a^{19} + \frac{3417499450}{10393224687} a^{18} + \frac{2323277005}{10393224687} a^{17} - \frac{31959542200}{93539022183} a^{16} + \frac{28847482274}{93539022183} a^{15} + \frac{1915894321}{93539022183} a^{14} - \frac{945227366}{31179674061} a^{13} - \frac{27158056108}{93539022183} a^{12} + \frac{46761304069}{93539022183} a^{11} + \frac{28887580891}{93539022183} a^{10} + \frac{11446416467}{31179674061} a^{9} + \frac{13660328665}{31179674061} a^{8} - \frac{4506624766}{31179674061} a^{7} - \frac{8979623549}{31179674061} a^{6} - \frac{13816046155}{31179674061} a^{5} + \frac{854289557}{10393224687} a^{4} - \frac{12810429941}{31179674061} a^{3} + \frac{4503560474}{31179674061} a^{2} - \frac{12995762059}{31179674061} a + \frac{5409737}{21788731}$, $\frac{1}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{35} + \frac{6602635527818719998693128214479193711142667914759397083883768340486541467489263760713503604185744933835328312221855601901611286455560901185297531737897}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{34} - \frac{3705525941026678272916596183115210862804623205336293352767280852434608486664574784831468594552277106282564792444901854206970347652472540542922189947616964540}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{33} + \frac{256448549780187556539359105769322814964754866595804281279798330646175877767106274327074800773614080396070048813474007837816315716584103222095364414394169563169}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{32} - \frac{2332565613853259848416373043327394497220780828763251206597169600000630016183761441449339994573015363680280348730283278588868970862848328557298462437357216322480}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{31} - \frac{905507009050399307151426723441525155331398294964744650307472770226244388459272694580740926004903767758341354779285880406105309920455997734651965680078728590967}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{30} + \frac{1500033645346765267730997448265314508679597712111195608626267038633750369482342382593150050699545891397417651039569874580775319850461032501409990362201926029}{137660532548445234619868212005738409362921668460744406317530140721485036282975125313468340707670351217669392380295429273571303944410436356911279888057684036811341} a^{29} + \frac{1049197337460435528203826727689172656426595584362230614615672931395500088079643870120761063457972043639176828204777833746628188448907396312755349918580785240146}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{28} + \frac{131348866009583022325256597477691551779211836931626522977229324955202495598657206811817795895118190696865148309390584295567938255011806301200324961783054992987544}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{27} - \frac{290589027416729910916313570279263480402485856082818129340668520292479349231860360997925720070070829147104335587754681543279034709898133351196936254469062076109065}{810667580563066381650335026256015077359427603157717059425455273137634102555297960179313561945169846059608644017295305722142123228194791879588648229673028216777897} a^{26} + \frac{314166846894620256683016476926979301479291158419639977862148570362797198445502458032473049531412455732544954984638912804675549448014638025792676584487464595099035}{810667580563066381650335026256015077359427603157717059425455273137634102555297960179313561945169846059608644017295305722142123228194791879588648229673028216777897} a^{25} - \frac{216714560895883841350487201271920247307752682618489239057768425592067856688065240507542737987763136835612151874415968315908000665227872022111131294118297790465775}{810667580563066381650335026256015077359427603157717059425455273137634102555297960179313561945169846059608644017295305722142123228194791879588648229673028216777897} a^{24} - \frac{763965830215569235613708750191373042524684347188812328639651841516966218659356058433128344160313825183394262970079723991118675329611210144017024260535430482984945}{2432002741689199144951005078768045232078282809473151178276365819412902307665893880537940685835509538178825932051885917166426369684584375638765944689019084650333691} a^{23} + \frac{589933019686035394823306621518470279619921740830818520840790055216888289517733379554426298903573172203851315897351990943618918724848223809869347386888011571536375}{2432002741689199144951005078768045232078282809473151178276365819412902307665893880537940685835509538178825932051885917166426369684584375638765944689019084650333691} a^{22} + \frac{301106318001143890080315308805681716853301505740338690798645051985205215569577155167610463707965448893091068208123904560607358117900620907929886295374638848904832}{2432002741689199144951005078768045232078282809473151178276365819412902307665893880537940685835509538178825932051885917166426369684584375638765944689019084650333691} a^{21} + \frac{5292452152271819023726380682471893956064752905694148795227353805295773416948312296788922840631356694513011210661311218460970106289572596744021691604821508604478}{810667580563066381650335026256015077359427603157717059425455273137634102555297960179313561945169846059608644017295305722142123228194791879588648229673028216777897} a^{20} + \frac{38610947487758374378733467466654855432102345334569691696550776193888049529289420253981500051005948683659549463177663228341979049346623403387758023957825630897196}{90074175618118486850037225139557230817714178128635228825050585904181566950588662242145951327241094006623182668588367302460235914243865764398738692185892024086433} a^{19} + \frac{238887959623567468693836733044230400681673615586994211907024696085162613598850612270260194687462689002312447718689192848329122425573528064776544174238072261090472}{810667580563066381650335026256015077359427603157717059425455273137634102555297960179313561945169846059608644017295305722142123228194791879588648229673028216777897} a^{18} - \frac{2855462858290200965382365869038045889086499301871611000960696305570834371946693906498513843193361570656129092452894262570237319471228249012935899484646153617005613}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{17} - \frac{2221624943316378103162606122622278874060036628778127843841136819076086966077400546603279623820166810749688572551054382170667808974495454595831942410242172570675135}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{16} - \frac{694740537362189131727489961404064954751168698935855319832570993499260823234053729444647408542809225290868875245968636581178834400543029422698446392446588909993228}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{15} - \frac{2842649032415583187244843967023737846395527116928119459861100771797998649736318457509525945356973917632987240204402905004102182777204457787603707116037273138745445}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{14} + \frac{1955031805307420589020525622207861466293552806569117803512478242348197189442885337730077662387759841727647016296251297232772322285789262405504487381554620004433585}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{13} + \frac{671918845570150137362828957317485935944322046963247296557572498076158681982183682608136560259054153886456268167415161176698408026238853339420811475929001837067686}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{12} + \frac{987696655400854852246449543102481740016591150834304843763487368749927209406689008299319021576326026418647136395490527172823827542700659136147792533180177310149469}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{11} - \frac{3639269721300298340501301883364599089253104433260530827891446311253809832190610927762531197843725640322016948356997581152929761818087470200898350826867479589372328}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{10} + \frac{3217025191739182874573271503327783576930080566752773282560421739515677334181164518753097113136199625100127974643627151406218009474908909393018976012626105039342803}{7296008225067597434853015236304135696234848428419453534829097458238706922997681641613822057506528614536477796155657751499279109053753126916297834067057253951001073} a^{9} + \frac{386223845917898883595861994561855018790234896596265043096771832728761267549306627323269300500959185874319695731871114061993239885839164466648593966804899493055013}{810667580563066381650335026256015077359427603157717059425455273137634102555297960179313561945169846059608644017295305722142123228194791879588648229673028216777897} a^{8} - \frac{35859757897584886923664366329126002716785588405133276539424423183634813728007500071324402016097188311875158032486115415366579107693015048475671511790331617570023}{810667580563066381650335026256015077359427603157717059425455273137634102555297960179313561945169846059608644017295305722142123228194791879588648229673028216777897} a^{7} - \frac{38905124729857647660767311526282891964166800666692111842005689197730265223846104305070344174600841847422129338079389861886423442953839003613498827557460639697613}{90074175618118486850037225139557230817714178128635228825050585904181566950588662242145951327241094006623182668588367302460235914243865764398738692185892024086433} a^{6} + \frac{176216337061066107756679800265679368601723871254827690604240995536689182037917235834151805020635035210091796873709601456920133427551861052857518521873594887362244}{2432002741689199144951005078768045232078282809473151178276365819412902307665893880537940685835509538178825932051885917166426369684584375638765944689019084650333691} a^{5} + \frac{1059291662116537032887159449212479208924073966554583966962490835814751827912303491960393342551749528789100677618551329589816781963460651646319736032488196328861841}{2432002741689199144951005078768045232078282809473151178276365819412902307665893880537940685835509538178825932051885917166426369684584375638765944689019084650333691} a^{4} - \frac{37374178317222693658698939079204344664004796309847098927578046825836788113590122553202000482047583142512855108726499077043326004390974491020104330462283782758697}{2432002741689199144951005078768045232078282809473151178276365819412902307665893880537940685835509538178825932051885917166426369684584375638765944689019084650333691} a^{3} - \frac{393413355550915723452461295000531726769500001041055229359687292039420821965354720606272985224525182911844933957768251972032593204481306874857284013712648136313624}{2432002741689199144951005078768045232078282809473151178276365819412902307665893880537940685835509538178825932051885917166426369684584375638765944689019084650333691} a^{2} + \frac{843545970862485782760840026313089971607085710384084444256259568584110856888613590930166229814996754388803091494761887664277144710153020276661527036009525726969919}{2432002741689199144951005078768045232078282809473151178276365819412902307665893880537940685835509538178825932051885917166426369684584375638765944689019084650333691} a - \frac{1729736982591397059719119083715839237189686736494274062058381091956214388817789418719956445496127351176929761856719921622494205584393687701003637543247402496232}{45886844182815078206622737335246136454307222820248135439176713573828345427658375104489446902556783739223130793431809757857101314803478785637093296019228012270447}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{9}\times C_{1229913}$, which has order $11069217$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
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:  \( 1230094119602891.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 4.0.2197.1, 6.6.14414517.1, \(\Q(\zeta_{27})^+\), 12.0.456488925854205933.1, 18.18.10443002414754749649962321483613.1

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 $36$ R $36$ $36$ $36$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ $18^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ $36$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ $36$ $18^{2}$ $36$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{36}$ $36$

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
$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$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
13Data not computed