Global number field 46.46.452622802195165363548725490731426798867221951354690902502568076897138423364177359263252889243492799448213.1

• Label: 46.46.4526228021...8213.1
• Degree: $46$
• Signature: $[46, 0]$
• Discriminant: $19^{23}\cdot 47^{45}$
• Root discriminant: $188.42$
• Ramified primes: $19, 47$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

\( x^{46} - x^{45} - 234 x^{44} + 234 x^{43} + 25616 x^{42} - 25616 x^{41} - 1742759 x^{40} + 1742759 x^{39} + 82563491 x^{38} - 82563491 x^{37} - 2892242759 x^{36} + 2892242759 x^{35} + 77645194741 x^{34} - 77645194741 x^{33} - 1633775352134 x^{32} + 1633775352134 x^{31} + 27328726210366 x^{30} - 27328726210366 x^{29} - 366459672227134 x^{28} + 366459672227134 x^{27} + 3954569780897866 x^{26} - 3954569780897866 x^{25} - 34345464008164634 x^{24} + 34345464008164634 x^{23} + 239226205913710366 x^{22} - 239226205913710366 x^{21} - 1326693081195664634 x^{20} + 1326693081195664634 x^{19} + 5791121860210585366 x^{18} - 5791121860210585366 x^{17} - 19566093868549180259 x^{16} + 19566093868549180259 x^{15} + 49961755710308241616 x^{14} - 49961755710308241616 x^{13} - 93183816952045274009 x^{12} + 93183816952045274009 x^{11} + 120711866336528944741 x^{10} - 120711866336528944741 x^{9} - 100420888943012070884 x^{8} + 100420888943012070884 x^{7} + 47000947910015272866 x^{6} - 47000947910015272866 x^{5} - 9699758571918320884 x^{4} + 9699758571918320884 x^{3} + 609460788433241616 x^{2} - 609460788433241616 x + 49177127544569741 \)

Invariants

Degree:		$46$
Signature:		$[46, 0]$
Discriminant:		\(452622802195165363548725490731426798867221951354690902502568076897138423364177359263252889243492799448213=19^{23}\cdot 47^{45}\)
Root discriminant:		$188.42$
Ramified primes:		$19, 47$
This field is Galois and abelian over $\Q$.
Conductor:		\(893=19\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{893}(1,·)$, $\chi_{893}(132,·)$, $\chi_{893}(647,·)$, $\chi_{893}(265,·)$, $\chi_{893}(778,·)$, $\chi_{893}(267,·)$, $\chi_{893}(780,·)$, $\chi_{893}(398,·)$, $\chi_{893}(400,·)$, $\chi_{893}(533,·)$, $\chi_{893}(151,·)$, $\chi_{893}(153,·)$, $\chi_{893}(666,·)$, $\chi_{893}(797,·)$, $\chi_{893}(286,·)$, $\chi_{893}(417,·)$, $\chi_{893}(550,·)$, $\chi_{893}(170,·)$, $\chi_{893}(685,·)$, $\chi_{893}(436,·)$, $\chi_{893}(569,·)$, $\chi_{893}(571,·)$, $\chi_{893}(702,·)$, $\chi_{893}(191,·)$, $\chi_{893}(322,·)$, $\chi_{893}(324,·)$, $\chi_{893}(457,·)$, $\chi_{893}(208,·)$, $\chi_{893}(723,·)$, $\chi_{893}(343,·)$, $\chi_{893}(476,·)$, $\chi_{893}(607,·)$, $\chi_{893}(96,·)$, $\chi_{893}(227,·)$, $\chi_{893}(740,·)$, $\chi_{893}(742,·)$, $\chi_{893}(360,·)$, $\chi_{893}(493,·)$, $\chi_{893}(495,·)$, $\chi_{893}(113,·)$, $\chi_{893}(626,·)$, $\chi_{893}(115,·)$, $\chi_{893}(628,·)$, $\chi_{893}(246,·)$, $\chi_{893}(761,·)$, $\chi_{893}(892,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4722739333912051} a^{24} - \frac{910522443917369}{4722739333912051} a^{23} - \frac{120}{4722739333912051} a^{22} + \frac{809815704432313}{4722739333912051} a^{21} + \frac{6300}{4722739333912051} a^{20} + \frac{2013868783592809}{4722739333912051} a^{19} - \frac{190000}{4722739333912051} a^{18} + \frac{1638981341505581}{4722739333912051} a^{17} + \frac{3633750}{4722739333912051} a^{16} + \frac{2238644173417523}{4722739333912051} a^{15} - \frac{45900000}{4722739333912051} a^{14} - \frac{929572242340163}{4722739333912051} a^{13} + \frac{386750000}{4722739333912051} a^{12} + \frac{441096911195998}{4722739333912051} a^{11} - \frac{2145000000}{4722739333912051} a^{10} + \frac{1713292121090266}{4722739333912051} a^{9} + \frac{7541015625}{4722739333912051} a^{8} - \frac{417137029358747}{4722739333912051} a^{7} - \frac{15640625000}{4722739333912051} a^{6} - \frac{879018062372772}{4722739333912051} a^{5} + \frac{16757812500}{4722739333912051} a^{4} + \frac{1766030663497221}{4722739333912051} a^{3} - \frac{7031250000}{4722739333912051} a^{2} + \frac{1959999061615748}{4722739333912051} a + \frac{488281250}{4722739333912051}$, $\frac{1}{4722739333912051} a^{25} - \frac{125}{4722739333912051} a^{23} + \frac{170127114325206}{4722739333912051} a^{22} + \frac{6875}{4722739333912051} a^{21} + \frac{176974759875544}{4722739333912051} a^{20} - \frac{218750}{4722739333912051} a^{19} + \frac{1039177573735762}{4722739333912051} a^{18} + \frac{4453125}{4722739333912051} a^{17} - \frac{1048669186459848}{4722739333912051} a^{16} - \frac{60562500}{4722739333912051} a^{15} - \frac{1413137561317404}{4722739333912051} a^{14} + \frac{557812500}{4722739333912051} a^{13} + \frac{405050086025225}{4722739333912051} a^{12} - \frac{3453125000}{4722739333912051} a^{11} + \frac{166992742271610}{4722739333912051} a^{10} + \frac{13964843750}{4722739333912051} a^{9} - \frac{2065037276567486}{4722739333912051} a^{8} - \frac{34912109375}{4722739333912051} a^{7} + \frac{439853857506664}{4722739333912051} a^{6} + \frac{48876953125}{4722739333912051} a^{5} - \frac{321723441218182}{4722739333912051} a^{4} - \frac{31738281250}{4722739333912051} a^{3} + \frac{160861720609091}{4722739333912051} a^{2} + \frac{6103515625}{4722739333912051} a + \frac{2289528954501819}{4722739333912051}$, $\frac{1}{4722739333912051} a^{26} - \frac{299434361456695}{4722739333912051} a^{23} - \frac{8125}{4722739333912051} a^{22} + \frac{2226411801761598}{4722739333912051} a^{21} + \frac{568750}{4722739333912051} a^{20} - \frac{2255148508413867}{4722739333912051} a^{19} - \frac{19296875}{4722739333912051} a^{18} + \frac{746207143519584}{4722739333912051} a^{17} + \frac{393656250}{4722739333912051} a^{16} - \frac{224236584938038}{4722739333912051} a^{15} - \frac{5179687500}{4722739333912051} a^{14} + \frac{2277003141306125}{4722739333912051} a^{13} + \frac{44890625000}{4722739333912051} a^{12} - \frac{1368765365173252}{4722739333912051} a^{11} - \frac{254160156250}{4722739333912051} a^{10} - \frac{426792166326531}{4722739333912051} a^{9} + \frac{907714843750}{4722739333912051} a^{8} + \frac{247857860695850}{4722739333912051} a^{7} - \frac{1906201171875}{4722739333912051} a^{6} - \frac{1575976557837509}{4722739333912051} a^{5} + \frac{2062988281250}{4722739333912051} a^{4} - \frac{1054054036104681}{4722739333912051} a^{3} - \frac{872802734375}{4722739333912051} a^{2} + \frac{1706966293043667}{4722739333912051} a + \frac{61035156250}{4722739333912051}$, $\frac{1}{4722739333912051} a^{27} - \frac{8775}{4722739333912051} a^{23} - \frac{646536235657445}{4722739333912051} a^{22} + \frac{643500}{4722739333912051} a^{21} - \frac{191665562143716}{4722739333912051} a^{20} - \frac{23034375}{4722739333912051} a^{19} - \frac{1664453323964070}{4722739333912051} a^{18} + \frac{500175000}{4722739333912051} a^{17} + \frac{2194306017000373}{4722739333912051} a^{16} - \frac{7085812500}{4722739333912051} a^{15} + \frac{809123015121458}{4722739333912051} a^{14} + \frac{67128750000}{4722739333912051} a^{13} - \frac{1332113303461691}{4722739333912051} a^{12} - \frac{424216406250}{4722739333912051} a^{11} + \frac{1632079945466382}{4722739333912051} a^{10} + \frac{1742812500000}{4722739333912051} a^{9} - \frac{1455904250397925}{4722739333912051} a^{8} - \frac{4411494140625}{4722739333912051} a^{7} - \frac{141242994344665}{4722739333912051} a^{6} + \frac{6238476562500}{4722739333912051} a^{5} + \frac{1456793717791887}{4722739333912051} a^{4} - \frac{4084716796875}{4722739333912051} a^{3} - \frac{766660676131244}{4722739333912051} a^{2} + \frac{791015625000}{4722739333912051} a + \frac{106186048777257}{4722739333912051}$, $\frac{1}{4722739333912051} a^{28} + \frac{393971368619872}{4722739333912051} a^{23} - \frac{409500}{4722739333912051} a^{22} - \frac{1781556706233896}{4722739333912051} a^{21} + \frac{32248125}{4722739333912051} a^{20} + \frac{2266274537952114}{4722739333912051} a^{19} - \frac{1167075000}{4722739333912051} a^{18} - \frac{1208433367633698}{4722739333912051} a^{17} + \frac{24800343750}{4722739333912051} a^{16} - \frac{1683884320246377}{4722739333912051} a^{15} - \frac{335643750000}{4722739333912051} a^{14} - \frac{2157710172279939}{4722739333912051} a^{13} + \frac{2969514843750}{4722739333912051} a^{12} - \frac{388778117532988}{4722739333912051} a^{11} - \frac{17079562500000}{4722739333912051} a^{10} + \frac{203158474627892}{4722739333912051} a^{9} + \frac{61760917968750}{4722739333912051} a^{8} - \frac{395691835510065}{4722739333912051} a^{7} - \frac{131008007812500}{4722739333912051} a^{6} + \frac{306628675096870}{4722739333912051} a^{5} + \frac{142965087890625}{4722739333912051} a^{4} + \frac{844656946543700}{4722739333912051} a^{3} - \frac{60908203125000}{4722739333912051} a^{2} - \frac{1118702380723785}{4722739333912051} a + \frac{4284667968750}{4722739333912051}$, $\frac{1}{4722739333912051} a^{29} - \frac{456750}{4722739333912051} a^{23} - \frac{1732385810969766}{4722739333912051} a^{22} + \frac{37681875}{4722739333912051} a^{21} - \frac{315197463414711}{4722739333912051} a^{20} - \frac{1438762500}{4722739333912051} a^{19} - \frac{2066838097962048}{4722739333912051} a^{18} + \frac{32543437500}{4722739333912051} a^{17} - \frac{615796687930849}{4722739333912051} a^{16} - \frac{474204375000}{4722739333912051} a^{15} - \frac{238299280342021}{4722739333912051} a^{14} + \frac{4586055468750}{4722739333912051} a^{13} - \frac{994043939670166}{4722739333912051} a^{12} - \frac{29441343750000}{4722739333912051} a^{11} - \frac{170472237785514}{4722739333912051} a^{10} + \frac{122466093750000}{4722739333912051} a^{9} - \frac{7763146085628}{16806901544171} a^{8} - \frac{313123535156250}{4722739333912051} a^{7} + \frac{337169341864481}{4722739333912051} a^{6} + \frac{446490966796875}{4722739333912051} a^{5} - \frac{1549434863377623}{4722739333912051} a^{4} - \frac{294389648437500}{4722739333912051} a^{3} - \frac{1104972810248835}{4722739333912051} a^{2} + \frac{57348632812500}{4722739333912051} a + \frac{1179731391083919}{4722739333912051}$, $\frac{1}{4722739333912051} a^{30} + \frac{1567099225950544}{4722739333912051} a^{23} - \frac{17128125}{4722739333912051} a^{22} - \frac{1936829996286281}{4722739333912051} a^{21} + \frac{1438762500}{4722739333912051} a^{20} - \frac{1271780130888415}{4722739333912051} a^{19} - \frac{54239062500}{4722739333912051} a^{18} - \frac{2022621746925160}{4722739333912051} a^{17} + \frac{1185510937500}{4722739333912051} a^{16} - \frac{914318789225577}{4722739333912051} a^{15} - \frac{16378769531250}{4722739333912051} a^{14} + \frac{595864552088586}{4722739333912051} a^{13} + \frac{147206718750000}{4722739333912051} a^{12} - \frac{1216268153794674}{4722739333912051} a^{11} - \frac{857262656250000}{4722739333912051} a^{10} + \frac{255452703819485}{4722739333912051} a^{9} - \frac{1591503982349551}{4722739333912051} a^{8} - \frac{2250781585866327}{4722739333912051} a^{7} - \frac{1974625168041074}{4722739333912051} a^{6} + \frac{1189570238203040}{4722739333912051} a^{5} - \frac{2085737456886602}{4722739333912051} a^{4} - \frac{1032173965043783}{4722739333912051} a^{3} + \frac{1568564529224551}{4722739333912051} a^{2} - \frac{2271533316580539}{4722739333912051} a + \frac{223022460937500}{4722739333912051}$, $\frac{1}{4722739333912051} a^{31} - \frac{19665625}{4722739333912051} a^{23} + \frac{1928243095209010}{4722739333912051} a^{22} + \frac{1730575000}{4722739333912051} a^{21} + \frac{1251043590783026}{4722739333912051} a^{20} - \frac{68829687500}{4722739333912051} a^{19} + \frac{1729002371179545}{4722739333912051} a^{18} + \frac{1601343750000}{4722739333912051} a^{17} - \frac{52012789080276}{4722739333912051} a^{16} - \frac{23819988281250}{4722739333912051} a^{15} - \frac{1099508325812801}{4722739333912051} a^{14} + \frac{234020937500000}{4722739333912051} a^{13} + \frac{912137254193400}{4722739333912051} a^{12} - \frac{1521136093750000}{4722739333912051} a^{11} - \frac{2143356935970202}{4722739333912051} a^{10} + \frac{1668588791087949}{4722739333912051} a^{9} + \frac{500487792906355}{4722739333912051} a^{8} - \frac{2309424820529472}{4722739333912051} a^{7} - \frac{1543189633454935}{4722739333912051} a^{6} + \frac{46508408564745}{4722739333912051} a^{5} + \frac{883767468751465}{4722739333912051} a^{4} - \frac{1524775162326347}{4722739333912051} a^{3} + \frac{1614288357459995}{4722739333912051} a^{2} - \frac{1649985427662051}{4722739333912051} a + \frac{386087291234970}{4722739333912051}$, $\frac{1}{4722739333912051} a^{32} + \frac{1257726759935927}{4722739333912051} a^{23} - \frac{629300000}{4722739333912051} a^{22} - \frac{1926146529317296}{4722739333912051} a^{21} + \frac{55063750000}{4722739333912051} a^{20} + \frac{15371023185911}{4722739333912051} a^{19} - \frac{2135125000000}{4722739333912051} a^{18} + \frac{1559087289617783}{4722739333912051} a^{17} + \frac{47639976562500}{4722739333912051} a^{16} - \frac{1344719028836706}{4722739333912051} a^{15} - \frac{668631250000000}{4722739333912051} a^{14} + \frac{624435900197591}{4722739333912051} a^{13} + \frac{1361805041087949}{4722739333912051} a^{12} + \frac{56712241644808}{4722739333912051} a^{11} + \frac{1990477171296408}{4722739333912051} a^{10} - \frac{1363952465119544}{4722739333912051} a^{9} - \frac{415558771412428}{4722739333912051} a^{8} - \frac{672608193289289}{4722739333912051} a^{7} - \frac{558100902776940}{4722739333912051} a^{6} + \frac{1007044846446072}{4722739333912051} a^{5} + \frac{2159067243054634}{4722739333912051} a^{4} - \frac{1023553882103884}{4722739333912051} a^{3} + \frac{1758268808449479}{4722739333912051} a^{2} - \frac{1805112715266724}{4722739333912051} a + \frac{156877289207148}{4722739333912051}$, $\frac{1}{4722739333912051} a^{33} - \frac{741675000}{4722739333912051} a^{23} - \frac{2126594022191688}{4722739333912051} a^{22} + \frac{67986875000}{4722739333912051} a^{21} + \frac{1093385731267389}{4722739333912051} a^{20} - \frac{2781281250000}{4722739333912051} a^{19} - \frac{966820834032817}{4722739333912051} a^{18} + \frac{66055429687500}{4722739333912051} a^{17} - \frac{264119503139491}{4722739333912051} a^{16} - \frac{998170937500000}{4722739333912051} a^{15} - \frac{1591241826506475}{4722739333912051} a^{14} + \frac{483695394675898}{4722739333912051} a^{13} + \frac{534556393434501}{4722739333912051} a^{12} + \frac{926803799768714}{4722739333912051} a^{11} + \frac{351376833612626}{4722739333912051} a^{10} + \frac{2277798320601042}{4722739333912051} a^{9} - \frac{947224195939262}{4722739333912051} a^{8} + \frac{807306489006752}{4722739333912051} a^{7} + \frac{2101367592747355}{4722739333912051} a^{6} + \frac{1457634701385831}{4722739333912051} a^{5} + \frac{1514681223504210}{4722739333912051} a^{4} - \frac{971756467590554}{4722739333912051} a^{3} - \frac{1086694409708926}{4722739333912051} a^{2} - \frac{1154692148293229}{4722739333912051} a - \frac{443451771267518}{4722739333912051}$, $\frac{1}{4722739333912051} a^{34} + \frac{1596474384397495}{4722739333912051} a^{23} - \frac{21014125000}{4722739333912051} a^{22} - \frac{2012940974404066}{4722739333912051} a^{21} + \frac{1891271250000}{4722739333912051} a^{20} - \frac{1500883569557750}{4722739333912051} a^{19} - \frac{74862820312500}{4722739333912051} a^{18} - \frac{1572919209371079}{4722739333912051} a^{17} + \frac{1696890593750000}{4722739333912051} a^{16} - \frac{44355577323347}{4722739333912051} a^{15} - \frac{500011767939745}{4722739333912051} a^{14} - \frac{195738956024472}{4722739333912051} a^{13} - \frac{317489318866397}{4722739333912051} a^{12} - \frac{1009216573336479}{4722739333912051} a^{11} - \frac{1774660484949822}{4722739333912051} a^{10} + \frac{1601864367441334}{4722739333912051} a^{9} + \frac{2066698809013368}{4722739333912051} a^{8} - \frac{823158763482382}{4722739333912051} a^{7} + \frac{244891914383087}{4722739333912051} a^{6} + \frac{91906038671037}{4722739333912051} a^{5} + \frac{2351641947303265}{4722739333912051} a^{4} - \frac{718827241977570}{4722739333912051} a^{3} - \frac{2152811259388925}{4722739333912051} a^{2} - \frac{97718006305367}{4722739333912051} a - \frac{1504932617477927}{4722739333912051}$, $\frac{1}{4722739333912051} a^{35} - \frac{25361875000}{4722739333912051} a^{23} + \frac{654411796813294}{4722739333912051} a^{22} + \frac{2391262500000}{4722739333912051} a^{21} + \frac{145275958892380}{4722739333912051} a^{20} - \frac{99862382812500}{4722739333912051} a^{19} + \frac{1180917145379344}{4722739333912051} a^{18} - \frac{2313361208912051}{4722739333912051} a^{17} + \frac{2190368894802406}{4722739333912051} a^{16} + \frac{918429358796408}{4722739333912051} a^{15} - \frac{584532004026104}{4722739333912051} a^{14} + \frac{2024988204860022}{4722739333912051} a^{13} + \frac{867394676629257}{4722739333912051} a^{12} - \frac{1074574762145531}{4722739333912051} a^{11} - \frac{2059461163754776}{4722739333912051} a^{10} + \frac{905812884807035}{4722739333912051} a^{9} - \frac{1676772382052956}{4722739333912051} a^{8} - \frac{995189646909965}{4722739333912051} a^{7} - \frac{236631345291647}{4722739333912051} a^{6} + \frac{1269951660183651}{4722739333912051} a^{5} - \frac{1041499949719273}{4722739333912051} a^{4} + \frac{2312134602943125}{4722739333912051} a^{3} - \frac{1778769061554742}{4722739333912051} a^{2} - \frac{2090353101288661}{4722739333912051} a - \frac{1523100389882700}{4722739333912051}$, $\frac{1}{4722739333912051} a^{36} - \frac{796268553032287}{4722739333912051} a^{23} - \frac{652162500000}{4722739333912051} a^{22} + \frac{1677639504212410}{4722739333912051} a^{21} + \frac{59917429687500}{4722739333912051} a^{20} + \frac{126569221970660}{4722739333912051} a^{19} + \frac{2313361208912051}{4722739333912051} a^{18} - \frac{819890625434457}{4722739333912051} a^{17} - \frac{1377644038194612}{4722739333912051} a^{16} + \frac{90311123518870}{4722739333912051} a^{15} - \frac{291198152775432}{4722739333912051} a^{14} - \frac{2283035810816722}{4722739333912051} a^{13} - \frac{1499015047475458}{4722739333912051} a^{12} - \frac{1303822380262718}{4722739333912051} a^{11} + \frac{918325217722504}{4722739333912051} a^{10} - \frac{1765444680248}{16806901544171} a^{9} + \frac{1248398547547739}{4722739333912051} a^{8} + \frac{944060707910600}{4722739333912051} a^{7} - \frac{1984086273828757}{4722739333912051} a^{6} + \frac{139172850885016}{4722739333912051} a^{5} - \frac{1622843706762518}{4722739333912051} a^{4} - \frac{870540702822637}{4722739333912051} a^{3} - \frac{1859437666154952}{4722739333912051} a^{2} + \frac{202513801551297}{4722739333912051} a + \frac{705493826352278}{4722739333912051}$, $\frac{1}{4722739333912051} a^{37} - \frac{804333750000}{4722739333912051} a^{23} + \frac{580199818578990}{4722739333912051} a^{22} + \frac{77417123437500}{4722739333912051} a^{21} + \frac{1069280710780598}{4722739333912051} a^{20} + \frac{1438376521412051}{4722739333912051} a^{19} + \frac{1109595112589328}{4722739333912051} a^{18} - \frac{54277114004867}{4722739333912051} a^{17} - \frac{1980901028584642}{4722739333912051} a^{16} + \frac{2126847943869413}{4722739333912051} a^{15} - \frac{1722574489137689}{4722739333912051} a^{14} + \frac{201129606444340}{4722739333912051} a^{13} - \frac{1740779306230672}{4722739333912051} a^{12} + \frac{1807628846312434}{4722739333912051} a^{11} - \frac{1128521531388161}{4722739333912051} a^{10} - \frac{1600694128377508}{4722739333912051} a^{9} - \frac{595979938587184}{4722739333912051} a^{8} + \frac{1840452420034933}{4722739333912051} a^{7} - \frac{1914156919984822}{4722739333912051} a^{6} - \frac{1700089084652600}{4722739333912051} a^{5} + \frac{2004132527953911}{4722739333912051} a^{4} - \frac{1436730503630885}{4722739333912051} a^{3} - \frac{1961401125232216}{4722739333912051} a^{2} - \frac{893265771375208}{4722739333912051} a - \frac{1358380972861939}{4722739333912051}$, $\frac{1}{4722739333912051} a^{38} - \frac{898406128210098}{4722739333912051} a^{23} - \frac{19102926562500}{4722739333912051} a^{22} + \frac{1090348570691882}{4722739333912051} a^{21} + \frac{1782939812500000}{4722739333912051} a^{20} + \frac{56202117025128}{4722739333912051} a^{19} - \frac{1750030928819235}{4722739333912051} a^{18} - \frac{295636658477248}{4722739333912051} a^{17} + \frac{1498964314809844}{4722739333912051} a^{16} - \frac{560718160905875}{4722739333912051} a^{15} - \frac{1064622203052993}{4722739333912051} a^{14} - \frac{1127551800532349}{4722739333912051} a^{13} + \frac{490995227337166}{4722739333912051} a^{12} + \frac{571680170917438}{4722739333912051} a^{11} - \frac{529197379642341}{4722739333912051} a^{10} - \frac{258717737535371}{4722739333912051} a^{9} + \frac{803805340180766}{4722739333912051} a^{8} + \frac{1796795810771790}{4722739333912051} a^{7} - \frac{2348666598384432}{4722739333912051} a^{6} - \frac{1032295800699075}{4722739333912051} a^{5} - \frac{67148976930772}{4722739333912051} a^{4} - \frac{2168700513759485}{4722739333912051} a^{3} - \frac{1666072258126810}{4722739333912051} a^{2} - \frac{1955277384639203}{4722739333912051} a - \frac{1914140938661160}{4722739333912051}$, $\frac{1}{4722739333912051} a^{39} - \frac{24032714062500}{4722739333912051} a^{23} + \frac{1904617865457295}{4722739333912051} a^{22} + \frac{2349865375000000}{4722739333912051} a^{21} + \frac{2173087814005430}{4722739333912051} a^{20} - \frac{1759873050346929}{4722739333912051} a^{19} + \frac{1230488340074096}{4722739333912051} a^{18} + \frac{1779464778349123}{4722739333912051} a^{17} - \frac{1411422754728023}{4722739333912051} a^{16} - \frac{1361364848264882}{4722739333912051} a^{15} + \frac{1664103138000803}{4722739333912051} a^{14} - \frac{2343993023740197}{4722739333912051} a^{13} - \frac{2310477499996268}{4722739333912051} a^{12} + \frac{314509831925504}{4722739333912051} a^{11} + \frac{1081897796830168}{4722739333912051} a^{10} - \frac{893695418706875}{4722739333912051} a^{9} + \frac{699541625294356}{4722739333912051} a^{8} - \frac{1978903655633560}{4722739333912051} a^{7} + \frac{1068574386442564}{4722739333912051} a^{6} + \frac{890825896455125}{4722739333912051} a^{5} + \frac{640445000637385}{4722739333912051} a^{4} - \frac{1313113083674719}{4722739333912051} a^{3} - \frac{2044076915021822}{4722739333912051} a^{2} - \frac{1779688963131542}{4722739333912051} a - \frac{1371299005003333}{4722739333912051}$, $\frac{1}{4722739333912051} a^{40} + \frac{77610659462373}{4722739333912051} a^{23} - \frac{534060312500000}{4722739333912051} a^{22} - \frac{244353522620835}{4722739333912051} a^{21} - \frac{1481433141782561}{4722739333912051} a^{20} - \frac{688667510325980}{4722739333912051} a^{19} - \frac{2270010537609611}{4722739333912051} a^{18} + \frac{1334386308641582}{4722739333912051} a^{17} - \frac{659663606624923}{4722739333912051} a^{16} + \frac{501661553756600}{4722739333912051} a^{15} + \frac{474978065748026}{4722739333912051} a^{14} + \frac{1805862315028665}{4722739333912051} a^{13} - \frac{786274579813760}{4722739333912051} a^{12} - \frac{1781750706716893}{4722739333912051} a^{11} + \frac{766564298518037}{4722739333912051} a^{10} + \frac{1521564179780959}{4722739333912051} a^{9} + \frac{449039610343698}{4722739333912051} a^{8} - \frac{1397493277017245}{4722739333912051} a^{7} + \frac{402914777454639}{4722739333912051} a^{6} + \frac{27818439158349}{4722739333912051} a^{5} - \frac{115674122758191}{4722739333912051} a^{4} - \frac{1572825713884102}{4722739333912051} a^{3} + \frac{1714053146996895}{4722739333912051} a^{2} - \frac{619721671823351}{4722739333912051} a + \frac{1003657455467872}{4722739333912051}$, $\frac{1}{4722739333912051} a^{41} - \frac{684264775390625}{4722739333912051} a^{23} - \frac{376553054960177}{4722739333912051} a^{22} + \frac{1623862088903161}{4722739333912051} a^{21} + \frac{1529068603577424}{4722739333912051} a^{20} - \frac{1684374052798028}{4722739333912051} a^{19} - \frac{1755255647823691}{4722739333912051} a^{18} - \frac{820715755062860}{4722739333912051} a^{17} + \frac{1147164713993315}{4722739333912051} a^{16} + \frac{1617007283196143}{4722739333912051} a^{15} + \frac{1854790713043722}{4722739333912051} a^{14} + \frac{58269218361028}{4722739333912051} a^{13} - \frac{1931643356626426}{4722739333912051} a^{12} + \frac{1531301293752795}{4722739333912051} a^{11} + \frac{11375341457268}{4722739333912051} a^{10} - \frac{2125680443491833}{4722739333912051} a^{9} - \frac{1475110195028952}{4722739333912051} a^{8} - \frac{1318403728862870}{4722739333912051} a^{7} - \frac{161030866736781}{4722739333912051} a^{6} + \frac{1178943609800307}{4722739333912051} a^{5} - \frac{695810124151884}{4722739333912051} a^{4} - \frac{371669570489346}{4722739333912051} a^{3} + \frac{69136533080513}{4722739333912051} a^{2} + \frac{1263015376082379}{4722739333912051} a + \frac{2051157884175921}{4722739333912051}$, $\frac{1}{4722739333912051} a^{42} - \frac{183001145176809}{4722739333912051} a^{23} - \frac{201342281466972}{4722739333912051} a^{22} - \frac{1838443685861692}{4722739333912051} a^{21} + \frac{2045438380348960}{4722739333912051} a^{20} + \frac{1324482137210324}{4722739333912051} a^{19} + \frac{1163083291039119}{4722739333912051} a^{18} + \frac{745315517896831}{4722739333912051} a^{17} + \frac{2049107614531209}{4722739333912051} a^{16} - \frac{151137112321611}{4722739333912051} a^{15} - \frac{1671361808572397}{4722739333912051} a^{14} - \frac{2142848730257796}{4722739333912051} a^{13} + \frac{1149702981822196}{4722739333912051} a^{12} - \frac{318182115325453}{4722739333912051} a^{11} - \frac{1557041352301704}{4722739333912051} a^{10} - \frac{1290436410408534}{4722739333912051} a^{9} - \frac{144435043856008}{4722739333912051} a^{8} - \frac{715052220598035}{4722739333912051} a^{7} + \frac{1192873399221953}{4722739333912051} a^{6} + \frac{227558798950206}{4722739333912051} a^{5} - \frac{386594344869681}{4722739333912051} a^{4} - \frac{1731010177131568}{4722739333912051} a^{3} + \frac{641780684603296}{4722739333912051} a^{2} + \frac{98912469540823}{4722739333912051} a - \frac{809575526159184}{4722739333912051}$, $\frac{1}{4722739333912051} a^{43} + \frac{166984845413829}{4722739333912051} a^{23} - \frac{184884437518517}{4722739333912051} a^{22} + \frac{2192472794265304}{4722739333912051} a^{21} + \frac{1883299276566580}{4722739333912051} a^{20} - \frac{1465898070866030}{4722739333912051} a^{19} - \frac{665291815293707}{4722739333912051} a^{18} - \frac{950122580720886}{4722739333912051} a^{17} - \frac{329023035046865}{4722739333912051} a^{16} - \frac{1193885549175903}{4722739333912051} a^{15} + \frac{849827030573631}{4722739333912051} a^{14} + \frac{1876132400745834}{4722739333912051} a^{13} - \frac{1658175319405256}{4722739333912051} a^{12} + \frac{2055580621758575}{4722739333912051} a^{11} + \frac{2252143925467844}{4722739333912051} a^{10} + \frac{921633323926325}{4722739333912051} a^{9} + \frac{851196183637975}{4722739333912051} a^{8} - \frac{2005331704125046}{4722739333912051} a^{7} + \frac{2051615467329203}{4722739333912051} a^{6} + \frac{221008160129933}{4722739333912051} a^{5} + \frac{700044202523554}{4722739333912051} a^{4} - \frac{2005331704125046}{4722739333912051} a^{3} - \frac{2209130165437340}{4722739333912051} a^{2} - \frac{637299013622020}{4722739333912051} a + \frac{357061544119819}{4722739333912051}$, $\frac{1}{4722739333912051} a^{44} - \frac{1758039391863885}{4722739333912051} a^{23} - \frac{1383042425635471}{4722739333912051} a^{22} - \frac{1390714996286089}{4722739333912051} a^{21} - \frac{299552715601357}{4722739333912051} a^{20} - \frac{2260454859582042}{4722739333912051} a^{19} - \frac{1192339174369504}{4722739333912051} a^{18} + \frac{267100055006198}{4722739333912051} a^{17} - \frac{103547696080122}{4722739333912051} a^{16} - \frac{372160123604447}{4722739333912051} a^{15} + \frac{1774531275597169}{4722739333912051} a^{14} + \frac{1578616366877535}{4722739333912051} a^{13} + \frac{200461966043186}{4722739333912051} a^{12} + \frac{1391712366625112}{4722739333912051} a^{11} + \frac{1884108696218970}{4722739333912051} a^{10} + \frac{2040483016868644}{4722739333912051} a^{9} + \frac{2211624430673331}{4722739333912051} a^{8} - \frac{1516245032362804}{4722739333912051} a^{7} + \frac{2319537610272083}{4722739333912051} a^{6} + \frac{1923739034764848}{4722739333912051} a^{5} - \frac{1555047924184749}{4722739333912051} a^{4} + \frac{482442528155550}{4722739333912051} a^{3} + \frac{1254418127916731}{4722739333912051} a^{2} - \frac{1113339401498938}{4722739333912051} a - \frac{1115273273838535}{4722739333912051}$, $\frac{1}{4722739333912051} a^{45} + \frac{253064142796605}{4722739333912051} a^{23} + \frac{167828006090006}{4722739333912051} a^{22} + \frac{457765271191943}{4722739333912051} a^{21} - \frac{1436024140866137}{4722739333912051} a^{20} - \frac{1276323842738096}{4722739333912051} a^{19} - \frac{2032484485512725}{4722739333912051} a^{18} + \frac{2290015352424750}{4722739333912051} a^{17} + \frac{512397791872490}{4722739333912051} a^{16} + \frac{1447004412823708}{4722739333912051} a^{15} + \frac{2336384172841407}{4722739333912051} a^{14} - \frac{701130635511820}{4722739333912051} a^{13} + \frac{779053928192401}{4722739333912051} a^{12} + \frac{1277819060132229}{4722739333912051} a^{11} + \frac{596620585173846}{4722739333912051} a^{10} - \frac{630149928542343}{4722739333912051} a^{9} - \frac{720148692293576}{4722739333912051} a^{8} + \frac{1399382020095003}{4722739333912051} a^{7} + \frac{447492526988377}{4722739333912051} a^{6} - \frac{1595701718526966}{4722739333912051} a^{5} + \frac{1726796691208051}{4722739333912051} a^{4} - \frac{167460286869887}{4722739333912051} a^{3} + \frac{1898351919797841}{4722739333912051} a^{2} + \frac{66560790233342}{4722739333912051} a - \frac{1455423665983512}{4722739333912051}$

Class group and class number

Not computed

Unit group

Rank:		$45$
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

\(\Q(\sqrt{893}) \), \(\Q(\zeta_{47})^+\)

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

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
19	Data not computed
47	Data not computed