Global number field 36.0.9191228435463896118643414243850705299591960444061012799480281884689056267053.1

• Label: 36.0.91912284354...7053.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $13^{33}\cdot 43^{24}$
• Root discriminant: $128.85$
• Ramified primes: $13, 43$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_3\times C_{12}$ (as 36T3)

Normalized defining polynomial

\( x^{36} - x^{35} + 15 x^{34} - 37 x^{33} + 255 x^{32} - 893 x^{31} + 4759 x^{30} - 19301 x^{29} + 93071 x^{28} - 401357 x^{27} + 1858759 x^{26} - 8222325 x^{25} + 37455807 x^{24} + 144852694 x^{23} + 445307204 x^{22} + 1282984056 x^{21} + 3792495248 x^{20} + 10606823904 x^{19} + 32224237120 x^{18} + 85931335552 x^{17} + 280353392896 x^{16} + 664891407872 x^{15} + 2572605408256 x^{14} + 4493047158784 x^{13} + 26204297293824 x^{12} + 16117519663104 x^{11} + 9913428803584 x^{10} + 6097468129280 x^{9} + 3750377816064 x^{8} + 2306745565184 x^{7} + 1418798825472 x^{6} + 872616558592 x^{5} + 536602476544 x^{4} + 329638739968 x^{3} + 201863462912 x^{2} + 120259084288 x + 68719476736 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(9191228435463896118643414243850705299591960444061012799480281884689056267053=13^{33}\cdot 43^{24}\)
Root discriminant:		$128.85$
Ramified primes:		$13, 43$
This field is Galois and abelian over $\Q$.
Conductor:		\(559=13\cdot 43\)
Dirichlet character group:		$\lbrace$$\chi_{559}(1,·)$, $\chi_{559}(259,·)$, $\chi_{559}(388,·)$, $\chi_{559}(517,·)$, $\chi_{559}(6,·)$, $\chi_{559}(135,·)$, $\chi_{559}(264,·)$, $\chi_{559}(393,·)$, $\chi_{559}(522,·)$, $\chi_{559}(36,·)$, $\chi_{559}(165,·)$, $\chi_{559}(294,·)$, $\chi_{559}(423,·)$, $\chi_{559}(552,·)$, $\chi_{559}(44,·)$, $\chi_{559}(173,·)$, $\chi_{559}(302,·)$, $\chi_{559}(431,·)$, $\chi_{559}(49,·)$, $\chi_{559}(178,·)$, $\chi_{559}(307,·)$, $\chi_{559}(436,·)$, $\chi_{559}(79,·)$, $\chi_{559}(337,·)$, $\chi_{559}(466,·)$, $\chi_{559}(87,·)$, $\chi_{559}(216,·)$, $\chi_{559}(345,·)$, $\chi_{559}(474,·)$, $\chi_{559}(92,·)$, $\chi_{559}(350,·)$, $\chi_{559}(479,·)$, $\chi_{559}(122,·)$, $\chi_{559}(251,·)$, $\chi_{559}(380,·)$, $\chi_{559}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{8} a^{12} - \frac{3}{8} a^{11} + \frac{1}{8} a^{10} - \frac{3}{8} a^{9} + \frac{1}{8} a^{8} - \frac{3}{8} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{48} a^{16} - \frac{1}{48} a^{15} - \frac{1}{48} a^{14} + \frac{1}{16} a^{13} - \frac{3}{16} a^{12} - \frac{7}{16} a^{11} + \frac{5}{16} a^{10} + \frac{1}{16} a^{9} - \frac{3}{16} a^{8} - \frac{7}{16} a^{7} + \frac{5}{16} a^{6} + \frac{1}{16} a^{5} - \frac{3}{16} a^{4} + \frac{7}{24} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{96} a^{17} - \frac{1}{96} a^{16} - \frac{1}{96} a^{15} + \frac{1}{32} a^{14} + \frac{7}{96} a^{13} - \frac{7}{32} a^{12} - \frac{11}{32} a^{11} + \frac{1}{32} a^{10} - \frac{3}{32} a^{9} + \frac{9}{32} a^{8} + \frac{5}{32} a^{7} - \frac{15}{32} a^{6} + \frac{13}{32} a^{5} - \frac{17}{48} a^{4} - \frac{5}{24} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{192} a^{18} - \frac{1}{192} a^{17} - \frac{1}{192} a^{16} + \frac{1}{64} a^{15} + \frac{7}{192} a^{14} + \frac{11}{192} a^{13} + \frac{21}{64} a^{12} - \frac{31}{64} a^{11} - \frac{3}{64} a^{10} - \frac{23}{64} a^{9} - \frac{27}{64} a^{8} - \frac{15}{64} a^{7} + \frac{13}{64} a^{6} - \frac{17}{96} a^{5} - \frac{5}{48} a^{4} + \frac{1}{12} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{384} a^{19} - \frac{1}{384} a^{18} - \frac{1}{384} a^{17} + \frac{1}{128} a^{16} + \frac{7}{384} a^{15} + \frac{11}{384} a^{14} - \frac{1}{384} a^{13} + \frac{33}{128} a^{12} + \frac{61}{128} a^{11} - \frac{23}{128} a^{10} - \frac{27}{128} a^{9} - \frac{15}{128} a^{8} - \frac{51}{128} a^{7} - \frac{17}{192} a^{6} + \frac{43}{96} a^{5} + \frac{1}{24} a^{4} + \frac{1}{8} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{768} a^{20} - \frac{1}{768} a^{19} - \frac{1}{768} a^{18} + \frac{1}{256} a^{17} + \frac{7}{768} a^{16} + \frac{11}{768} a^{15} - \frac{1}{768} a^{14} - \frac{29}{768} a^{13} + \frac{61}{256} a^{12} - \frac{23}{256} a^{11} + \frac{101}{256} a^{10} + \frac{113}{256} a^{9} - \frac{51}{256} a^{8} - \frac{17}{384} a^{7} - \frac{53}{192} a^{6} - \frac{23}{48} a^{5} - \frac{7}{16} a^{4} - \frac{1}{3} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{1536} a^{21} - \frac{1}{1536} a^{20} - \frac{1}{1536} a^{19} + \frac{1}{512} a^{18} + \frac{7}{1536} a^{17} + \frac{11}{1536} a^{16} - \frac{1}{1536} a^{15} - \frac{29}{1536} a^{14} - \frac{73}{1536} a^{13} + \frac{233}{512} a^{12} - \frac{155}{512} a^{11} - \frac{143}{512} a^{10} + \frac{205}{512} a^{9} - \frac{17}{768} a^{8} - \frac{53}{384} a^{7} - \frac{23}{96} a^{6} + \frac{9}{32} a^{5} + \frac{1}{3} a^{4} - \frac{5}{24} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3072} a^{22} - \frac{1}{3072} a^{21} - \frac{1}{3072} a^{20} + \frac{1}{1024} a^{19} + \frac{7}{3072} a^{18} + \frac{11}{3072} a^{17} - \frac{1}{3072} a^{16} - \frac{29}{3072} a^{15} - \frac{73}{3072} a^{14} + \frac{187}{3072} a^{13} + \frac{357}{1024} a^{12} + \frac{369}{1024} a^{11} - \frac{307}{1024} a^{10} + \frac{751}{1536} a^{9} + \frac{331}{768} a^{8} - \frac{23}{192} a^{7} - \frac{23}{64} a^{6} + \frac{1}{6} a^{5} - \frac{5}{48} a^{4} - \frac{5}{12} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6144} a^{23} - \frac{1}{6144} a^{22} - \frac{1}{6144} a^{21} + \frac{1}{2048} a^{20} + \frac{7}{6144} a^{19} + \frac{11}{6144} a^{18} - \frac{1}{6144} a^{17} - \frac{29}{6144} a^{16} - \frac{73}{6144} a^{15} + \frac{187}{6144} a^{14} + \frac{47}{6144} a^{13} + \frac{369}{2048} a^{12} + \frac{717}{2048} a^{11} - \frac{785}{3072} a^{10} - \frac{437}{1536} a^{9} + \frac{169}{384} a^{8} - \frac{23}{128} a^{7} + \frac{1}{12} a^{6} - \frac{5}{96} a^{5} - \frac{5}{24} a^{4} + \frac{1}{12} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12288} a^{24} - \frac{1}{12288} a^{23} - \frac{1}{12288} a^{22} + \frac{1}{4096} a^{21} + \frac{7}{12288} a^{20} + \frac{11}{12288} a^{19} - \frac{1}{12288} a^{18} - \frac{29}{12288} a^{17} - \frac{73}{12288} a^{16} + \frac{187}{12288} a^{15} + \frac{47}{12288} a^{14} - \frac{941}{12288} a^{13} - \frac{1331}{4096} a^{12} - \frac{785}{6144} a^{11} - \frac{437}{3072} a^{10} + \frac{169}{768} a^{9} + \frac{105}{256} a^{8} - \frac{11}{24} a^{7} - \frac{5}{192} a^{6} + \frac{19}{48} a^{5} + \frac{1}{24} a^{4} - \frac{7}{24} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{495122927498678059008} a^{25} + \frac{15060421794919187}{495122927498678059008} a^{24} + \frac{9805163304805667}{495122927498678059008} a^{23} + \frac{39867913862253679}{495122927498678059008} a^{22} - \frac{23079001954327837}{495122927498678059008} a^{21} - \frac{47300940753267683}{165040975832892686336} a^{20} - \frac{500146515998816101}{495122927498678059008} a^{19} + \frac{1276904267385144335}{495122927498678059008} a^{18} - \frac{662067472837416303}{165040975832892686336} a^{17} + \frac{3233912310783212615}{495122927498678059008} a^{16} + \frac{4263649187319563}{495122927498678059008} a^{15} + \frac{6633294697217492245}{165040975832892686336} a^{14} + \frac{36808996427150639395}{495122927498678059008} a^{13} - \frac{37942083602363693675}{247561463749339029504} a^{12} + \frac{41187874420394686705}{123780731874669514752} a^{11} + \frac{30000372658491894509}{61890365937334757376} a^{10} + \frac{14438569759422937643}{30945182968667378688} a^{9} - \frac{5296220932719282121}{15472591484333689344} a^{8} + \frac{679623865869391827}{2578765247388948224} a^{7} + \frac{818552017006288217}{3868147871083422336} a^{6} + \frac{287598979875515161}{1934073935541711168} a^{5} - \frac{138576306171249971}{322345655923618528} a^{4} + \frac{174296233367402065}{483518483885427792} a^{3} + \frac{84129020689144121}{241759241942713896} a^{2} + \frac{8912519643947525}{40293206990452316} a - \frac{12083181570144478}{30219905242839237}$, $\frac{1}{5941475129984136708096} a^{26} + \frac{1}{1980491709994712236032} a^{25} - \frac{45379388690759287}{1980491709994712236032} a^{24} + \frac{70244973790484141}{1980491709994712236032} a^{23} - \frac{60865103613877111}{1980491709994712236032} a^{22} + \frac{39315740837418423}{660163903331570745344} a^{21} + \frac{1046746783958540273}{1980491709994712236032} a^{20} + \frac{363811720041350127}{660163903331570745344} a^{19} + \frac{2304381045641678393}{1980491709994712236032} a^{18} + \frac{1533912308142234919}{660163903331570745344} a^{17} + \frac{18928116433564389697}{1980491709994712236032} a^{16} + \frac{27061152143276049757}{1980491709994712236032} a^{15} + \frac{36077606698319081609}{1980491709994712236032} a^{14} - \frac{119350795424399716255}{2970737564992068354048} a^{13} - \frac{178879670393665669423}{495122927498678059008} a^{12} + \frac{119125938254474675591}{247561463749339029504} a^{11} - \frac{21369469630511074475}{123780731874669514752} a^{10} - \frac{16528523990286364921}{61890365937334757376} a^{9} + \frac{1382110520801793575}{10315060989555792896} a^{8} + \frac{3734190619785812531}{15472591484333689344} a^{7} - \frac{590531273285029165}{2578765247388948224} a^{6} - \frac{1533405251039820073}{3868147871083422336} a^{5} + \frac{167434437983282371}{644691311847237056} a^{4} + \frac{21671655324216019}{967036967770855584} a^{3} - \frac{103998364995817585}{483518483885427792} a^{2} - \frac{14068436547405275}{30219905242839237} a - \frac{3113825997155987}{90659715728517711}$, $\frac{1}{23765900519936546832384} a^{27} - \frac{1}{23765900519936546832384} a^{26} + \frac{5}{7921966839978848944128} a^{25} - \frac{62201245487781373}{2640655613326282981376} a^{24} + \frac{49919611333386375}{2640655613326282981376} a^{23} - \frac{183445897098028607}{7921966839978848944128} a^{22} - \frac{1384631029970887139}{7921966839978848944128} a^{21} - \frac{2381682201402786359}{7921966839978848944128} a^{20} + \frac{1698178979235393687}{2640655613326282981376} a^{19} + \frac{13899204440645078609}{7921966839978848944128} a^{18} - \frac{6042717617982596051}{7921966839978848944128} a^{17} - \frac{5165691547528438247}{7921966839978848944128} a^{16} + \frac{139455961036396837333}{7921966839978848944128} a^{15} - \frac{156409108321168709941}{11882950259968273416192} a^{14} - \frac{407005415666790459583}{5941475129984136708096} a^{13} + \frac{75982933532490117005}{990245854997356118016} a^{12} - \frac{79245759475512154199}{165040975832892686336} a^{11} + \frac{18252850014070991911}{82520487916446343168} a^{10} + \frac{46543064517839169275}{123780731874669514752} a^{9} - \frac{18851782812938283827}{61890365937334757376} a^{8} - \frac{6212664697976256109}{30945182968667378688} a^{7} + \frac{2034217475830585119}{5157530494777896448} a^{6} + \frac{1268614207566474331}{7736295742166844672} a^{5} + \frac{384417683706086449}{3868147871083422336} a^{4} + \frac{603174083555452853}{1934073935541711168} a^{3} + \frac{109242960261038645}{483518483885427792} a^{2} + \frac{172670827504863709}{362638862914070844} a - \frac{43035028009962778}{90659715728517711}$, $\frac{1}{95063602079746187329536} a^{28} - \frac{1}{95063602079746187329536} a^{27} - \frac{1}{95063602079746187329536} a^{26} - \frac{7}{31687867359915395776512} a^{25} - \frac{44983142078755835}{31687867359915395776512} a^{24} - \frac{2546668373046457103}{31687867359915395776512} a^{23} + \frac{638968127981291823}{10562622453305131925504} a^{22} + \frac{4049847488258943353}{31687867359915395776512} a^{21} + \frac{633305637701266151}{10562622453305131925504} a^{20} + \frac{39462712850970404737}{31687867359915395776512} a^{19} - \frac{45262655973588773219}{31687867359915395776512} a^{18} - \frac{77622602749226693431}{31687867359915395776512} a^{17} - \frac{70530793452402874729}{10562622453305131925504} a^{16} + \frac{309453383798206367507}{47531801039873093664768} a^{15} + \frac{69780640840278658901}{23765900519936546832384} a^{14} - \frac{652485023312433793181}{11882950259968273416192} a^{13} + \frac{747171026931232690423}{1980491709994712236032} a^{12} - \frac{8407064261424223535}{990245854997356118016} a^{11} - \frac{19394185074211255801}{495122927498678059008} a^{10} - \frac{8071422687031585461}{82520487916446343168} a^{9} + \frac{43374852674730822055}{123780731874669514752} a^{8} - \frac{2877532891188531877}{20630121979111585792} a^{7} - \frac{5328546052568840737}{30945182968667378688} a^{6} + \frac{6438282415742460325}{15472591484333689344} a^{5} - \frac{3404911212626212007}{7736295742166844672} a^{4} + \frac{240055272440806885}{644691311847237056} a^{3} + \frac{144686574045238609}{362638862914070844} a^{2} + \frac{22814898151384781}{90659715728517711} a - \frac{184479213929521}{90659715728517711}$, $\frac{1}{380254408318984749318144} a^{29} - \frac{1}{380254408318984749318144} a^{28} - \frac{1}{380254408318984749318144} a^{27} - \frac{7}{126751469439661583106048} a^{26} + \frac{5}{126751469439661583106048} a^{25} + \frac{3278612064715555057}{126751469439661583106048} a^{24} + \frac{2164419183522774925}{126751469439661583106048} a^{23} + \frac{2475905764271824249}{126751469439661583106048} a^{22} + \frac{1597066287322584245}{126751469439661583106048} a^{21} + \frac{15750260944300755841}{126751469439661583106048} a^{20} - \frac{13198579035959170403}{126751469439661583106048} a^{19} + \frac{73641900652529692739}{42250489813220527702016} a^{18} - \frac{177235965338474503529}{42250489813220527702016} a^{17} - \frac{1247594707101744276205}{190127204159492374659072} a^{16} + \frac{1637277073622239751765}{95063602079746187329536} a^{15} + \frac{15920119105539774691}{47531801039873093664768} a^{14} - \frac{317287478250678474697}{7921966839978848944128} a^{13} + \frac{951785993537530547921}{3960983419989424472064} a^{12} - \frac{390143334306208402489}{1980491709994712236032} a^{11} - \frac{264955517106482844095}{990245854997356118016} a^{10} - \frac{171277765329135399433}{495122927498678059008} a^{9} + \frac{43560834596764206881}{247561463749339029504} a^{8} + \frac{15045116780135386895}{123780731874669514752} a^{7} + \frac{6766298341068643189}{61890365937334757376} a^{6} + \frac{1904641654391775219}{10315060989555792896} a^{5} + \frac{963327488669533573}{2578765247388948224} a^{4} + \frac{140718036781147733}{725277725828141688} a^{3} + \frac{86187507822415903}{181319431457035422} a^{2} + \frac{3456461212963085}{90659715728517711} a + \frac{12019794454350395}{30219905242839237}$, $\frac{1}{1521017633275938997272576} a^{30} - \frac{1}{1521017633275938997272576} a^{29} - \frac{1}{1521017633275938997272576} a^{28} - \frac{7}{507005877758646332424192} a^{27} + \frac{5}{507005877758646332424192} a^{26} + \frac{241}{507005877758646332424192} a^{25} + \frac{4560905598897792303}{169001959252882110808064} a^{24} - \frac{35250664369129071239}{507005877758646332424192} a^{23} + \frac{61767723689943659701}{507005877758646332424192} a^{22} - \frac{1524951966575642325}{169001959252882110808064} a^{21} - \frac{168999504151170757987}{507005877758646332424192} a^{20} - \frac{389190267964555497271}{507005877758646332424192} a^{19} + \frac{700451670372263282629}{507005877758646332424192} a^{18} + \frac{1805275356291926387987}{760508816637969498636288} a^{17} + \frac{865279628571285313109}{380254408318984749318144} a^{16} - \frac{531481475633045005085}{190127204159492374659072} a^{15} + \frac{496314693841364903479}{31687867359915395776512} a^{14} - \frac{385548537116554465125}{5281311226652565962752} a^{13} + \frac{798507718705488577607}{7921966839978848944128} a^{12} - \frac{193841638303370799477}{1320327806663141490688} a^{11} - \frac{352774341513580050025}{1980491709994712236032} a^{10} + \frac{379360674322318205185}{990245854997356118016} a^{9} + \frac{5072839506698547429}{165040975832892686336} a^{8} + \frac{72158915064956336981}{247561463749339029504} a^{7} + \frac{9166878589546918585}{123780731874669514752} a^{6} - \frac{9498856181933996257}{30945182968667378688} a^{5} + \frac{297920397648822745}{1450555451656283376} a^{4} + \frac{254696391625295555}{1450555451656283376} a^{3} - \frac{77218451079266669}{725277725828141688} a^{2} - \frac{3159229109283173}{40293206990452316} a + \frac{5535642493358608}{30219905242839237}$, $\frac{1}{6084070533103755989090304} a^{31} - \frac{1}{6084070533103755989090304} a^{30} - \frac{1}{6084070533103755989090304} a^{29} - \frac{7}{2028023511034585329696768} a^{28} + \frac{5}{2028023511034585329696768} a^{27} - \frac{301}{6084070533103755989090304} a^{26} - \frac{1139}{2028023511034585329696768} a^{25} + \frac{37785967307068778873}{2028023511034585329696768} a^{24} + \frac{55255270719586988213}{2028023511034585329696768} a^{23} - \frac{186415631752194820223}{2028023511034585329696768} a^{22} + \frac{657701683369862424221}{2028023511034585329696768} a^{21} - \frac{1068907080331002831671}{2028023511034585329696768} a^{20} + \frac{1205433248221503407045}{2028023511034585329696768} a^{19} - \frac{7302730640472028119277}{3042035266551877994545152} a^{18} + \frac{6877923228590118935381}{1521017633275938997272576} a^{17} - \frac{5242901637948627188381}{760508816637969498636288} a^{16} + \frac{769681799865454070263}{126751469439661583106048} a^{15} + \frac{1768421899568966858801}{63375734719830791553024} a^{14} - \frac{5172005992775253250315}{95063602079746187329536} a^{13} - \frac{6173983817952288509759}{15843933679957697888256} a^{12} + \frac{1502792432165513775415}{7921966839978848944128} a^{11} - \frac{47578753037372481055}{3960983419989424472064} a^{10} - \frac{293241833989316855473}{1980491709994712236032} a^{9} + \frac{457794559193053204661}{990245854997356118016} a^{8} + \frac{145542118045587465113}{495122927498678059008} a^{7} + \frac{53291730002432148079}{123780731874669514752} a^{6} + \frac{5419400517016289033}{11604443613250267008} a^{5} + \frac{31210564144571054}{90659715728517711} a^{4} - \frac{155965083693249679}{1450555451656283376} a^{3} + \frac{22114172465330573}{80586413980904632} a^{2} + \frac{12069431382004507}{40293206990452316} a - \frac{39064088455097713}{90659715728517711}$, $\frac{1}{24336282132415023956361216} a^{32} - \frac{1}{24336282132415023956361216} a^{31} - \frac{1}{24336282132415023956361216} a^{30} - \frac{7}{8112094044138341318787072} a^{29} + \frac{5}{8112094044138341318787072} a^{28} - \frac{301}{24336282132415023956361216} a^{27} + \frac{679}{24336282132415023956361216} a^{26} - \frac{557}{2704031348046113772929024} a^{25} - \frac{47508548095515770699}{8112094044138341318787072} a^{24} + \frac{611484470046392944513}{8112094044138341318787072} a^{23} - \frac{1276604143383613720931}{8112094044138341318787072} a^{22} - \frac{115055781502630271933}{2704031348046113772929024} a^{21} - \frac{1293921516623580982331}{8112094044138341318787072} a^{20} - \frac{12675295159750429124845}{12168141066207511978180608} a^{19} - \frac{5177524277625040857259}{6084070533103755989090304} a^{18} - \frac{10521089407780197643421}{3042035266551877994545152} a^{17} - \frac{20610567560790561539}{169001959252882110808064} a^{16} + \frac{44775421444434136241}{253502938879323166212096} a^{15} + \frac{9804693397448643750581}{380254408318984749318144} a^{14} - \frac{4246709665510648390909}{190127204159492374659072} a^{13} + \frac{4033939495499141657085}{10562622453305131925504} a^{12} + \frac{6525543261948502827169}{15843933679957697888256} a^{11} + \frac{2488850165252910972047}{7921966839978848944128} a^{10} + \frac{1571191007717030493685}{3960983419989424472064} a^{9} - \frac{17336007233935840653}{660163903331570745344} a^{8} - \frac{231531034049601611569}{495122927498678059008} a^{7} + \frac{16134397574352534611}{46417774453001068032} a^{6} + \frac{3664261368910434199}{11604443613250267008} a^{5} - \frac{501283381481759921}{2901110903312566752} a^{4} - \frac{22634267547213383}{120879620971356948} a^{3} - \frac{12071502965238787}{40293206990452316} a^{2} + \frac{10963267050008881}{181319431457035422} a + \frac{22050775538807200}{90659715728517711}$, $\frac{1}{97345128529660095825444864} a^{33} - \frac{1}{97345128529660095825444864} a^{32} - \frac{1}{97345128529660095825444864} a^{31} - \frac{7}{32448376176553365275148288} a^{30} + \frac{5}{32448376176553365275148288} a^{29} - \frac{301}{97345128529660095825444864} a^{28} + \frac{679}{97345128529660095825444864} a^{27} - \frac{557}{10816125392184455091716096} a^{26} - \frac{5401}{10816125392184455091716096} a^{25} + \frac{797132307686717032321}{32448376176553365275148288} a^{24} - \frac{2526642930011210893667}{32448376176553365275148288} a^{23} + \frac{3123872784320117550281}{32448376176553365275148288} a^{22} - \frac{874480817583426206057}{10816125392184455091716096} a^{21} + \frac{13793559669528167536403}{48672564264830047912722432} a^{20} + \frac{10189572425268076992341}{24336282132415023956361216} a^{19} + \frac{15049182231797633100643}{12168141066207511978180608} a^{18} + \frac{1593928078112365042941}{676007837011528443232256} a^{17} - \frac{3118427298368088376261}{338003918505764221616128} a^{16} - \frac{14183259648432528107851}{1521017633275938997272576} a^{15} + \frac{21267286662271777921283}{760508816637969498636288} a^{14} + \frac{10388842768164616533751}{126751469439661583106048} a^{13} - \frac{8492650796705811312501}{21125244906610263851008} a^{12} + \frac{13297809238599169728911}{31687867359915395776512} a^{11} + \frac{4243120280576151191797}{15843933679957697888256} a^{10} - \frac{870330099168453718439}{7921966839978848944128} a^{9} + \frac{155163840262019588293}{660163903331570745344} a^{8} + \frac{38357170093492690715}{185671097812004272128} a^{7} + \frac{19913190474942249163}{46417774453001068032} a^{6} + \frac{400860994525563457}{11604443613250267008} a^{5} + \frac{154122757001037301}{483518483885427792} a^{4} + \frac{1788657935607355}{483518483885427792} a^{3} - \frac{200065448128207799}{725277725828141688} a^{2} - \frac{141084994275779465}{362638862914070844} a + \frac{10000879244607137}{30219905242839237}$, $\frac{1}{389380514118640383301779456} a^{34} - \frac{1}{389380514118640383301779456} a^{33} - \frac{1}{389380514118640383301779456} a^{32} - \frac{7}{129793504706213461100593152} a^{31} + \frac{5}{129793504706213461100593152} a^{30} - \frac{301}{389380514118640383301779456} a^{29} + \frac{679}{389380514118640383301779456} a^{28} - \frac{557}{43264501568737820366864384} a^{27} + \frac{16927}{389380514118640383301779456} a^{26} - \frac{30847}{129793504706213461100593152} a^{25} + \frac{1702943799133024499357}{129793504706213461100593152} a^{24} + \frac{9003759947069002624201}{129793504706213461100593152} a^{23} + \frac{14837453240793340613573}{129793504706213461100593152} a^{22} + \frac{60324805797208500524819}{194690257059320191650889728} a^{21} - \frac{55143172992349772671147}{97345128529660095825444864} a^{20} + \frac{31440759867195282590563}{48672564264830047912722432} a^{19} - \frac{1727039147189686661891}{2704031348046113772929024} a^{18} + \frac{5134004361136140956731}{1352015674023056886464512} a^{17} + \frac{17805956810878536360629}{6084070533103755989090304} a^{16} + \frac{41610041855394768807683}{3042035266551877994545152} a^{15} - \frac{1563403779949258584073}{507005877758646332424192} a^{14} + \frac{3422825913265459236323}{760508816637969498636288} a^{13} - \frac{19912398167164904519537}{126751469439661583106048} a^{12} - \frac{7938907795309725326347}{63375734719830791553024} a^{11} + \frac{7097170664331003249497}{31687867359915395776512} a^{10} + \frac{2210784615686531070415}{7921966839978848944128} a^{9} - \frac{267955791467956061437}{742684391248017088512} a^{8} - \frac{47729873839140222185}{185671097812004272128} a^{7} - \frac{17631415486296266621}{46417774453001068032} a^{6} - \frac{1810915625602253467}{3868147871083422336} a^{5} + \frac{331041292226531639}{967036967770855584} a^{4} - \frac{93326031354669041}{725277725828141688} a^{3} + \frac{37748068460066585}{181319431457035422} a^{2} + \frac{15215346914657141}{60439810485678474} a - \frac{12789420089501801}{90659715728517711}$, $\frac{1}{1557522056474561533207117824} a^{35} - \frac{1}{1557522056474561533207117824} a^{34} - \frac{1}{1557522056474561533207117824} a^{33} - \frac{7}{519174018824853844402372608} a^{32} + \frac{5}{519174018824853844402372608} a^{31} - \frac{301}{1557522056474561533207117824} a^{30} + \frac{679}{1557522056474561533207117824} a^{29} - \frac{557}{173058006274951281467457536} a^{28} + \frac{16927}{1557522056474561533207117824} a^{27} - \frac{30847}{519174018824853844402372608} a^{26} + \frac{472733}{519174018824853844402372608} a^{25} - \frac{1418269985846885339959}{519174018824853844402372608} a^{24} + \frac{41658254722281398637509}{519174018824853844402372608} a^{23} + \frac{75169239452662220773139}{778761028237280766603558912} a^{22} - \frac{98669202985941445077163}{389380514118640383301779456} a^{21} + \frac{24696888213614965590883}{194690257059320191650889728} a^{20} + \frac{8238442638709063453949}{10816125392184455091716096} a^{19} - \frac{11718624335546680789957}{5408062696092227545858048} a^{18} + \frac{34044925536357430551221}{24336282132415023956361216} a^{17} + \frac{46700764672365601179395}{12168141066207511978180608} a^{16} - \frac{17409487606111610920969}{2028023511034585329696768} a^{15} - \frac{97980956653233576253981}{3042035266551877994545152} a^{14} + \frac{24331010889800812829839}{507005877758646332424192} a^{13} - \frac{78776618541124574806027}{253502938879323166212096} a^{12} - \frac{36316405562645623062695}{126751469439661583106048} a^{11} + \frac{14272474813497526197199}{31687867359915395776512} a^{10} + \frac{1196555334509913619907}{2970737564992068354048} a^{9} + \frac{343115923709299646167}{742684391248017088512} a^{8} - \frac{2718425554174543517}{185671097812004272128} a^{7} + \frac{2980143105394379681}{15472591484333689344} a^{6} - \frac{99909387765359305}{3868147871083422336} a^{5} - \frac{36431527512353047}{1450555451656283376} a^{4} - \frac{607806720302821475}{1450555451656283376} a^{3} - \frac{44557402819417747}{241759241942713896} a^{2} - \frac{144210141992848991}{362638862914070844} a + \frac{28873610860276}{10073301747613079}$

Class group and class number

Not computed

Unit group

Rank:		$17$
Torsion generator:		\( -\frac{15005268779904977}{1557522056474561533207117824} a^{35} + \frac{23579708082707821}{1557522056474561533207117824} a^{34} - \frac{233653471001377499}{1557522056474561533207117824} a^{33} + \frac{683811534398526809}{1557522056474561533207117824} a^{32} - \frac{4143597793079474363}{1557522056474561533207117824} a^{31} + \frac{15586187042669869681}{1557522056474561533207117824} a^{30} - \frac{26355682806990241745}{519174018824853844402372608} a^{29} + \frac{36713605484776077297}{173058006274951281467457536} a^{28} - \frac{1562050623597933806411}{1557522056474561533207117824} a^{27} + \frac{6820501302047485347713}{1557522056474561533207117824} a^{26} - \frac{10444196542440812067617}{519174018824853844402372608} a^{25} + \frac{46438670934633110453315}{519174018824853844402372608} a^{24} - \frac{70281808671518299015971}{173058006274951281467457536} a^{23} - \frac{926195531652165549271465}{778761028237280766603558912} a^{22} - \frac{1359980913274375747323143}{389380514118640383301779456} a^{21} - \frac{1929157626101726857644817}{194690257059320191650889728} a^{20} - \frac{2869158851757286137537127}{97345128529660095825444864} a^{19} - \frac{3957491352203025185665153}{48672564264830047912722432} a^{18} - \frac{6134152678947262030902551}{24336282132415023956361216} a^{17} - \frac{2638328180493223665613211}{4056047022069170659393536} a^{16} - \frac{4518183584457518808024653}{2028023511034585329696768} a^{15} - \frac{14791017839545308245999185}{3042035266551877994545152} a^{14} - \frac{32130434175551622364419631}{1521017633275938997272576} a^{13} - \frac{7382931532420070418237791}{253502938879323166212096} a^{12} - \frac{28863702525987749086028567}{126751469439661583106048} a^{11} - \frac{58188741020319022945021}{5281311226652565962752} a^{10} - \frac{80528027841286870517761}{11882950259968273416192} a^{9} - \frac{12382611461186148207853}{2970737564992068354048} a^{8} - \frac{1904037847970573837929}{742684391248017088512} a^{7} - \frac{292772086384377407669}{185671097812004272128} a^{6} - \frac{45013662729889230289}{46417774453001068032} a^{5} - \frac{6917428907536194397}{11604443613250267008} a^{4} - \frac{117898540413539105}{322345655923618528} a^{3} - \frac{53590245642517775}{241759241942713896} a^{2} - \frac{23579708082707821}{181319431457035422} a - \frac{4287219651401422}{90659715728517711} \) (order $26$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

$C_3\times C_{12}$ (as 36T3):

An abelian group of order 36

The 36 conjugacy class representatives for $C_3\times C_{12}$

Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.312481.2, 3.3.312481.1, 3.3.1849.1, 4.0.2197.1, \(\Q(\zeta_{13})^+\), 6.6.1269376879693.1, 6.6.1269376879693.2, 6.6.7511105797.1, 9.9.30512012057180641.1, \(\Q(\zeta_{13})\), 12.0.20947129615088780945065237.2, 12.0.20947129615088780945065237.1, 12.0.123947512515318230444173.1, 18.18.2045369386871248375939854322260425557.1

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	${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/3.3.0.1}{3} }^{12}$	${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$	R	${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$	${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$	R	${\href{/LocalNumberField/47.4.0.1}{4} }^{9}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$	${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
13	Data not computed
43	Data not computed