Global number field 36.0.3571176196381911298524292556211763388204839472594427825914130981966768013.1

• Label: 36.0.35711761963...8013.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $13^{33}\cdot 31^{24}$
• Root discriminant: $103.60$
• Ramified primes: $13, 31$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_3\times C_{12}$ (as 36T3)

Normalized defining polynomial

\( x^{36} - x^{35} + 11 x^{34} - 13 x^{33} + 115 x^{32} - 157 x^{31} + 1203 x^{30} - 1853 x^{29} + 12627 x^{28} - 21533 x^{27} + 132979 x^{26} - 247293 x^{25} + 1404819 x^{24} + 345266 x^{23} + 11724580 x^{22} + 2966632 x^{21} + 117041296 x^{20} + 6421664 x^{19} + 1187724352 x^{18} - 187177344 x^{17} + 12115794176 x^{16} - 4485772800 x^{15} + 124146295808 x^{14} - 72077670400 x^{13} + 1277654446080 x^{12} - 1005260783616 x^{11} + 790941073408 x^{10} - 622313340928 x^{9} + 489637806080 x^{8} - 385242628096 x^{7} + 303113961472 x^{6} - 238437793792 x^{5} + 187636383744 x^{4} - 147102629888 x^{3} + 115964116992 x^{2} - 85899345920 x + 68719476736 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(3571176196381911298524292556211763388204839472594427825914130981966768013=13^{33}\cdot 31^{24}\)
Root discriminant:		$103.60$
Ramified primes:		$13, 31$
This field is Galois and abelian over $\Q$.
Conductor:		\(403=13\cdot 31\)
Dirichlet character group:		$\lbrace$$\chi_{403}(1,·)$, $\chi_{403}(5,·)$, $\chi_{403}(129,·)$, $\chi_{403}(397,·)$, $\chi_{403}(149,·)$, $\chi_{403}(280,·)$, $\chi_{403}(25,·)$, $\chi_{403}(284,·)$, $\chi_{403}(218,·)$, $\chi_{403}(32,·)$, $\chi_{403}(36,·)$, $\chi_{403}(304,·)$, $\chi_{403}(180,·)$, $\chi_{403}(222,·)$, $\chi_{403}(311,·)$, $\chi_{403}(56,·)$, $\chi_{403}(187,·)$, $\chi_{403}(191,·)$, $\chi_{403}(160,·)$, $\chi_{403}(67,·)$, $\chi_{403}(335,·)$, $\chi_{403}(211,·)$, $\chi_{403}(342,·)$, $\chi_{403}(87,·)$, $\chi_{403}(346,·)$, $\chi_{403}(94,·)$, $\chi_{403}(98,·)$, $\chi_{403}(315,·)$, $\chi_{403}(366,·)$, $\chi_{403}(125,·)$, $\chi_{403}(242,·)$, $\chi_{403}(373,·)$, $\chi_{403}(118,·)$, $\chi_{403}(249,·)$, $\chi_{403}(63,·)$, $\chi_{403}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{18} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{2}{9}$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{7}{18} a - \frac{1}{9}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{14} - \frac{1}{72} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{11}{36} a^{2} + \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{144} a^{16} - \frac{1}{144} a^{15} - \frac{1}{144} a^{14} - \frac{1}{144} a^{13} + \frac{7}{16} a^{12} + \frac{7}{16} a^{11} + \frac{7}{16} a^{10} + \frac{7}{16} a^{9} + \frac{7}{16} a^{8} + \frac{7}{16} a^{7} + \frac{7}{16} a^{6} + \frac{7}{16} a^{5} + \frac{7}{16} a^{4} - \frac{25}{72} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{288} a^{17} - \frac{1}{288} a^{16} - \frac{1}{288} a^{15} - \frac{1}{288} a^{14} - \frac{1}{288} a^{13} - \frac{9}{32} a^{12} + \frac{7}{32} a^{11} - \frac{9}{32} a^{10} + \frac{7}{32} a^{9} - \frac{9}{32} a^{8} + \frac{7}{32} a^{7} - \frac{9}{32} a^{6} + \frac{7}{32} a^{5} + \frac{47}{144} a^{4} - \frac{7}{18} a^{3} + \frac{1}{9} a^{2} - \frac{7}{18} a + \frac{1}{9}$, $\frac{1}{576} a^{18} - \frac{1}{576} a^{17} - \frac{1}{576} a^{16} - \frac{1}{576} a^{15} - \frac{1}{576} a^{14} + \frac{5}{192} a^{13} - \frac{25}{64} a^{12} - \frac{9}{64} a^{11} + \frac{7}{64} a^{10} + \frac{23}{64} a^{9} - \frac{25}{64} a^{8} - \frac{9}{64} a^{7} + \frac{7}{64} a^{6} + \frac{47}{288} a^{5} - \frac{7}{36} a^{4} + \frac{1}{18} a^{3} + \frac{11}{36} a^{2} - \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{1152} a^{19} - \frac{1}{1152} a^{18} - \frac{1}{1152} a^{17} - \frac{1}{1152} a^{16} - \frac{1}{1152} a^{15} + \frac{5}{384} a^{14} + \frac{31}{1152} a^{13} + \frac{55}{128} a^{12} + \frac{7}{128} a^{11} - \frac{41}{128} a^{10} + \frac{39}{128} a^{9} - \frac{9}{128} a^{8} - \frac{57}{128} a^{7} + \frac{47}{576} a^{6} - \frac{7}{72} a^{5} - \frac{17}{36} a^{4} + \frac{11}{72} a^{3} - \frac{2}{9} a^{2} - \frac{1}{6} a - \frac{1}{9}$, $\frac{1}{2304} a^{20} - \frac{1}{2304} a^{19} - \frac{1}{2304} a^{18} - \frac{1}{2304} a^{17} - \frac{1}{2304} a^{16} + \frac{5}{768} a^{15} + \frac{31}{2304} a^{14} - \frac{17}{2304} a^{13} - \frac{121}{256} a^{12} - \frac{41}{256} a^{11} + \frac{39}{256} a^{10} + \frac{119}{256} a^{9} - \frac{57}{256} a^{8} + \frac{47}{1152} a^{7} + \frac{65}{144} a^{6} - \frac{17}{72} a^{5} + \frac{11}{144} a^{4} + \frac{7}{18} a^{3} - \frac{1}{12} a^{2} + \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{4608} a^{21} - \frac{1}{4608} a^{20} - \frac{1}{4608} a^{19} - \frac{1}{4608} a^{18} - \frac{1}{4608} a^{17} + \frac{5}{1536} a^{16} + \frac{31}{4608} a^{15} - \frac{17}{4608} a^{14} - \frac{65}{4608} a^{13} - \frac{41}{512} a^{12} - \frac{217}{512} a^{11} - \frac{137}{512} a^{10} + \frac{199}{512} a^{9} + \frac{47}{2304} a^{8} - \frac{79}{288} a^{7} - \frac{17}{144} a^{6} - \frac{133}{288} a^{5} - \frac{11}{36} a^{4} - \frac{1}{24} a^{3} - \frac{5}{18} a^{2} - \frac{4}{9} a - \frac{1}{9}$, $\frac{1}{9216} a^{22} - \frac{1}{9216} a^{21} - \frac{1}{9216} a^{20} - \frac{1}{9216} a^{19} - \frac{1}{9216} a^{18} + \frac{5}{3072} a^{17} + \frac{31}{9216} a^{16} - \frac{17}{9216} a^{15} - \frac{65}{9216} a^{14} + \frac{143}{9216} a^{13} - \frac{217}{1024} a^{12} - \frac{137}{1024} a^{11} - \frac{313}{1024} a^{10} - \frac{2257}{4608} a^{9} + \frac{209}{576} a^{8} + \frac{127}{288} a^{7} + \frac{155}{576} a^{6} - \frac{11}{72} a^{5} - \frac{1}{48} a^{4} - \frac{5}{36} a^{3} + \frac{5}{18} a^{2} + \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{18432} a^{23} - \frac{1}{18432} a^{22} - \frac{1}{18432} a^{21} - \frac{1}{18432} a^{20} - \frac{1}{18432} a^{19} + \frac{5}{6144} a^{18} + \frac{31}{18432} a^{17} - \frac{17}{18432} a^{16} - \frac{65}{18432} a^{15} + \frac{143}{18432} a^{14} + \frac{95}{18432} a^{13} - \frac{137}{2048} a^{12} - \frac{313}{2048} a^{11} + \frac{2351}{9216} a^{10} - \frac{367}{1152} a^{9} + \frac{127}{576} a^{8} - \frac{421}{1152} a^{7} - \frac{11}{144} a^{6} - \frac{1}{96} a^{5} - \frac{5}{72} a^{4} + \frac{5}{36} a^{3} + \frac{2}{9} a^{2} - \frac{7}{18} a + \frac{4}{9}$, $\frac{1}{36864} a^{24} - \frac{1}{36864} a^{23} - \frac{1}{36864} a^{22} - \frac{1}{36864} a^{21} - \frac{1}{36864} a^{20} + \frac{5}{12288} a^{19} + \frac{31}{36864} a^{18} - \frac{17}{36864} a^{17} - \frac{65}{36864} a^{16} + \frac{143}{36864} a^{15} + \frac{95}{36864} a^{14} + \frac{815}{36864} a^{13} - \frac{313}{4096} a^{12} + \frac{2351}{18432} a^{11} - \frac{367}{2304} a^{10} + \frac{127}{1152} a^{9} + \frac{731}{2304} a^{8} + \frac{133}{288} a^{7} + \frac{95}{192} a^{6} + \frac{67}{144} a^{5} - \frac{31}{72} a^{4} - \frac{7}{18} a^{3} - \frac{7}{36} a^{2} - \frac{5}{18} a + \frac{2}{9}$, $\frac{1}{76359470929625088} a^{25} + \frac{218204997739}{76359470929625088} a^{24} - \frac{1609495714609}{76359470929625088} a^{23} + \frac{3791545692007}{76359470929625088} a^{22} - \frac{1569797116249}{76359470929625088} a^{21} - \frac{6532843160425}{76359470929625088} a^{20} + \frac{21167237533991}{76359470929625088} a^{19} + \frac{33514479851255}{76359470929625088} a^{18} + \frac{125895150205255}{76359470929625088} a^{17} - \frac{151686555098729}{76359470929625088} a^{16} - \frac{147447506250163}{25453156976541696} a^{15} - \frac{22453943531587}{25453156976541696} a^{14} - \frac{1327362968276345}{76359470929625088} a^{13} - \frac{16768513579941709}{38179735464812544} a^{12} - \frac{5373107096668805}{19089867732406272} a^{11} - \frac{3784582371954365}{9544933866203136} a^{10} + \frac{325811462972869}{4772466933101568} a^{9} + \frac{454532198644795}{2386233466550784} a^{8} + \frac{440546961875735}{1193116733275392} a^{7} + \frac{48751745371597}{596558366637696} a^{6} - \frac{18388761840055}{298279183318848} a^{5} - \frac{24937660412941}{149139591659424} a^{4} - \frac{804548957633}{74569795829712} a^{3} - \frac{194128173383}{12428299304952} a^{2} - \frac{2671797196507}{6214149652476} a - \frac{421021742555}{4660612239357}$, $\frac{1}{2748940953466503168} a^{26} - \frac{1}{305437883718500352} a^{25} - \frac{139321845165}{33937542635388928} a^{24} - \frac{1497579111629}{305437883718500352} a^{23} - \frac{1226820772581}{33937542635388928} a^{22} - \frac{13965577014941}{305437883718500352} a^{21} + \frac{24139600509107}{305437883718500352} a^{20} + \frac{13010585554627}{305437883718500352} a^{19} + \frac{38886934472305}{101812627906166784} a^{18} + \frac{206561856202211}{305437883718500352} a^{17} - \frac{1056965552307853}{305437883718500352} a^{16} - \frac{186322287758333}{305437883718500352} a^{15} - \frac{27383636317253}{33937542635388928} a^{14} - \frac{35199084420921371}{1374470476733251584} a^{13} + \frac{30403319314813001}{76359470929625088} a^{12} - \frac{144050828444503}{4242192829423616} a^{11} + \frac{7325788288360885}{19089867732406272} a^{10} + \frac{489934866469441}{1060548207355904} a^{9} + \frac{495905566484497}{4772466933101568} a^{8} - \frac{1105530883513511}{2386233466550784} a^{7} + \frac{119326338027709}{1193116733275392} a^{6} + \frac{19549551745883}{198852788879232} a^{5} + \frac{58298183893481}{298279183318848} a^{4} + \frac{55552852644541}{149139591659424} a^{3} - \frac{4628342881469}{74569795829712} a^{2} + \frac{438418851531}{2071383217492} a + \frac{7356604595077}{41945510154213}$, $\frac{1}{10995763813866012672} a^{27} - \frac{1}{10995763813866012672} a^{26} + \frac{1}{407250511624667136} a^{25} - \frac{1653417681997}{135750170541555712} a^{24} + \frac{14849653422347}{1221751534874001408} a^{23} - \frac{31088718882565}{1221751534874001408} a^{22} + \frac{20179726667417}{407250511624667136} a^{21} - \frac{84209713816375}{407250511624667136} a^{20} - \frac{451237016944789}{1221751534874001408} a^{19} + \frac{530355457183355}{1221751534874001408} a^{18} + \frac{1420626900622987}{1221751534874001408} a^{17} + \frac{273031535652251}{1221751534874001408} a^{16} + \frac{1207309810349995}{1221751534874001408} a^{15} + \frac{68123234520641425}{5497881906933006336} a^{14} - \frac{1982578885705307}{2748940953466503168} a^{13} + \frac{25215946451003963}{50906313953083392} a^{12} + \frac{543613713932677}{8484385658847232} a^{11} + \frac{16957643169154177}{38179735464812544} a^{10} + \frac{7579256031819917}{19089867732406272} a^{9} - \frac{1409221999548701}{3181644622067712} a^{8} - \frac{53373072261841}{1590822311033856} a^{7} - \frac{289549586275199}{2386233466550784} a^{6} + \frac{55211382052661}{1193116733275392} a^{5} - \frac{186614747377651}{596558366637696} a^{4} - \frac{58213757454721}{298279183318848} a^{3} + \frac{493643785957}{74569795829712} a^{2} - \frac{19037996607116}{41945510154213} a - \frac{1730224079224}{41945510154213}$, $\frac{1}{43983055255464050688} a^{28} - \frac{1}{43983055255464050688} a^{27} - \frac{5}{43983055255464050688} a^{26} - \frac{7}{1629002046498668544} a^{25} + \frac{5722406277049}{1629002046498668544} a^{24} - \frac{42045897981}{543000682166222848} a^{23} + \frac{57350200464377}{1629002046498668544} a^{22} - \frac{4277442395805}{543000682166222848} a^{21} - \frac{365120723872885}{4887006139496005632} a^{20} + \frac{1356555719395483}{4887006139496005632} a^{19} + \frac{3168646848224939}{4887006139496005632} a^{18} + \frac{830660506082979}{543000682166222848} a^{17} + \frac{375142349092505}{1629002046498668544} a^{16} + \frac{150113627208892705}{21991527627732025344} a^{15} + \frac{84832296944740333}{10995763813866012672} a^{14} - \frac{2949685304891423}{5497881906933006336} a^{13} + \frac{38404869481784759}{101812627906166784} a^{12} + \frac{3630628316867387}{50906313953083392} a^{11} - \frac{237880377068571}{8484385658847232} a^{10} - \frac{3068053113527309}{12726578488270848} a^{9} - \frac{994245732839555}{2121096414711808} a^{8} - \frac{1119215607734495}{9544933866203136} a^{7} - \frac{2380278926995259}{4772466933101568} a^{6} + \frac{906923574206125}{2386233466550784} a^{5} - \frac{40519368074817}{132568525919488} a^{4} - \frac{9843099104629}{99426394439616} a^{3} - \frac{32141924592907}{335564081233704} a^{2} - \frac{56943815001127}{167782040616852} a - \frac{6730987621421}{41945510154213}$, $\frac{1}{175932221021856202752} a^{29} - \frac{1}{175932221021856202752} a^{28} - \frac{5}{175932221021856202752} a^{27} + \frac{1}{58644073673952067584} a^{26} + \frac{107}{19548024557984022528} a^{25} - \frac{228325126167205}{19548024557984022528} a^{24} + \frac{77579792518841}{6516008185994674176} a^{23} - \frac{394894224515909}{19548024557984022528} a^{22} + \frac{298562330247833}{6516008185994674176} a^{21} - \frac{2982714215450405}{19548024557984022528} a^{20} - \frac{8188340990936341}{19548024557984022528} a^{19} - \frac{14473305237619717}{19548024557984022528} a^{18} - \frac{23396733124569077}{19548024557984022528} a^{17} - \frac{219790803728596415}{87966110510928101376} a^{16} - \frac{66174820278660227}{43983055255464050688} a^{15} - \frac{156592994428032215}{21991527627732025344} a^{14} + \frac{101314803408589943}{3665254604622004224} a^{13} + \frac{72153698773708873}{610875767437000704} a^{12} - \frac{46767703300682779}{305437883718500352} a^{11} + \frac{6604053652025371}{50906313953083392} a^{10} - \frac{36518037437936243}{76359470929625088} a^{9} - \frac{5719339218778973}{12726578488270848} a^{8} - \frac{5634320613377219}{19089867732406272} a^{7} + \frac{699814871841541}{9544933866203136} a^{6} + \frac{1813541906010239}{4772466933101568} a^{5} - \frac{477676333991651}{1193116733275392} a^{4} - \frac{7754670346580}{41945510154213} a^{3} - \frac{29207756257013}{83891020308426} a^{2} + \frac{12641923946321}{83891020308426} a - \frac{5440361952731}{13981836718071}$, $\frac{1}{703728884087424811008} a^{30} - \frac{1}{703728884087424811008} a^{29} - \frac{5}{703728884087424811008} a^{28} + \frac{1}{234576294695808270336} a^{27} - \frac{61}{703728884087424811008} a^{26} + \frac{347}{78192098231936090112} a^{25} - \frac{213315255809351}{26064032743978696704} a^{24} - \frac{382601179448599}{26064032743978696704} a^{23} + \frac{3232731522915275}{78192098231936090112} a^{22} - \frac{953853909367735}{26064032743978696704} a^{21} - \frac{7931093984899349}{78192098231936090112} a^{20} + \frac{1725776295729833}{26064032743978696704} a^{19} + \frac{28369407980547083}{78192098231936090112} a^{18} - \frac{170074778304873407}{351864442043712405504} a^{17} - \frac{405210546189844355}{175932221021856202752} a^{16} - \frac{7765565404087895}{87966110510928101376} a^{15} + \frac{14177724598853687}{14661018418488016896} a^{14} - \frac{445891046598344623}{21991527627732025344} a^{13} - \frac{399254188625279035}{1221751534874001408} a^{12} + \frac{43817500376158811}{203625255812333568} a^{11} - \frac{42573310375773841}{101812627906166784} a^{10} + \frac{29544234795772585}{152718941859250176} a^{9} - \frac{4531099064087905}{25453156976541696} a^{8} - \frac{7010429648332891}{38179735464812544} a^{7} - \frac{311264453573419}{6363289244135424} a^{6} - \frac{1945101731369587}{4772466933101568} a^{5} + \frac{135135803962271}{335564081233704} a^{4} - \frac{57038467448623}{671128162467408} a^{3} - \frac{134280610543027}{335564081233704} a^{2} - \frac{20520610182767}{55927346872284} a - \frac{13197498649168}{41945510154213}$, $\frac{1}{2814915536349699244032} a^{31} - \frac{1}{2814915536349699244032} a^{30} - \frac{5}{2814915536349699244032} a^{29} + \frac{1}{938305178783233081344} a^{28} - \frac{61}{2814915536349699244032} a^{27} + \frac{17}{938305178783233081344} a^{26} - \frac{109}{34752043658638262272} a^{25} - \frac{3962227034833733}{312768392927744360448} a^{24} - \frac{2540030231951413}{312768392927744360448} a^{23} - \frac{1048232493668279}{104256130975914786816} a^{22} + \frac{4640554717866233}{104256130975914786816} a^{21} - \frac{65688880819258373}{312768392927744360448} a^{20} - \frac{30584132858634919}{104256130975914786816} a^{19} - \frac{810055257731063231}{1407457768174849622016} a^{18} - \frac{398298124090973315}{703728884087424811008} a^{17} - \frac{248766164279061719}{351864442043712405504} a^{16} + \frac{254028527976345719}{58644073673952067584} a^{15} + \frac{1042246742994856625}{87966110510928101376} a^{14} + \frac{378441474682034287}{14661018418488016896} a^{13} - \frac{71935099331196183}{271500341083111424} a^{12} + \frac{187812182824527661}{1221751534874001408} a^{11} - \frac{115505053418734775}{610875767437000704} a^{10} - \frac{15839500045893185}{101812627906166784} a^{9} - \frac{1525596860356649}{50906313953083392} a^{8} - \frac{2934495683523425}{76359470929625088} a^{7} - \frac{2594738737759105}{6363289244135424} a^{6} - \frac{408394177149799}{5369025299739264} a^{5} + \frac{400790505458095}{1342256324934816} a^{4} + \frac{27168923187767}{335564081233704} a^{3} - \frac{15267168733079}{55927346872284} a^{2} + \frac{27893722274413}{83891020308426} a - \frac{5796894379702}{13981836718071}$, $\frac{1}{11259662145398796976128} a^{32} - \frac{1}{11259662145398796976128} a^{31} - \frac{5}{11259662145398796976128} a^{30} + \frac{1}{3753220715132932325376} a^{29} - \frac{61}{11259662145398796976128} a^{28} + \frac{17}{3753220715132932325376} a^{27} - \frac{637}{11259662145398796976128} a^{26} + \frac{7355}{1251073571710977441792} a^{25} - \frac{16543526301685813}{1251073571710977441792} a^{24} + \frac{2354241133697243}{1251073571710977441792} a^{23} - \frac{32039333608940821}{1251073571710977441792} a^{22} - \frac{76766465467573253}{1251073571710977441792} a^{21} - \frac{224792941552257013}{1251073571710977441792} a^{20} - \frac{2364460042963070399}{5629831072699398488064} a^{19} - \frac{370401402364560515}{2814915536349699244032} a^{18} + \frac{1984417834234552105}{1407457768174849622016} a^{17} + \frac{201444557834752631}{234576294695808270336} a^{16} - \frac{598529793026314831}{351864442043712405504} a^{15} - \frac{364163074995883793}{58644073673952067584} a^{14} - \frac{345746196567703495}{87966110510928101376} a^{13} + \frac{1854359436131162669}{4887006139496005632} a^{12} - \frac{598023785490572855}{2443503069748002816} a^{11} + \frac{259420957282480445}{1221751534874001408} a^{10} - \frac{236057453528711291}{610875767437000704} a^{9} - \frac{136880549238516449}{305437883718500352} a^{8} + \frac{21577740899230397}{76359470929625088} a^{7} + \frac{9226634916474521}{21476101198957056} a^{6} - \frac{2067909962369051}{5369025299739264} a^{5} + \frac{273930312257723}{1342256324934816} a^{4} - \frac{45119150091913}{111854693744568} a^{3} - \frac{11343324225421}{167782040616852} a^{2} - \frac{5441078744177}{27963673436142} a + \frac{12227897893826}{41945510154213}$, $\frac{1}{45038648581595187904512} a^{33} - \frac{1}{45038648581595187904512} a^{32} - \frac{5}{45038648581595187904512} a^{31} + \frac{1}{15012882860531729301504} a^{30} - \frac{61}{45038648581595187904512} a^{29} + \frac{17}{15012882860531729301504} a^{28} - \frac{637}{45038648581595187904512} a^{27} + \frac{659}{45038648581595187904512} a^{26} - \frac{2509}{556032698538212196352} a^{25} + \frac{37889239743224027}{5004294286843909767168} a^{24} - \frac{23751982865900821}{5004294286843909767168} a^{23} - \frac{140356301868262405}{5004294286843909767168} a^{22} + \frac{205950391155046411}{5004294286843909767168} a^{21} + \frac{2908009227564503617}{22519324290797593952256} a^{20} + \frac{653465753577569149}{11259662145398796976128} a^{19} + \frac{3869329008460136233}{5629831072699398488064} a^{18} + \frac{869003035759452791}{938305178783233081344} a^{17} - \frac{750728557903280719}{1407457768174849622016} a^{16} + \frac{329401258871887087}{234576294695808270336} a^{15} + \frac{236085824212325945}{351864442043712405504} a^{14} + \frac{1601737971529709461}{175932221021856202752} a^{13} - \frac{448421381252474943}{1086001364332445696} a^{12} - \frac{2001865848203724739}{4887006139496005632} a^{11} + \frac{427992056879046277}{2443503069748002816} a^{10} + \frac{229101558481303583}{1221751534874001408} a^{9} + \frac{87657524813620285}{305437883718500352} a^{8} + \frac{1001572539712793}{85904404795828224} a^{7} + \frac{7696805865253657}{21476101198957056} a^{6} - \frac{462264442825381}{5369025299739264} a^{5} - \frac{26479912982107}{111854693744568} a^{4} + \frac{18515086319447}{671128162467408} a^{3} + \frac{54330063826375}{111854693744568} a^{2} - \frac{17402254492525}{167782040616852} a + \frac{7618295360696}{41945510154213}$, $\frac{1}{180154594326380751618048} a^{34} - \frac{1}{180154594326380751618048} a^{33} - \frac{5}{180154594326380751618048} a^{32} + \frac{1}{60051531442126917206016} a^{31} - \frac{61}{180154594326380751618048} a^{30} + \frac{17}{60051531442126917206016} a^{29} - \frac{637}{180154594326380751618048} a^{28} + \frac{659}{180154594326380751618048} a^{27} - \frac{2207}{60051531442126917206016} a^{26} + \frac{88283}{20017177147375639068672} a^{25} - \frac{167147233767974165}{20017177147375639068672} a^{24} + \frac{31616412539346601}{6672392382458546356224} a^{23} + \frac{135227051122605401}{6672392382458546356224} a^{22} + \frac{7431241042015998529}{90077297163190375809024} a^{21} + \frac{7119491706892581757}{45038648581595187904512} a^{20} - \frac{5806052130137727191}{22519324290797593952256} a^{19} + \frac{2861657272110764663}{3753220715132932325376} a^{18} + \frac{7859900031366139313}{5629831072699398488064} a^{17} - \frac{2607199054324694801}{938305178783233081344} a^{16} - \frac{6950528639733360583}{1407457768174849622016} a^{15} - \frac{8218828556202391403}{703728884087424811008} a^{14} - \frac{513493308740755877}{117288147347904135168} a^{13} - \frac{5585346710153792707}{19548024557984022528} a^{12} + \frac{55332581908321157}{9774012278992011264} a^{11} + \frac{633531275189145013}{1629002046498668544} a^{10} - \frac{154898458015250881}{407250511624667136} a^{9} + \frac{110051085957564257}{343617619183312896} a^{8} - \frac{1076294849062115}{85904404795828224} a^{7} - \frac{312047748259963}{21476101198957056} a^{6} - \frac{630688044008803}{1789675099913088} a^{5} - \frac{95799962208995}{335564081233704} a^{4} + \frac{96180497589851}{223709387489136} a^{3} + \frac{11688175307863}{335564081233704} a^{2} - \frac{3254285799733}{167782040616852} a - \frac{6579678081902}{13981836718071}$, $\frac{1}{720618377305523006472192} a^{35} - \frac{1}{720618377305523006472192} a^{34} - \frac{5}{720618377305523006472192} a^{33} + \frac{1}{240206125768507668824064} a^{32} - \frac{61}{720618377305523006472192} a^{31} + \frac{17}{240206125768507668824064} a^{30} - \frac{637}{720618377305523006472192} a^{29} + \frac{659}{720618377305523006472192} a^{28} - \frac{2207}{240206125768507668824064} a^{27} + \frac{2705}{240206125768507668824064} a^{26} - \frac{124181}{80068708589502556274688} a^{25} - \frac{305069906388394327}{26689569529834185424896} a^{24} - \frac{28110513158843277}{8896523176611395141632} a^{23} - \frac{9492046066514079167}{360309188652761503236096} a^{22} - \frac{17420130826382017667}{180154594326380751618048} a^{21} + \frac{16871533092949794601}{90077297163190375809024} a^{20} + \frac{5571319517664023159}{15012882860531729301504} a^{19} + \frac{16401722629496434097}{22519324290797593952256} a^{18} + \frac{3786506348237902063}{3753220715132932325376} a^{17} + \frac{12942456488408256569}{5629831072699398488064} a^{16} - \frac{766757118891473771}{2814915536349699244032} a^{15} + \frac{1667663236404013147}{469152589391616540672} a^{14} + \frac{2841356278857850807}{234576294695808270336} a^{13} + \frac{8366732048886651269}{39096049115968045056} a^{12} - \frac{2415496597447275595}{6516008185994674176} a^{11} - \frac{128602332134104811}{543000682166222848} a^{10} - \frac{189523289037117295}{1374470476733251584} a^{9} - \frac{67380341908357595}{343617619183312896} a^{8} - \frac{16991135796237919}{85904404795828224} a^{7} - \frac{446743580270413}{7158700399652352} a^{6} - \frac{2599869026772797}{5369025299739264} a^{5} - \frac{4611077321473}{13981836718071} a^{4} - \frac{152400424321129}{671128162467408} a^{3} - \frac{125923269163493}{335564081233704} a^{2} - \frac{22090541806097}{55927346872284} a - \frac{2436280496431}{13981836718071}$

Class group and class number

Not computed

Unit group

Rank:		$17$
Torsion generator:		\( -\frac{407213}{2443503069748002816} a^{29} - \frac{1306077081251}{2443503069748002816} a^{16} + \frac{68043143861319}{33142131479872} a^{3} \) (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, 3.3.961.1, 3.3.162409.2, 3.3.162409.1, 4.0.2197.1, \(\Q(\zeta_{13})^+\), 6.6.2028975637.1, 6.6.342896882653.1, 6.6.342896882653.2, 9.9.4283810754983929.1, \(\Q(\zeta_{13})\), 12.0.9044482471780404024493.1, 12.0.1528517537730888280139317.1, 12.0.1528517537730888280139317.2, 18.18.40317222982181607576029567064659077.1

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	${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$	R	${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{9}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$	${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$31$	31.12.8.1	$x^{12} - 186 x^{9} + 11532 x^{6} - 238328 x^{3} + 4537258673$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$
	31.12.8.1	$x^{12} - 186 x^{9} + 11532 x^{6} - 238328 x^{3} + 4537258673$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$
	31.12.8.1	$x^{12} - 186 x^{9} + 11532 x^{6} - 238328 x^{3} + 4537258673$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$