Global number field 46.0.35053291047098822445710921889723167962319821671472348329989416363445958210915442479511871796900053441791.1

• Label: 46.0.35053291047...1791.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-\,17^{23}\cdot 47^{45}$
• Root discriminant: $178.23$
• Ramified primes: $17, 47$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

\( x^{46} - x^{45} + 189 x^{44} - 189 x^{43} + 16733 x^{42} - 16733 x^{41} + 922141 x^{40} - 922141 x^{39} + 35453981 x^{38} - 35453981 x^{37} + 1010238493 x^{36} - 1010238493 x^{35} + 22122644509 x^{34} - 22122644509 x^{33} + 381033546781 x^{32} - 381033546781 x^{31} + 5240134992925 x^{30} - 5240134992925 x^{29} + 58093519143965 x^{28} - 58093519143965 x^{27} + 522060523799581 x^{26} - 522060523799581 x^{25} + 3812008374993949 x^{24} - 3812008374993949 x^{23} + 22611710381818909 x^{22} - 22611710381818909 x^{21} + 108699033598592029 x^{20} - 108699033598592029 x^{19} + 421743845295948829 x^{18} - 421743845295948829 x^{17} + 1313921558633415709 x^{16} - 1313921558633415709 x^{15} + 3270956542728504349 x^{14} - 3270956542728504349 x^{13} + 6494308281238062109 x^{12} - 6494308281238062109 x^{11} + 10347510359456384029 x^{10} - 10347510359456384029 x^{9} + 13534369221140710429 x^{8} - 13534369221140710429 x^{7} + 15234027280705684509 x^{6} - 15234027280705684509 x^{5} + 15756998991341061149 x^{4} - 15756998991341061149 x^{3} + 15833067603797115933 x^{2} - 15833067603797115933 x + 15836374934773466141 \)

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		\(-35053291047098822445710921889723167962319821671472348329989416363445958210915442479511871796900053441791=-\,17^{23}\cdot 47^{45}\)
Root discriminant:		$178.23$
Ramified primes:		$17, 47$
This field is Galois and abelian over $\Q$.
Conductor:		\(799=17\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{799}(256,·)$, $\chi_{799}(1,·)$, $\chi_{799}(135,·)$, $\chi_{799}(392,·)$, $\chi_{799}(781,·)$, $\chi_{799}(664,·)$, $\chi_{799}(18,·)$, $\chi_{799}(407,·)$, $\chi_{799}(152,·)$, $\chi_{799}(798,·)$, $\chi_{799}(543,·)$, $\chi_{799}(33,·)$, $\chi_{799}(290,·)$, $\chi_{799}(426,·)$, $\chi_{799}(647,·)$, $\chi_{799}(560,·)$, $\chi_{799}(305,·)$, $\chi_{799}(307,·)$, $\chi_{799}(696,·)$, $\chi_{799}(186,·)$, $\chi_{799}(577,·)$, $\chi_{799}(322,·)$, $\chi_{799}(67,·)$, $\chi_{799}(324,·)$, $\chi_{799}(545,·)$, $\chi_{799}(458,·)$, $\chi_{799}(203,·)$, $\chi_{799}(460,·)$, $\chi_{799}(205,·)$, $\chi_{799}(254,·)$, $\chi_{799}(594,·)$, $\chi_{799}(339,·)$, $\chi_{799}(596,·)$, $\chi_{799}(341,·)$, $\chi_{799}(475,·)$, $\chi_{799}(732,·)$, $\chi_{799}(477,·)$, $\chi_{799}(222,·)$, $\chi_{799}(613,·)$, $\chi_{799}(103,·)$, $\chi_{799}(492,·)$, $\chi_{799}(494,·)$, $\chi_{799}(239,·)$, $\chi_{799}(373,·)$, $\chi_{799}(509,·)$, $\chi_{799}(766,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{3840885093721692949} a^{24} - \frac{1861928776928916393}{3840885093721692949} a^{23} + \frac{96}{3840885093721692949} a^{22} + \frac{1542381740015874549}{3840885093721692949} a^{21} + \frac{4032}{3840885093721692949} a^{20} + \frac{241108101087894776}{3840885093721692949} a^{19} + \frac{97280}{3840885093721692949} a^{18} + \frac{888202592337969354}{3840885093721692949} a^{17} + \frac{1488384}{3840885093721692949} a^{16} - \frac{424548735578019863}{3840885093721692949} a^{15} + \frac{15040512}{3840885093721692949} a^{14} - \frac{1401864802913723422}{3840885093721692949} a^{13} + \frac{101384192}{3840885093721692949} a^{12} - \frac{1120278318247727563}{3840885093721692949} a^{11} + \frac{449839104}{3840885093721692949} a^{10} + \frac{262838506403126012}{3840885093721692949} a^{9} + \frac{1265172480}{3840885093721692949} a^{8} - \frac{137364603376836161}{3840885093721692949} a^{7} + \frac{2099249152}{3840885093721692949} a^{6} - \frac{1463447835743012531}{3840885093721692949} a^{5} + \frac{1799356416}{3840885093721692949} a^{4} - \frac{309677884500054950}{3840885093721692949} a^{3} + \frac{603979776}{3840885093721692949} a^{2} - \frac{56305069909100900}{3840885093721692949} a + \frac{33554432}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{25} + \frac{100}{3840885093721692949} a^{23} - \frac{234055079727720326}{3840885093721692949} a^{22} + \frac{4400}{3840885093721692949} a^{21} - \frac{1392421547430923943}{3840885093721692949} a^{20} + \frac{112000}{3840885093721692949} a^{19} + \frac{860372509728591452}{3840885093721692949} a^{18} + \frac{1824000}{3840885093721692949} a^{17} + \frac{686005037987175416}{3840885093721692949} a^{16} + \frac{19845120}{3840885093721692949} a^{15} - \frac{1095626410095508984}{3840885093721692949} a^{14} + \frac{146227200}{3840885093721692949} a^{13} + \frac{1370760303014498606}{3840885093721692949} a^{12} + \frac{724172800}{3840885093721692949} a^{11} - \frac{142588520863643565}{3840885093721692949} a^{10} + \frac{2342912000}{3840885093721692949} a^{9} - \frac{1037572799207019787}{3840885093721692949} a^{8} + \frac{4685824000}{3840885093721692949} a^{7} + \frac{1765739495307653375}{3840885093721692949} a^{6} + \frac{5248122880}{3840885093721692949} a^{5} + \frac{1909499100154559691}{3840885093721692949} a^{4} + \frac{2726297600}{3840885093721692949} a^{3} - \frac{1540731416171191893}{3840885093721692949} a^{2} + \frac{419430400}{3840885093721692949} a - \frac{1149380656381697541}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{26} + \frac{1596338114522657422}{3840885093721692949} a^{23} - \frac{5200}{3840885093721692949} a^{22} + \frac{1845693293571032066}{3840885093721692949} a^{21} - \frac{291200}{3840885093721692949} a^{20} - \frac{205127036730728454}{3840885093721692949} a^{19} - \frac{7904000}{3840885093721692949} a^{18} + \frac{206102959789177843}{3840885093721692949} a^{17} - \frac{128993280}{3840885093721692949} a^{16} - \frac{890488883232145123}{3840885093721692949} a^{15} - \frac{1357824000}{3840885093721692949} a^{14} - \frac{555507873315798307}{3840885093721692949} a^{13} - \frac{9414246400}{3840885093721692949} a^{12} + \frac{499575585980017214}{3840885093721692949} a^{11} - \frac{42640998400}{3840885093721692949} a^{10} - \frac{435227783467770344}{3840885093721692949} a^{9} - \frac{121831424000}{3840885093721692949} a^{8} + \frac{138659458104497679}{3840885093721692949} a^{7} - \frac{204676792320}{3840885093721692949} a^{6} - \frac{1540235980690212220}{3840885093721692949} a^{5} - \frac{177209344000}{3840885093721692949} a^{4} - \frac{1300023715939240485}{3840885093721692949} a^{3} - \frac{59978547200}{3840885093721692949} a^{2} + \frac{640241240806699510}{3840885093721692949} a - \frac{3355443200}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{27} - \frac{5616}{3840885093721692949} a^{23} - \frac{1608247045458055435}{3840885093721692949} a^{22} - \frac{329472}{3840885093721692949} a^{21} + \frac{683012285471928566}{3840885093721692949} a^{20} - \frac{9434880}{3840885093721692949} a^{19} - \frac{740453542557213298}{3840885093721692949} a^{18} - \frac{163897344}{3840885093721692949} a^{17} - \frac{1161528522361665669}{3840885093721692949} a^{16} - \frac{1857503232}{3840885093721692949} a^{15} - \frac{1677219976636556267}{3840885093721692949} a^{14} - \frac{14077919232}{3840885093721692949} a^{13} - \frac{1229139745938076779}{3840885093721692949} a^{12} - \frac{71171702784}{3840885093721692949} a^{11} - \frac{520552978662356775}{3840885093721692949} a^{10} - \frac{233916334080}{3840885093721692949} a^{9} + \frac{1579069575122964007}{3840885093721692949} a^{8} - \frac{473680576512}{3840885093721692949} a^{7} + \frac{622169837549513068}{3840885093721692949} a^{6} - \frac{535881056256}{3840885093721692949} a^{5} - \frac{1641324496717586209}{3840885093721692949} a^{4} - \frac{280699600896}{3840885093721692949} a^{3} + \frac{1385177503196444822}{3840885093721692949} a^{2} - \frac{43486543872}{3840885093721692949} a + \frac{468150358324062845}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{28} + \frac{529851925917381604}{3840885093721692949} a^{23} + \frac{209664}{3840885093721692949} a^{22} + \frac{1502977872205795755}{3840885093721692949} a^{21} + \frac{13208832}{3840885093721692949} a^{20} + \frac{1331089177023930670}{3840885093721692949} a^{19} + \frac{382427136}{3840885093721692949} a^{18} + \frac{1515378396916778593}{3840885093721692949} a^{17} + \frac{6501261312}{3840885093721692949} a^{16} - \frac{753275781624785546}{3840885093721692949} a^{15} + \frac{70389596160}{3840885093721692949} a^{14} - \frac{287430779938269281}{3840885093721692949} a^{13} + \frac{498201919488}{3840885093721692949} a^{12} - \frac{633804741767300121}{3840885093721692949} a^{11} + \frac{2292380073984}{3840885093721692949} a^{10} - \frac{1060639547773137966}{3840885093721692949} a^{9} + \frac{6631528071168}{3840885093721692949} a^{8} + \frac{1200461111297915641}{3840885093721692949} a^{7} + \frac{11253502181376}{3840885093721692949} a^{6} - \frac{870269465053049445}{3840885093721692949} a^{5} + \frac{9824486031360}{3840885093721692949} a^{4} - \frac{1685759486906941430}{3840885093721692949} a^{3} + \frac{3348463878144}{3840885093721692949} a^{2} - \frac{788544566007769737}{3840885093721692949} a + \frac{188441690112}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{29} + \frac{233856}{3840885093721692949} a^{23} + \frac{568699202519170108}{3840885093721692949} a^{22} + \frac{15434496}{3840885093721692949} a^{21} + \frac{500235987402582986}{3840885093721692949} a^{20} + \frac{471453696}{3840885093721692949} a^{19} - \frac{1642902194567975896}{3840885093721692949} a^{18} + \frac{8531066880}{3840885093721692949} a^{17} - \frac{1832082178564705955}{3840885093721692949} a^{16} + \frac{99447865344}{3840885093721692949} a^{15} + \frac{535753810766650325}{3840885093721692949} a^{14} + \frac{769412431872}{3840885093721692949} a^{13} + \frac{1216439075696824013}{3840885093721692949} a^{12} + \frac{3951550267392}{3840885093721692949} a^{11} + \frac{1242274430966254749}{3840885093721692949} a^{10} + \frac{13149696688128}{3840885093721692949} a^{9} - \frac{724529466013745674}{3840885093721692949} a^{8} + \frac{26897106862080}{3840885093721692949} a^{7} - \frac{1645992845434295622}{3840885093721692949} a^{6} + \frac{30682625605632}{3840885093721692949} a^{5} - \frac{155873759392756468}{3840885093721692949} a^{4} + \frac{16184242077696}{3840885093721692949} a^{3} + \frac{1390262362184019361}{3840885093721692949} a^{2} + \frac{2522219544576}{3840885093721692949} a + \frac{547809839757509722}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{30} + \frac{1846106931470008131}{3840885093721692949} a^{23} - \frac{7015680}{3840885093721692949} a^{22} - \frac{1045690854492800317}{3840885093721692949} a^{21} - \frac{471453696}{3840885093721692949} a^{20} + \frac{1815070722885472117}{3840885093721692949} a^{19} - \frac{14218444800}{3840885093721692949} a^{18} + \frac{1728352502428726941}{3840885093721692949} a^{17} - \frac{248619663360}{3840885093721692949} a^{16} + \frac{766073532138693352}{3840885093721692949} a^{15} - \frac{2747901542400}{3840885093721692949} a^{14} + \frac{805499746021430299}{3840885093721692949} a^{13} - \frac{19757751336960}{3840885093721692949} a^{12} - \frac{1723576185132823613}{3840885093721692949} a^{11} - \frac{92047876816896}{3840885093721692949} a^{10} - \frac{1402128047198145099}{3840885093721692949} a^{9} - \frac{268971068620800}{3840885093721692949} a^{8} + \frac{568655653444838707}{3840885093721692949} a^{7} - \frac{460239384084480}{3840885093721692949} a^{6} + \frac{1516695874538858321}{3840885093721692949} a^{5} - \frac{404606051942400}{3840885093721692949} a^{4} + \frac{1533177884513853166}{3840885093721692949} a^{3} - \frac{138722074951680}{3840885093721692949} a^{2} + \frac{1272137224494150950}{3840885093721692949} a - \frac{7846905249792}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{31} - \frac{8055040}{3840885093721692949} a^{23} - \frac{1591241964415705239}{3840885093721692949} a^{22} - \frac{567074816}{3840885093721692949} a^{21} - \frac{1893650425268069862}{3840885093721692949} a^{20} - \frac{18043289600}{3840885093721692949} a^{19} + \frac{710386245234959654}{3840885093721692949} a^{18} - \frac{335826124800}{3840885093721692949} a^{17} + \frac{170641662588656362}{3840885093721692949} a^{16} - \frac{3996330885120}{3840885093721692949} a^{15} - \frac{562003506505315188}{3840885093721692949} a^{14} - \frac{31409758535680}{3840885093721692949} a^{13} + \frac{438839205158574805}{3840885093721692949} a^{12} - \frac{163330744385536}{3840885093721692949} a^{11} + \frac{801171445336230358}{3840885093721692949} a^{10} - \frac{549010905497600}{3840885093721692949} a^{9} + \frac{764052517257624447}{3840885093721692949} a^{8} - \frac{1132334992588800}{3840885093721692949} a^{7} + \frac{1556399923108095923}{3840885093721692949} a^{6} - \frac{1300733529948160}{3840885093721692949} a^{5} - \frac{1176279426513723853}{3840885093721692949} a^{4} - \frac{690185138339840}{3840885093721692949} a^{3} + \frac{1410183152646754643}{3840885093721692949} a^{2} - \frac{108112916774912}{3840885093721692949} a - \frac{632478297389581953}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{32} - \frac{1207280434160563065}{3840885093721692949} a^{23} + \frac{206209024}{3840885093721692949} a^{22} - \frac{1137155485290226395}{3840885093721692949} a^{21} + \frac{14434631680}{3840885093721692949} a^{20} + \frac{243103094575154742}{3840885093721692949} a^{19} + \frac{447768166400}{3840885093721692949} a^{18} - \frac{1265289947522832554}{3840885093721692949} a^{17} + \frac{7992661770240}{3840885093721692949} a^{16} + \frac{1320356885746418085}{3840885093721692949} a^{15} + \frac{89742167244800}{3840885093721692949} a^{14} - \frac{1196889269100379392}{3840885093721692949} a^{13} + \frac{653322977542144}{3840885093721692949} a^{12} + \frac{961410733250638959}{3840885093721692949} a^{11} + \frac{3074461070786560}{3840885093721692949} a^{10} - \frac{1075576411899715802}{3840885093721692949} a^{9} + \frac{9058679940710400}{3840885093721692949} a^{8} - \frac{1322355469380843495}{3840885093721692949} a^{7} + \frac{15608802359377920}{3840885093721692949} a^{6} + \frac{1386855206370511838}{3840885093721692949} a^{5} + \frac{13803702766796800}{3840885093721692949} a^{4} + \frac{328422477229727642}{3840885093721692949} a^{3} + \frac{4756968338096128}{3840885093721692949} a^{2} - \frac{829162056556314135}{3840885093721692949} a + \frac{270282291937280}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{33} + \frac{243032064}{3840885093721692949} a^{23} - \frac{464786617526960625}{3840885093721692949} a^{22} + \frac{17822351360}{3840885093721692949} a^{21} + \frac{1596399884580466439}{3840885093721692949} a^{20} + \frac{583276953600}{3840885093721692949} a^{19} + \frac{231834463846829073}{3840885093721692949} a^{18} + \frac{11082262118400}{3840885093721692949} a^{17} + \frac{1565138326744252579}{3840885093721692949} a^{16} + \frac{133972235386880}{3840885093721692949} a^{15} + \frac{63780526123518774}{3840885093721692949} a^{14} + \frac{1066136946868224}{3840885093721692949} a^{13} + \frac{70428753440184573}{3840885093721692949} a^{12} + \frac{5599911236075520}{3840885093721692949} a^{11} - \frac{122836505128477042}{3840885093721692949} a^{10} + \frac{18980091304345600}{3840885093721692949} a^{9} - \frac{563217028447580957}{3840885093721692949} a^{8} + \frac{39420189632102400}{3840885093721692949} a^{7} - \frac{568500384899862197}{3840885093721692949} a^{6} + \frac{45552219130429440}{3840885093721692949} a^{5} + \frac{298497010592989733}{3840885093721692949} a^{4} + \frac{24294516869562368}{3840885093721692949} a^{3} + \frac{315744493314101090}{3840885093721692949} a^{2} + \frac{3822563843112960}{3840885093721692949} a + \frac{1557283455884570326}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{34} + \frac{680336094538360699}{3840885093721692949} a^{23} - \frac{5508726784}{3840885093721692949} a^{22} - \frac{597093229617374325}{3840885093721692949} a^{21} - \frac{396628328448}{3840885093721692949} a^{20} + \frac{832467051532136340}{3840885093721692949} a^{19} - \frac{12559897067520}{3840885093721692949} a^{18} - \frac{1640333086406190709}{3840885093721692949} a^{17} - \frac{227752800157696}{3840885093721692949} a^{16} - \frac{543059651942868266}{3840885093721692949} a^{15} - \frac{2589189728108544}{3840885093721692949} a^{14} + \frac{808062247690060448}{3840885093721692949} a^{13} - \frac{19039698202656768}{3840885093721692949} a^{12} - \frac{128943474095933982}{3840885093721692949} a^{11} - \frac{90345234608685056}{3840885093721692949} a^{10} + \frac{1058884919346706614}{3840885093721692949} a^{9} - \frac{268057289498296320}{3840885093721692949} a^{8} + \frac{1020013233573412204}{3840885093721692949} a^{7} - \frac{464632635130380288}{3840885093721692949} a^{6} - \frac{1424369925164337348}{3840885093721692949} a^{5} - \frac{413006786782560256}{3840885093721692949} a^{4} + \frac{141876785245367413}{3840885093721692949} a^{3} - \frac{142963887732424704}{3840885093721692949} a^{2} - \frac{1616871204082039119}{3840885093721692949} a - \frac{8154802865307648}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{35} - \frac{6648463360}{3840885093721692949} a^{23} - \frac{614311712031221296}{3840885093721692949} a^{22} - \frac{501484093440}{3840885093721692949} a^{21} + \frac{109290790150563558}{3840885093721692949} a^{20} - \frac{16754127667200}{3840885093721692949} a^{19} + \frac{1396325234077907739}{3840885093721692949} a^{18} - \frac{323381257830400}{3840885093721692949} a^{17} + \frac{1363545564292199780}{3840885093721692949} a^{16} - \frac{3958186595844096}{3840885093721692949} a^{15} - \frac{544678701914129866}{3840885093721692949} a^{14} - \frac{31817002301521920}{3840885093721692949} a^{13} - \frac{285354286077266625}{3840885093721692949} a^{12} - \frac{168512271448801280}{3840885093721692949} a^{11} - \frac{86974744840245796}{3840885093721692949} a^{10} - \frac{575142077084467200}{3840885093721692949} a^{9} + \frac{197836111623666644}{3840885093721692949} a^{8} - \frac{1201636125337190400}{3840885093721692949} a^{7} - \frac{370239868500627013}{3840885093721692949} a^{6} - \frac{1395678107058307072}{3840885093721692949} a^{5} - \frac{600676777780749278}{3840885093721692949} a^{4} - \frac{747684700209807360}{3840885093721692949} a^{3} + \frac{1222703724258807650}{3840885093721692949} a^{2} - \frac{118104041497559040}{3840885093721692949} a + \frac{1651431699132927634}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{36} + \frac{155014821534365173}{3840885093721692949} a^{23} + \frac{136768389120}{3840885093721692949} a^{22} + \frac{1751400191194045624}{3840885093721692949} a^{21} + \frac{10052476600320}{3840885093721692949} a^{20} + \frac{1785654682548410246}{3840885093721692949} a^{19} + \frac{323381257830400}{3840885093721692949} a^{18} + \frac{1022132764487593862}{3840885093721692949} a^{17} + \frac{5937279893766144}{3840885093721692949} a^{16} - \frac{984931832702317488}{3840885093721692949} a^{15} + \frac{68179290646118400}{3840885093721692949} a^{14} - \frac{1833793444285889131}{3840885093721692949} a^{13} + \frac{505536814346403840}{3840885093721692949} a^{12} - \frac{1200205567545781179}{3840885093721692949} a^{11} - \frac{1425288369966930709}{3840885093721692949} a^{10} - \frac{1706430274469848322}{3840885093721692949} a^{9} - \frac{471953435420243498}{3840885093721692949} a^{8} + \frac{1204936954829645787}{3840885093721692949} a^{7} + \frac{1038447682359684801}{3840885093721692949} a^{6} + \frac{219263955419050139}{3840885093721692949} a^{5} - \frac{307384778017968447}{3840885093721692949} a^{4} + \frac{1431829937480092449}{3840885093721692949} a^{3} + \frac{56548275697755371}{3840885093721692949} a^{2} + \frac{1689454646991343052}{3840885093721692949} a + \frac{223085411717611520}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{37} + \frac{168681013248}{3840885093721692949} a^{23} - \frac{1607367394939932137}{3840885093721692949} a^{22} + \frac{12988438020096}{3840885093721692949} a^{21} - \frac{1010720561097709552}{3840885093721692949} a^{20} + \frac{440819714621440}{3840885093721692949} a^{19} + \frac{495171852810082196}{3840885093721692949} a^{18} + \frac{8614876708601856}{3840885093721692949} a^{17} - \frac{597486575180521490}{3840885093721692949} a^{16} + \frac{106511202942713856}{3840885093721692949} a^{15} + \frac{1474987301136203120}{3840885093721692949} a^{14} + \frac{863301329114628096}{3840885093721692949} a^{13} - \frac{661587925790160124}{3840885093721692949} a^{12} + \frac{763388661556323563}{3840885093721692949} a^{11} - \frac{349440427008724434}{3840885093721692949} a^{10} + \frac{444650429549155244}{3840885093721692949} a^{9} + \frac{1180585807092363773}{3840885093721692949} a^{8} - \frac{1370765154179789757}{3840885093721692949} a^{7} + \frac{1430703118608704351}{3840885093721692949} a^{6} + \frac{321216533651091758}{3840885093721692949} a^{5} - \frac{821980860118797897}{3840885093721692949} a^{4} + \frac{1625190650177697943}{3840885093721692949} a^{3} - \frac{1000941472118507500}{3840885093721692949} a^{2} - \frac{539221000301042453}{3840885093721692949} a - \frac{149466451061658364}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{38} + \frac{1699550824713534975}{3840885093721692949} a^{23} - \frac{3204939251712}{3840885093721692949} a^{22} - \frac{204166756992478219}{3840885093721692949} a^{21} - \frac{239302130794496}{3840885093721692949} a^{20} - \frac{1810641826475901025}{3840885093721692949} a^{19} - \frac{7794412260163584}{3840885093721692949} a^{18} + \frac{1370529867947100947}{3840885093721692949} a^{17} - \frac{144550918279397376}{3840885093721692949} a^{16} + \frac{134526714154420610}{3840885093721692949} a^{15} - \frac{1673747474814074880}{3840885093721692949} a^{14} - \frac{1649898123121718037}{3840885093721692949} a^{13} - \frac{974659197446680257}{3840885093721692949} a^{12} - \frac{747212966975765565}{3840885093721692949} a^{11} + \frac{1383036442690564432}{3840885093721692949} a^{10} + \frac{989652647836735383}{3840885093721692949} a^{9} + \frac{308224234350000347}{3840885093721692949} a^{8} + \frac{972463536395196215}{3840885093721692949} a^{7} - \frac{420828863317922630}{3840885093721692949} a^{6} - \frac{1432966969736808745}{3840885093721692949} a^{5} + \frac{1537849609021641746}{3840885093721692949} a^{4} - \frac{490574825639969649}{3840885093721692949} a^{3} + \frac{1284755933204594722}{3840885093721692949} a^{2} + \frac{641670228974747236}{3840885093721692949} a - \frac{1819110494999422187}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{39} - \frac{4032020348928}{3840885093721692949} a^{23} + \frac{1797013100540960988}{3840885093721692949} a^{22} - \frac{315393591738368}{3840885093721692949} a^{21} + \frac{1580325221772993740}{3840885093721692949} a^{20} - \frac{10838070697918464}{3840885093721692949} a^{19} - \frac{34839014462277348}{3840885093721692949} a^{18} - \frac{213946330660208640}{3840885093721692949} a^{17} - \frac{242749171239756084}{3840885093721692949} a^{16} + \frac{1173687504824425237}{3840885093721692949} a^{15} - \frac{408574052208012854}{3840885093721692949} a^{14} + \frac{1275795018941932158}{3840885093721692949} a^{13} - \frac{1626344455331902542}{3840885093721692949} a^{12} - \frac{1568625817955881866}{3840885093721692949} a^{11} - \frac{1523148754028874088}{3840885093721692949} a^{10} + \frac{235063491546897309}{3840885093721692949} a^{9} - \frac{573518084126896808}{3840885093721692949} a^{8} - \frac{1364591664789708511}{3840885093721692949} a^{7} + \frac{923871831869887870}{3840885093721692949} a^{6} - \frac{1000795824181303745}{3840885093721692949} a^{5} + \frac{980874848062671314}{3840885093721692949} a^{4} + \frac{313417988729196412}{3840885093721692949} a^{3} - \frac{1174319517335314494}{3840885093721692949} a^{2} - \frac{948203277525396530}{3840885093721692949} a - \frac{1807948360893533629}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{40} + \frac{1303295315261419042}{3840885093721692949} a^{23} + \frac{71680361758720}{3840885093721692949} a^{22} - \frac{1591974993006242853}{3840885093721692949} a^{21} + \frac{5419035348959232}{3840885093721692949} a^{20} - \frac{177639512815873751}{3840885093721692949} a^{19} + \frac{178288608883507200}{3840885093721692949} a^{18} - \frac{1194069477468738303}{3840885093721692949} a^{17} - \frac{506888107600108309}{3840885093721692949} a^{16} - \frac{1696212608905901995}{3840885093721692949} a^{15} + \frac{465283961690616110}{3840885093721692949} a^{14} + \frac{438941025800672955}{3840885093721692949} a^{13} + \frac{80679451168011716}{3840885093721692949} a^{12} + \frac{1170424535031911908}{3840885093721692949} a^{11} + \frac{1097720326446705893}{3840885093721692949} a^{10} - \frac{965580430592931678}{3840885093721692949} a^{9} - \frac{858811863494843343}{3840885093721692949} a^{8} - \frac{17077799648595818}{3840885093721692949} a^{7} + \frac{1744641040777238664}{3840885093721692949} a^{6} - \frac{273724660628640270}{3840885093721692949} a^{5} - \frac{76839765393262201}{3840885093721692949} a^{4} - \frac{1355475354418938092}{3840885093721692949} a^{3} - \frac{810605524103446068}{3840885093721692949} a^{2} - \frac{1142543041504793747}{3840885093721692949} a + \frac{861174340461595681}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{41} + \frac{91840463503360}{3840885093721692949} a^{23} + \frac{40882834713396432}{3840885093721692949} a^{22} + \frac{7273764709466112}{3840885093721692949} a^{21} - \frac{733542435581496863}{3840885093721692949} a^{20} + \frac{252477783303782400}{3840885093721692949} a^{19} + \frac{1854605644771102427}{3840885093721692949} a^{18} + \frac{1184625069182166251}{3840885093721692949} a^{17} - \frac{1142104363296654214}{3840885093721692949} a^{16} + \frac{1635319930138283696}{3840885093721692949} a^{15} + \frac{1388858183767864330}{3840885093721692949} a^{14} - \frac{521640124485502995}{3840885093721692949} a^{13} + \frac{34054638097804647}{3840885093721692949} a^{12} + \frac{1027892422422091613}{3840885093721692949} a^{11} - \frac{1650318396380999625}{3840885093721692949} a^{10} - \frac{35740031099630029}{3840885093721692949} a^{9} - \frac{1222737373354719491}{3840885093721692949} a^{8} - \frac{1657225692684619511}{3840885093721692949} a^{7} + \frac{371716592748735593}{3840885093721692949} a^{6} + \frac{1788813079527081814}{3840885093721692949} a^{5} - \frac{253542123849516931}{3840885093721692949} a^{4} - \frac{1139774963276193193}{3840885093721692949} a^{3} + \frac{1832692267294761720}{3840885093721692949} a^{2} - \frac{1642874253574526582}{3840885093721692949} a + \frac{378673355695624912}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{42} + \frac{190786561995850016}{3840885093721692949} a^{23} - \frac{1542919786856448}{3840885093721692949} a^{22} + \frac{1534945193239154018}{3840885093721692949} a^{21} - \frac{117822965541765120}{3840885093721692949} a^{20} + \frac{412868548276506891}{3840885093721692949} a^{19} - \frac{67845032981308651}{3840885093721692949} a^{18} + \frac{1322432698866142696}{3840885093721692949} a^{17} - \frac{627578220587433329}{3840885093721692949} a^{16} + \frac{1698382378211436927}{3840885093721692949} a^{15} + \frac{869400207475838325}{3840885093721692949} a^{14} + \frac{362357057090020253}{3840885093721692949} a^{13} + \frac{162174410162914869}{3840885093721692949} a^{12} + \frac{584054348849124844}{3840885093721692949} a^{11} - \frac{907485256733660025}{3840885093721692949} a^{10} - \frac{400963568374165900}{3840885093721692949} a^{9} - \frac{1228345319489059163}{3840885093721692949} a^{8} + \frac{39442668469048442}{3840885093721692949} a^{7} - \frac{999036275524493851}{3840885093721692949} a^{6} - \frac{696574022889099799}{3840885093721692949} a^{5} - \frac{785870772090620228}{3840885093721692949} a^{4} + \frac{1264604552984118959}{3840885093721692949} a^{3} - \frac{1362925220433004484}{3840885093721692949} a^{2} - \frac{1121300808259653026}{3840885093721692949} a - \frac{1264742307177146422}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{43} - \frac{2010471237419008}{3840885093721692949} a^{23} - \frac{1417024383475675722}{3840885093721692949} a^{22} - \frac{160837698993520640}{3840885093721692949} a^{21} - \frac{661530674652167821}{3840885093721692949} a^{20} - \frac{1788434371051529451}{3840885093721692949} a^{19} + \frac{762454605796915784}{3840885093721692949} a^{18} - \frac{1448164191371538079}{3840885093721692949} a^{17} - \frac{1492043314537364698}{3840885093721692949} a^{16} + \frac{38263944722939759}{3840885093721692949} a^{15} + \frac{1881624301498062910}{3840885093721692949} a^{14} + \frac{1366967843153145134}{3840885093721692949} a^{13} - \frac{1266634335746435789}{3840885093721692949} a^{12} + \frac{362786478589765736}{3840885093721692949} a^{11} - \frac{349678202606961020}{3840885093721692949} a^{10} + \frac{1292867509031882288}{3840885093721692949} a^{9} + \frac{183682759689312167}{3840885093721692949} a^{8} - \frac{1093541637028943087}{3840885093721692949} a^{7} - \frac{1737537347216041342}{3840885093721692949} a^{6} + \frac{368417110477345107}{3840885093721692949} a^{5} + \frac{1469746016052938479}{3840885093721692949} a^{4} + \frac{529216582292418168}{3840885093721692949} a^{3} + \frac{1069445433917389491}{3840885093721692949} a^{2} + \frac{1872654452388707114}{3840885093721692949} a - \frac{945204401468412346}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{44} - \frac{1625845174826941638}{3840885093721692949} a^{23} + \frac{32167539798704128}{3840885093721692949} a^{22} - \frac{668618010360167348}{3840885093721692949} a^{21} - \frac{1363984529221475093}{3840885093721692949} a^{20} + \frac{478961177476934704}{3840885093721692949} a^{19} - \frac{1754661995056780238}{3840885093721692949} a^{18} - \frac{273923292789547424}{3840885093721692949} a^{17} + \frac{341998170176935560}{3840885093721692949} a^{16} + \frac{1530611547176869174}{3840885093721692949} a^{15} + \frac{595397027703409753}{3840885093721692949} a^{14} - \frac{701868282355884376}{3840885093721692949} a^{13} - \frac{1566307271641823209}{3840885093721692949} a^{12} + \frac{1297885844619145060}{3840885093721692949} a^{11} - \frac{294782237845372216}{3840885093721692949} a^{10} + \frac{71881355983255283}{3840885093721692949} a^{9} + \frac{202521090461318044}{3840885093721692949} a^{8} - \frac{469506678621900684}{3840885093721692949} a^{7} + \frac{641155145826876653}{3840885093721692949} a^{6} + \frac{1880158542937895040}{3840885093721692949} a^{5} + \frac{1860387782226786050}{3840885093721692949} a^{4} + \frac{1912912387957623213}{3840885093721692949} a^{3} + \frac{1540560395597941819}{3840885093721692949} a^{2} - \frac{442675401304579583}{3840885093721692949} a - \frac{1085362195855512780}{3840885093721692949}$, $\frac{1}{3840885093721692949} a^{45} + \frac{42574685027696640}{3840885093721692949} a^{23} + \frac{1777115024158511940}{3840885093721692949} a^{22} - \frac{406527168154163989}{3840885093721692949} a^{21} - \frac{504148903224244823}{3840885093721692949} a^{20} - \frac{1865218489581265568}{3840885093721692949} a^{19} - \frac{1862590493500949655}{3840885093721692949} a^{18} + \frac{1662195188740362985}{3840885093721692949} a^{17} - \frac{882951589493935151}{3840885093721692949} a^{16} - \frac{922276941593365418}{3840885093721692949} a^{15} - \frac{1035489129088145927}{3840885093721692949} a^{14} - \frac{367723943524953009}{3840885093721692949} a^{13} - \frac{1519985802794128054}{3840885093721692949} a^{12} + \frac{1841235936775543253}{3840885093721692949} a^{11} - \frac{843736132992546696}{3840885093721692949} a^{10} - \frac{1556889542649559693}{3840885093721692949} a^{9} + \frac{1516370182428166627}{3840885093721692949} a^{8} + \frac{1027631738826801623}{3840885093721692949} a^{7} - \frac{1112428041041949879}{3840885093721692949} a^{6} - \frac{1465195187495006939}{3840885093721692949} a^{5} - \frac{1749557567945584703}{3840885093721692949} a^{4} + \frac{1266763995948389033}{3840885093721692949} a^{3} + \frac{1900255784580953160}{3840885093721692949} a^{2} + \frac{610713501354428963}{3840885093721692949} a + \frac{343545348867068094}{3840885093721692949}$

Class group and class number

Not computed

Unit group

Rank:		$22$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

$C_{46}$ (as 46T1):

A cyclic group of order 46

The 46 conjugacy class representatives for $C_{46}$

Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{-799}) \), \(\Q(\zeta_{47})^+\)

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	$23^{2}$	$46$	$23^{2}$	$46$	$23^{2}$	$46$	R	$46$	$23^{2}$	$23^{2}$	$23^{2}$	$46$	$23^{2}$	$46$	R	$23^{2}$	$23^{2}$

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
17	Data not computed
47	Data not computed