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Properties

Label 27.27.4934496944...7289.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 163^{26}$
Root discriminant $584.00$
Ramified primes $3, 163$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3091197936923, 13068822764814, 29863417968666, -197708702495375, 245143951250058, 77698876603932, -329718615030309, 118272316935156, 132948531325152, -86802211852084, -21143039489730, 23285753737722, 1147496197436, -3313822863408, 47349669510, 279857702460, -8421714480, -14492079306, 385246262, 457201308, -8185860, -8577060, 83130, 90954, -326, -489, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 489*x^25 - 326*x^24 + 90954*x^23 + 83130*x^22 - 8577060*x^21 - 8185860*x^20 + 457201308*x^19 + 385246262*x^18 - 14492079306*x^17 - 8421714480*x^16 + 279857702460*x^15 + 47349669510*x^14 - 3313822863408*x^13 + 1147496197436*x^12 + 23285753737722*x^11 - 21143039489730*x^10 - 86802211852084*x^9 + 132948531325152*x^8 + 118272316935156*x^7 - 329718615030309*x^6 + 77698876603932*x^5 + 245143951250058*x^4 - 197708702495375*x^3 + 29863417968666*x^2 + 13068822764814*x - 3091197936923)
 
gp: K = bnfinit(x^27 - 489*x^25 - 326*x^24 + 90954*x^23 + 83130*x^22 - 8577060*x^21 - 8185860*x^20 + 457201308*x^19 + 385246262*x^18 - 14492079306*x^17 - 8421714480*x^16 + 279857702460*x^15 + 47349669510*x^14 - 3313822863408*x^13 + 1147496197436*x^12 + 23285753737722*x^11 - 21143039489730*x^10 - 86802211852084*x^9 + 132948531325152*x^8 + 118272316935156*x^7 - 329718615030309*x^6 + 77698876603932*x^5 + 245143951250058*x^4 - 197708702495375*x^3 + 29863417968666*x^2 + 13068822764814*x - 3091197936923, 1)
 

Normalized defining polynomial

\( x^{27} - 489 x^{25} - 326 x^{24} + 90954 x^{23} + 83130 x^{22} - 8577060 x^{21} - 8185860 x^{20} + 457201308 x^{19} + 385246262 x^{18} - 14492079306 x^{17} - 8421714480 x^{16} + 279857702460 x^{15} + 47349669510 x^{14} - 3313822863408 x^{13} + 1147496197436 x^{12} + 23285753737722 x^{11} - 21143039489730 x^{10} - 86802211852084 x^{9} + 132948531325152 x^{8} + 118272316935156 x^{7} - 329718615030309 x^{6} + 77698876603932 x^{5} + 245143951250058 x^{4} - 197708702495375 x^{3} + 29863417968666 x^{2} + 13068822764814 x - 3091197936923 \)

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

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:  \(493449694474834715367307491089920591429543598652017227708295844390723767289=3^{36}\cdot 163^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $584.00$
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:  $3, 163$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1467=3^{2}\cdot 163\)
Dirichlet character group:    $\lbrace$$\chi_{1467}(1,·)$, $\chi_{1467}(1093,·)$, $\chi_{1467}(880,·)$, $\chi_{1467}(961,·)$, $\chi_{1467}(778,·)$, $\chi_{1467}(1291,·)$, $\chi_{1467}(1036,·)$, $\chi_{1467}(1039,·)$, $\chi_{1467}(403,·)$, $\chi_{1467}(919,·)$, $\chi_{1467}(25,·)$, $\chi_{1467}(673,·)$, $\chi_{1467}(379,·)$, $\chi_{1467}(1063,·)$, $\chi_{1467}(553,·)$, $\chi_{1467}(622,·)$, $\chi_{1467}(688,·)$, $\chi_{1467}(625,·)$, $\chi_{1467}(1462,·)$, $\chi_{1467}(169,·)$, $\chi_{1467}(1273,·)$, $\chi_{1467}(1018,·)$, $\chi_{1467}(955,·)$, $\chi_{1467}(1276,·)$, $\chi_{1467}(970,·)$, $\chi_{1467}(1342,·)$, $\chi_{1467}(511,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} + \frac{10}{23} a^{15} + \frac{7}{23} a^{14} - \frac{7}{23} a^{13} + \frac{1}{23} a^{12} + \frac{6}{23} a^{11} + \frac{2}{23} a^{10} - \frac{3}{23} a^{9} - \frac{4}{23} a^{8} - \frac{2}{23} a^{7} - \frac{9}{23} a^{6} - \frac{11}{23} a^{5} - \frac{6}{23} a^{4} + \frac{4}{23} a^{3} - \frac{8}{23} a^{2} - \frac{4}{23} a$, $\frac{1}{23} a^{17} - \frac{1}{23} a^{15} - \frac{8}{23} a^{14} + \frac{2}{23} a^{13} - \frac{4}{23} a^{12} + \frac{11}{23} a^{11} + \frac{3}{23} a^{9} - \frac{8}{23} a^{8} + \frac{11}{23} a^{7} + \frac{10}{23} a^{6} - \frac{11}{23} a^{5} - \frac{5}{23} a^{4} - \frac{2}{23} a^{3} + \frac{7}{23} a^{2} - \frac{6}{23} a$, $\frac{1}{23} a^{18} + \frac{2}{23} a^{15} + \frac{9}{23} a^{14} - \frac{11}{23} a^{13} - \frac{11}{23} a^{12} + \frac{6}{23} a^{11} + \frac{5}{23} a^{10} - \frac{11}{23} a^{9} + \frac{7}{23} a^{8} + \frac{8}{23} a^{7} + \frac{3}{23} a^{6} + \frac{7}{23} a^{5} - \frac{8}{23} a^{4} + \frac{11}{23} a^{3} + \frac{9}{23} a^{2} - \frac{4}{23} a$, $\frac{1}{23} a^{19} - \frac{11}{23} a^{15} - \frac{2}{23} a^{14} + \frac{3}{23} a^{13} + \frac{4}{23} a^{12} - \frac{7}{23} a^{11} + \frac{8}{23} a^{10} - \frac{10}{23} a^{9} - \frac{7}{23} a^{8} + \frac{7}{23} a^{7} + \frac{2}{23} a^{6} - \frac{9}{23} a^{5} + \frac{1}{23} a^{3} - \frac{11}{23} a^{2} + \frac{8}{23} a$, $\frac{1}{23} a^{20} - \frac{7}{23} a^{15} + \frac{11}{23} a^{14} - \frac{4}{23} a^{13} + \frac{4}{23} a^{12} + \frac{5}{23} a^{11} - \frac{11}{23} a^{10} + \frac{6}{23} a^{9} + \frac{9}{23} a^{8} + \frac{3}{23} a^{7} + \frac{7}{23} a^{6} - \frac{6}{23} a^{5} + \frac{4}{23} a^{4} + \frac{10}{23} a^{3} - \frac{11}{23} a^{2} + \frac{2}{23} a$, $\frac{1}{23} a^{21} - \frac{11}{23} a^{15} - \frac{1}{23} a^{14} + \frac{1}{23} a^{13} - \frac{11}{23} a^{12} + \frac{8}{23} a^{11} - \frac{3}{23} a^{10} + \frac{11}{23} a^{9} - \frac{2}{23} a^{8} - \frac{7}{23} a^{7} - \frac{4}{23} a^{5} - \frac{9}{23} a^{4} - \frac{6}{23} a^{3} - \frac{8}{23} a^{2} - \frac{5}{23} a$, $\frac{1}{529} a^{22} + \frac{8}{529} a^{21} + \frac{11}{529} a^{20} - \frac{1}{529} a^{19} - \frac{9}{529} a^{18} + \frac{3}{529} a^{17} - \frac{4}{529} a^{16} - \frac{83}{529} a^{15} + \frac{60}{529} a^{14} - \frac{178}{529} a^{13} + \frac{31}{529} a^{12} + \frac{259}{529} a^{11} + \frac{126}{529} a^{10} + \frac{42}{529} a^{9} + \frac{175}{529} a^{8} + \frac{101}{529} a^{7} + \frac{218}{529} a^{6} + \frac{235}{529} a^{5} + \frac{27}{529} a^{4} + \frac{68}{529} a^{3} - \frac{134}{529} a^{2} + \frac{79}{529} a + \frac{7}{23}$, $\frac{1}{127489} a^{23} - \frac{35}{127489} a^{22} - \frac{1}{529} a^{21} - \frac{2406}{127489} a^{20} - \frac{449}{127489} a^{19} - \frac{1864}{127489} a^{18} - \frac{1352}{127489} a^{17} - \frac{2602}{127489} a^{16} - \frac{62703}{127489} a^{15} + \frac{2509}{127489} a^{14} - \frac{44203}{127489} a^{13} + \frac{52792}{127489} a^{12} - \frac{33712}{127489} a^{11} + \frac{33701}{127489} a^{10} - \frac{5679}{127489} a^{9} + \frac{49777}{127489} a^{8} + \frac{13953}{127489} a^{7} + \frac{8088}{127489} a^{6} - \frac{8537}{127489} a^{5} + \frac{23747}{127489} a^{4} + \frac{59847}{127489} a^{3} - \frac{36985}{127489} a^{2} + \frac{10909}{127489} a - \frac{968}{5543}$, $\frac{1}{24605377} a^{24} - \frac{76}{24605377} a^{23} + \frac{2399}{24605377} a^{22} + \frac{277636}{24605377} a^{21} - \frac{237757}{24605377} a^{20} - \frac{45633}{24605377} a^{19} + \frac{385721}{24605377} a^{18} + \frac{95246}{24605377} a^{17} - \frac{249077}{24605377} a^{16} + \frac{2739381}{24605377} a^{15} - \frac{5019128}{24605377} a^{14} + \frac{6528465}{24605377} a^{13} - \frac{6312536}{24605377} a^{12} - \frac{2745213}{24605377} a^{11} - \frac{6556870}{24605377} a^{10} + \frac{1707890}{24605377} a^{9} - \frac{3235037}{24605377} a^{8} + \frac{7994166}{24605377} a^{7} + \frac{3342576}{24605377} a^{6} + \frac{8655488}{24605377} a^{5} + \frac{202926}{1069799} a^{4} + \frac{645421}{24605377} a^{3} - \frac{7436460}{24605377} a^{2} + \frac{8372516}{24605377} a - \frac{1764}{1069799}$, $\frac{1}{565923671} a^{25} - \frac{2}{565923671} a^{24} + \frac{56}{565923671} a^{23} + \frac{200788}{565923671} a^{22} - \frac{5135304}{565923671} a^{21} + \frac{8234701}{565923671} a^{20} - \frac{7534148}{565923671} a^{19} - \frac{2966308}{565923671} a^{18} + \frac{6223794}{565923671} a^{17} - \frac{6554539}{565923671} a^{16} - \frac{180457168}{565923671} a^{15} - \frac{126462141}{565923671} a^{14} + \frac{45290264}{565923671} a^{13} + \frac{282052421}{565923671} a^{12} + \frac{12628350}{565923671} a^{11} + \frac{101598117}{565923671} a^{10} - \frac{152747379}{565923671} a^{9} + \frac{2350561}{24605377} a^{8} - \frac{39887563}{565923671} a^{7} + \frac{129096453}{565923671} a^{6} + \frac{196034811}{565923671} a^{5} + \frac{149512680}{565923671} a^{4} - \frac{57372485}{565923671} a^{3} - \frac{197399101}{565923671} a^{2} + \frac{204607071}{565923671} a + \frac{1065678}{24605377}$, $\frac{1}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{26} + \frac{769525849901358967463310430310559060244552567105120466305052385019272107210126086332574766383258182357845782326214801123879837020441907961315943695997}{1197031784379454678178173414013588332005136076234851002300792881692904026611007961807826592018037975814794305316163310640574019531115697698373676400884906669711} a^{25} + \frac{4443789701772786297093765554889128899362075489519737318105191202514652851716248209533026287831640459580712543967795120718860361887651138847464829420625218}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{24} - \frac{902767793050329745802871049296723159144731688760069933466278986661632196298269300981602657662752375040171547503928485401227800281739932487038346256747025527}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{23} - \frac{21285292539731280005854193798137814941474535091927033216569447577261369395630877817252356018969001715125691048805100256634617246756523458384605729569725836116}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{22} + \frac{2645151478293547191466628037029915194540141428727405616775699151501306384513161619633710679089757058027838844965116132282660629580225995301290985298957362531691}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{21} + \frac{3823216256362034530020457508113424159683973690407057503682889920170385219466268265930852075263641917379411414908983387237746898079219234694438358139137294635536}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{20} + \frac{4036923017411784934466552486464456369784264991043799759328481494406899855945376766140398249811168483799856604213437396323586356042597902911254296583803238235984}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{19} - \frac{565754469431884239907605089318751568945829090036178612730606455303769817585952372007283384453563366145646061869490414353891152456619417979486392027449007372053}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{18} + \frac{2627238069624827007125033693884493891476865294480769954251542281925097531031333219734729319767119690302427477137586035749922885527047493377295026157107878468399}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{17} - \frac{3457667782725997082562368908236246214003782312402804508860907611136384554977840827596494312300752057142078802772927625931887490423487272480811973961188708269429}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{16} + \frac{1765967232847766783826413553727450280340258540672310345810089504614348425091840047554916411876389095202587617504885125556738245681795383212519473251210681767153}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{15} - \frac{108516598254054723487799710008442241269573598182234519526736354201974432517951470408443825148619934337619303913313026987078014803347990325293022887191507200995490}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{14} + \frac{96905049349016213478555320614676649043498414687693583831674936054443726862923010318602169825023007985018224446606298541647634284714268647610228436924366495316698}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{13} + \frac{1761478654384154123020904002270178300239194299804626797407399351802717387003537207827449588707111089365438204843742076566029326321950683981479455516486240886887}{10044658016749337082103802995853154264217011422318532323654479398553499005909762462126544880847014318793708735913892128418729816065449115468961719363947260315401} a^{12} - \frac{6591834321707304450031921927783830818492995168093410040330153816403529282035288011901133307811074985549969068415485527028667534853123600962111454042086108269524}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{11} + \frac{26296063923911446296636329137586542583084887135172757619789310776322747737368313512640703885791483113800873718465295785377245904105568141711179216753768778204583}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{10} + \frac{99021080267637130344776696855463551428776839764624334309423241726592422840439837452972079683468881241807767761013210642167461327407125419548907782539529050453562}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{9} - \frac{103668376348682093471783550660637756709864987339888488233802884832894562787883688610777122901349617591942753936688859060889694929402608916786367125700706048255739}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{8} - \frac{26951419051929993287153447369699530620960230981614712667320687832400009751054408453249957114409386180614662619375674245058062104827613134034125138341120325905105}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{7} - \frac{84331339307558846902236641360942171467702154222057469416783739804626796773748778958887008420023749507930907512691960971767941314242214862081016835318030992874817}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{6} - \frac{88895370979901198510222306981602714254886581786398493920154647435628044685244060638868366951241456023549517012358948123257330049133723839682124057789364822744856}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{5} - \frac{59536609739050253138543827916060167071558225043270774745936427964870998890405756799744491562682160946130036385564690479841939315418033517677521210467312651291919}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{4} - \frac{62086836117036123068101255116516332496111691743665563784222643189786199505734771232400018279708163244098480775721720212277859560651858638064194798809691699592507}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{3} - \frac{85802468678505233765978615970672735526788555695403223638363683043577606982121666867138136673417873921042843837079732897453895824716608194547127904541693887750573}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a^{2} + \frac{20858165262405512935206521937298004838513635091713211840679221917698978406107753057709785115146058980677725305082981238963133480306896750862703275081142261886546}{231027134385234752888387468904622548076991262713326243444053026166730477135924536628910532259481329332255300926019518953630785769505329655786119545370786987254223} a + \frac{4052713278724291125497525258532972062140945150996728305141718300808833779730916288245594182687402309417274215106674551840171252635213588697938702461679328398407}{10044658016749337082103802995853154264217011422318532323654479398553499005909762462126544880847014318793708735913892128418729816065449115468961719363947260315401}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  $26$
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:  \( 12819995733502660000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

3.3.26569.1, 9.9.498311414318121121.1

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 $27$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ $27$ $27$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{27}$ $27$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{9}$

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
3Data not computed
163Data not computed