Number field 46.46.452622802195165363548725490731426798867221951354690902502568076897138423364177359263252889243492799448213.1

• Label: 46.46.452...213.1
• Degree: $46$
• Signature: $[46, 0]$
• Discriminant: $4.526\times 10^{104}$
• 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
Root discriminant:		\(188.42\)
Galois root discriminant:		$19^{1/2}47^{45/46}\approx 188.41900453568252$
Ramified primes:		\(19\), \(47\)
Discriminant root field:		\(\Q(\sqrt{893}) \)
$\card{ \Gal(K/\Q) }$:		$46$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{47\!\cdots\!51}a^{24}-\frac{910522443917369}{47\!\cdots\!51}a^{23}-\frac{120}{47\!\cdots\!51}a^{22}+\frac{809815704432313}{47\!\cdots\!51}a^{21}+\frac{6300}{47\!\cdots\!51}a^{20}+\frac{20\!\cdots\!09}{47\!\cdots\!51}a^{19}-\frac{190000}{47\!\cdots\!51}a^{18}+\frac{16\!\cdots\!81}{47\!\cdots\!51}a^{17}+\frac{3633750}{47\!\cdots\!51}a^{16}+\frac{22\!\cdots\!23}{47\!\cdots\!51}a^{15}-\frac{45900000}{47\!\cdots\!51}a^{14}-\frac{929572242340163}{47\!\cdots\!51}a^{13}+\frac{386750000}{47\!\cdots\!51}a^{12}+\frac{441096911195998}{47\!\cdots\!51}a^{11}-\frac{2145000000}{47\!\cdots\!51}a^{10}+\frac{17\!\cdots\!66}{47\!\cdots\!51}a^{9}+\frac{7541015625}{47\!\cdots\!51}a^{8}-\frac{417137029358747}{47\!\cdots\!51}a^{7}-\frac{15640625000}{47\!\cdots\!51}a^{6}-\frac{879018062372772}{47\!\cdots\!51}a^{5}+\frac{16757812500}{47\!\cdots\!51}a^{4}+\frac{17\!\cdots\!21}{47\!\cdots\!51}a^{3}-\frac{7031250000}{47\!\cdots\!51}a^{2}+\frac{19\!\cdots\!48}{47\!\cdots\!51}a+\frac{488281250}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{25}-\frac{125}{47\!\cdots\!51}a^{23}+\frac{170127114325206}{47\!\cdots\!51}a^{22}+\frac{6875}{47\!\cdots\!51}a^{21}+\frac{176974759875544}{47\!\cdots\!51}a^{20}-\frac{218750}{47\!\cdots\!51}a^{19}+\frac{10\!\cdots\!62}{47\!\cdots\!51}a^{18}+\frac{4453125}{47\!\cdots\!51}a^{17}-\frac{10\!\cdots\!48}{47\!\cdots\!51}a^{16}-\frac{60562500}{47\!\cdots\!51}a^{15}-\frac{14\!\cdots\!04}{47\!\cdots\!51}a^{14}+\frac{557812500}{47\!\cdots\!51}a^{13}+\frac{405050086025225}{47\!\cdots\!51}a^{12}-\frac{3453125000}{47\!\cdots\!51}a^{11}+\frac{166992742271610}{47\!\cdots\!51}a^{10}+\frac{13964843750}{47\!\cdots\!51}a^{9}-\frac{20\!\cdots\!86}{47\!\cdots\!51}a^{8}-\frac{34912109375}{47\!\cdots\!51}a^{7}+\frac{439853857506664}{47\!\cdots\!51}a^{6}+\frac{48876953125}{47\!\cdots\!51}a^{5}-\frac{321723441218182}{47\!\cdots\!51}a^{4}-\frac{31738281250}{47\!\cdots\!51}a^{3}+\frac{160861720609091}{47\!\cdots\!51}a^{2}+\frac{6103515625}{47\!\cdots\!51}a+\frac{22\!\cdots\!19}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{26}-\frac{299434361456695}{47\!\cdots\!51}a^{23}-\frac{8125}{47\!\cdots\!51}a^{22}+\frac{22\!\cdots\!98}{47\!\cdots\!51}a^{21}+\frac{568750}{47\!\cdots\!51}a^{20}-\frac{22\!\cdots\!67}{47\!\cdots\!51}a^{19}-\frac{19296875}{47\!\cdots\!51}a^{18}+\frac{746207143519584}{47\!\cdots\!51}a^{17}+\frac{393656250}{47\!\cdots\!51}a^{16}-\frac{224236584938038}{47\!\cdots\!51}a^{15}-\frac{5179687500}{47\!\cdots\!51}a^{14}+\frac{22\!\cdots\!25}{47\!\cdots\!51}a^{13}+\frac{44890625000}{47\!\cdots\!51}a^{12}-\frac{13\!\cdots\!52}{47\!\cdots\!51}a^{11}-\frac{254160156250}{47\!\cdots\!51}a^{10}-\frac{426792166326531}{47\!\cdots\!51}a^{9}+\frac{907714843750}{47\!\cdots\!51}a^{8}+\frac{247857860695850}{47\!\cdots\!51}a^{7}-\frac{1906201171875}{47\!\cdots\!51}a^{6}-\frac{15\!\cdots\!09}{47\!\cdots\!51}a^{5}+\frac{2062988281250}{47\!\cdots\!51}a^{4}-\frac{10\!\cdots\!81}{47\!\cdots\!51}a^{3}-\frac{872802734375}{47\!\cdots\!51}a^{2}+\frac{17\!\cdots\!67}{47\!\cdots\!51}a+\frac{61035156250}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{27}-\frac{8775}{47\!\cdots\!51}a^{23}-\frac{646536235657445}{47\!\cdots\!51}a^{22}+\frac{643500}{47\!\cdots\!51}a^{21}-\frac{191665562143716}{47\!\cdots\!51}a^{20}-\frac{23034375}{47\!\cdots\!51}a^{19}-\frac{16\!\cdots\!70}{47\!\cdots\!51}a^{18}+\frac{500175000}{47\!\cdots\!51}a^{17}+\frac{21\!\cdots\!73}{47\!\cdots\!51}a^{16}-\frac{7085812500}{47\!\cdots\!51}a^{15}+\frac{809123015121458}{47\!\cdots\!51}a^{14}+\frac{67128750000}{47\!\cdots\!51}a^{13}-\frac{13\!\cdots\!91}{47\!\cdots\!51}a^{12}-\frac{424216406250}{47\!\cdots\!51}a^{11}+\frac{16\!\cdots\!82}{47\!\cdots\!51}a^{10}+\frac{1742812500000}{47\!\cdots\!51}a^{9}-\frac{14\!\cdots\!25}{47\!\cdots\!51}a^{8}-\frac{4411494140625}{47\!\cdots\!51}a^{7}-\frac{141242994344665}{47\!\cdots\!51}a^{6}+\frac{6238476562500}{47\!\cdots\!51}a^{5}+\frac{14\!\cdots\!87}{47\!\cdots\!51}a^{4}-\frac{4084716796875}{47\!\cdots\!51}a^{3}-\frac{766660676131244}{47\!\cdots\!51}a^{2}+\frac{791015625000}{47\!\cdots\!51}a+\frac{106186048777257}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{28}+\frac{393971368619872}{47\!\cdots\!51}a^{23}-\frac{409500}{47\!\cdots\!51}a^{22}-\frac{17\!\cdots\!96}{47\!\cdots\!51}a^{21}+\frac{32248125}{47\!\cdots\!51}a^{20}+\frac{22\!\cdots\!14}{47\!\cdots\!51}a^{19}-\frac{1167075000}{47\!\cdots\!51}a^{18}-\frac{12\!\cdots\!98}{47\!\cdots\!51}a^{17}+\frac{24800343750}{47\!\cdots\!51}a^{16}-\frac{16\!\cdots\!77}{47\!\cdots\!51}a^{15}-\frac{335643750000}{47\!\cdots\!51}a^{14}-\frac{21\!\cdots\!39}{47\!\cdots\!51}a^{13}+\frac{2969514843750}{47\!\cdots\!51}a^{12}-\frac{388778117532988}{47\!\cdots\!51}a^{11}-\frac{17079562500000}{47\!\cdots\!51}a^{10}+\frac{203158474627892}{47\!\cdots\!51}a^{9}+\frac{61760917968750}{47\!\cdots\!51}a^{8}-\frac{395691835510065}{47\!\cdots\!51}a^{7}-\frac{131008007812500}{47\!\cdots\!51}a^{6}+\frac{306628675096870}{47\!\cdots\!51}a^{5}+\frac{142965087890625}{47\!\cdots\!51}a^{4}+\frac{844656946543700}{47\!\cdots\!51}a^{3}-\frac{60908203125000}{47\!\cdots\!51}a^{2}-\frac{11\!\cdots\!85}{47\!\cdots\!51}a+\frac{4284667968750}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{29}-\frac{456750}{47\!\cdots\!51}a^{23}-\frac{17\!\cdots\!66}{47\!\cdots\!51}a^{22}+\frac{37681875}{47\!\cdots\!51}a^{21}-\frac{315197463414711}{47\!\cdots\!51}a^{20}-\frac{1438762500}{47\!\cdots\!51}a^{19}-\frac{20\!\cdots\!48}{47\!\cdots\!51}a^{18}+\frac{32543437500}{47\!\cdots\!51}a^{17}-\frac{615796687930849}{47\!\cdots\!51}a^{16}-\frac{474204375000}{47\!\cdots\!51}a^{15}-\frac{238299280342021}{47\!\cdots\!51}a^{14}+\frac{4586055468750}{47\!\cdots\!51}a^{13}-\frac{994043939670166}{47\!\cdots\!51}a^{12}-\frac{29441343750000}{47\!\cdots\!51}a^{11}-\frac{170472237785514}{47\!\cdots\!51}a^{10}+\frac{122466093750000}{47\!\cdots\!51}a^{9}-\frac{7763146085628}{16806901544171}a^{8}-\frac{313123535156250}{47\!\cdots\!51}a^{7}+\frac{337169341864481}{47\!\cdots\!51}a^{6}+\frac{446490966796875}{47\!\cdots\!51}a^{5}-\frac{15\!\cdots\!23}{47\!\cdots\!51}a^{4}-\frac{294389648437500}{47\!\cdots\!51}a^{3}-\frac{11\!\cdots\!35}{47\!\cdots\!51}a^{2}+\frac{57348632812500}{47\!\cdots\!51}a+\frac{11\!\cdots\!19}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{30}+\frac{15\!\cdots\!44}{47\!\cdots\!51}a^{23}-\frac{17128125}{47\!\cdots\!51}a^{22}-\frac{19\!\cdots\!81}{47\!\cdots\!51}a^{21}+\frac{1438762500}{47\!\cdots\!51}a^{20}-\frac{12\!\cdots\!15}{47\!\cdots\!51}a^{19}-\frac{54239062500}{47\!\cdots\!51}a^{18}-\frac{20\!\cdots\!60}{47\!\cdots\!51}a^{17}+\frac{1185510937500}{47\!\cdots\!51}a^{16}-\frac{914318789225577}{47\!\cdots\!51}a^{15}-\frac{16378769531250}{47\!\cdots\!51}a^{14}+\frac{595864552088586}{47\!\cdots\!51}a^{13}+\frac{147206718750000}{47\!\cdots\!51}a^{12}-\frac{12\!\cdots\!74}{47\!\cdots\!51}a^{11}-\frac{857262656250000}{47\!\cdots\!51}a^{10}+\frac{255452703819485}{47\!\cdots\!51}a^{9}-\frac{15\!\cdots\!51}{47\!\cdots\!51}a^{8}-\frac{22\!\cdots\!27}{47\!\cdots\!51}a^{7}-\frac{19\!\cdots\!74}{47\!\cdots\!51}a^{6}+\frac{11\!\cdots\!40}{47\!\cdots\!51}a^{5}-\frac{20\!\cdots\!02}{47\!\cdots\!51}a^{4}-\frac{10\!\cdots\!83}{47\!\cdots\!51}a^{3}+\frac{15\!\cdots\!51}{47\!\cdots\!51}a^{2}-\frac{22\!\cdots\!39}{47\!\cdots\!51}a+\frac{223022460937500}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{31}-\frac{19665625}{47\!\cdots\!51}a^{23}+\frac{19\!\cdots\!10}{47\!\cdots\!51}a^{22}+\frac{1730575000}{47\!\cdots\!51}a^{21}+\frac{12\!\cdots\!26}{47\!\cdots\!51}a^{20}-\frac{68829687500}{47\!\cdots\!51}a^{19}+\frac{17\!\cdots\!45}{47\!\cdots\!51}a^{18}+\frac{1601343750000}{47\!\cdots\!51}a^{17}-\frac{52012789080276}{47\!\cdots\!51}a^{16}-\frac{23819988281250}{47\!\cdots\!51}a^{15}-\frac{10\!\cdots\!01}{47\!\cdots\!51}a^{14}+\frac{234020937500000}{47\!\cdots\!51}a^{13}+\frac{912137254193400}{47\!\cdots\!51}a^{12}-\frac{15\!\cdots\!00}{47\!\cdots\!51}a^{11}-\frac{21\!\cdots\!02}{47\!\cdots\!51}a^{10}+\frac{16\!\cdots\!49}{47\!\cdots\!51}a^{9}+\frac{500487792906355}{47\!\cdots\!51}a^{8}-\frac{23\!\cdots\!72}{47\!\cdots\!51}a^{7}-\frac{15\!\cdots\!35}{47\!\cdots\!51}a^{6}+\frac{46508408564745}{47\!\cdots\!51}a^{5}+\frac{883767468751465}{47\!\cdots\!51}a^{4}-\frac{15\!\cdots\!47}{47\!\cdots\!51}a^{3}+\frac{16\!\cdots\!95}{47\!\cdots\!51}a^{2}-\frac{16\!\cdots\!51}{47\!\cdots\!51}a+\frac{386087291234970}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{32}+\frac{12\!\cdots\!27}{47\!\cdots\!51}a^{23}-\frac{629300000}{47\!\cdots\!51}a^{22}-\frac{19\!\cdots\!96}{47\!\cdots\!51}a^{21}+\frac{55063750000}{47\!\cdots\!51}a^{20}+\frac{15371023185911}{47\!\cdots\!51}a^{19}-\frac{2135125000000}{47\!\cdots\!51}a^{18}+\frac{15\!\cdots\!83}{47\!\cdots\!51}a^{17}+\frac{47639976562500}{47\!\cdots\!51}a^{16}-\frac{13\!\cdots\!06}{47\!\cdots\!51}a^{15}-\frac{668631250000000}{47\!\cdots\!51}a^{14}+\frac{624435900197591}{47\!\cdots\!51}a^{13}+\frac{13\!\cdots\!49}{47\!\cdots\!51}a^{12}+\frac{56712241644808}{47\!\cdots\!51}a^{11}+\frac{19\!\cdots\!08}{47\!\cdots\!51}a^{10}-\frac{13\!\cdots\!44}{47\!\cdots\!51}a^{9}-\frac{415558771412428}{47\!\cdots\!51}a^{8}-\frac{672608193289289}{47\!\cdots\!51}a^{7}-\frac{558100902776940}{47\!\cdots\!51}a^{6}+\frac{10\!\cdots\!72}{47\!\cdots\!51}a^{5}+\frac{21\!\cdots\!34}{47\!\cdots\!51}a^{4}-\frac{10\!\cdots\!84}{47\!\cdots\!51}a^{3}+\frac{17\!\cdots\!79}{47\!\cdots\!51}a^{2}-\frac{18\!\cdots\!24}{47\!\cdots\!51}a+\frac{156877289207148}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{33}-\frac{741675000}{47\!\cdots\!51}a^{23}-\frac{21\!\cdots\!88}{47\!\cdots\!51}a^{22}+\frac{67986875000}{47\!\cdots\!51}a^{21}+\frac{10\!\cdots\!89}{47\!\cdots\!51}a^{20}-\frac{2781281250000}{47\!\cdots\!51}a^{19}-\frac{966820834032817}{47\!\cdots\!51}a^{18}+\frac{66055429687500}{47\!\cdots\!51}a^{17}-\frac{264119503139491}{47\!\cdots\!51}a^{16}-\frac{998170937500000}{47\!\cdots\!51}a^{15}-\frac{15\!\cdots\!75}{47\!\cdots\!51}a^{14}+\frac{483695394675898}{47\!\cdots\!51}a^{13}+\frac{534556393434501}{47\!\cdots\!51}a^{12}+\frac{926803799768714}{47\!\cdots\!51}a^{11}+\frac{351376833612626}{47\!\cdots\!51}a^{10}+\frac{22\!\cdots\!42}{47\!\cdots\!51}a^{9}-\frac{947224195939262}{47\!\cdots\!51}a^{8}+\frac{807306489006752}{47\!\cdots\!51}a^{7}+\frac{21\!\cdots\!55}{47\!\cdots\!51}a^{6}+\frac{14\!\cdots\!31}{47\!\cdots\!51}a^{5}+\frac{15\!\cdots\!10}{47\!\cdots\!51}a^{4}-\frac{971756467590554}{47\!\cdots\!51}a^{3}-\frac{10\!\cdots\!26}{47\!\cdots\!51}a^{2}-\frac{11\!\cdots\!29}{47\!\cdots\!51}a-\frac{443451771267518}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{34}+\frac{15\!\cdots\!95}{47\!\cdots\!51}a^{23}-\frac{21014125000}{47\!\cdots\!51}a^{22}-\frac{20\!\cdots\!66}{47\!\cdots\!51}a^{21}+\frac{1891271250000}{47\!\cdots\!51}a^{20}-\frac{15\!\cdots\!50}{47\!\cdots\!51}a^{19}-\frac{74862820312500}{47\!\cdots\!51}a^{18}-\frac{15\!\cdots\!79}{47\!\cdots\!51}a^{17}+\frac{16\!\cdots\!00}{47\!\cdots\!51}a^{16}-\frac{44355577323347}{47\!\cdots\!51}a^{15}-\frac{500011767939745}{47\!\cdots\!51}a^{14}-\frac{195738956024472}{47\!\cdots\!51}a^{13}-\frac{317489318866397}{47\!\cdots\!51}a^{12}-\frac{10\!\cdots\!79}{47\!\cdots\!51}a^{11}-\frac{17\!\cdots\!22}{47\!\cdots\!51}a^{10}+\frac{16\!\cdots\!34}{47\!\cdots\!51}a^{9}+\frac{20\!\cdots\!68}{47\!\cdots\!51}a^{8}-\frac{823158763482382}{47\!\cdots\!51}a^{7}+\frac{244891914383087}{47\!\cdots\!51}a^{6}+\frac{91906038671037}{47\!\cdots\!51}a^{5}+\frac{23\!\cdots\!65}{47\!\cdots\!51}a^{4}-\frac{718827241977570}{47\!\cdots\!51}a^{3}-\frac{21\!\cdots\!25}{47\!\cdots\!51}a^{2}-\frac{97718006305367}{47\!\cdots\!51}a-\frac{15\!\cdots\!27}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{35}-\frac{25361875000}{47\!\cdots\!51}a^{23}+\frac{654411796813294}{47\!\cdots\!51}a^{22}+\frac{2391262500000}{47\!\cdots\!51}a^{21}+\frac{145275958892380}{47\!\cdots\!51}a^{20}-\frac{99862382812500}{47\!\cdots\!51}a^{19}+\frac{11\!\cdots\!44}{47\!\cdots\!51}a^{18}-\frac{23\!\cdots\!51}{47\!\cdots\!51}a^{17}+\frac{21\!\cdots\!06}{47\!\cdots\!51}a^{16}+\frac{918429358796408}{47\!\cdots\!51}a^{15}-\frac{584532004026104}{47\!\cdots\!51}a^{14}+\frac{20\!\cdots\!22}{47\!\cdots\!51}a^{13}+\frac{867394676629257}{47\!\cdots\!51}a^{12}-\frac{10\!\cdots\!31}{47\!\cdots\!51}a^{11}-\frac{20\!\cdots\!76}{47\!\cdots\!51}a^{10}+\frac{905812884807035}{47\!\cdots\!51}a^{9}-\frac{16\!\cdots\!56}{47\!\cdots\!51}a^{8}-\frac{995189646909965}{47\!\cdots\!51}a^{7}-\frac{236631345291647}{47\!\cdots\!51}a^{6}+\frac{12\!\cdots\!51}{47\!\cdots\!51}a^{5}-\frac{10\!\cdots\!73}{47\!\cdots\!51}a^{4}+\frac{23\!\cdots\!25}{47\!\cdots\!51}a^{3}-\frac{17\!\cdots\!42}{47\!\cdots\!51}a^{2}-\frac{20\!\cdots\!61}{47\!\cdots\!51}a-\frac{15\!\cdots\!00}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{36}-\frac{796268553032287}{47\!\cdots\!51}a^{23}-\frac{652162500000}{47\!\cdots\!51}a^{22}+\frac{16\!\cdots\!10}{47\!\cdots\!51}a^{21}+\frac{59917429687500}{47\!\cdots\!51}a^{20}+\frac{126569221970660}{47\!\cdots\!51}a^{19}+\frac{23\!\cdots\!51}{47\!\cdots\!51}a^{18}-\frac{819890625434457}{47\!\cdots\!51}a^{17}-\frac{13\!\cdots\!12}{47\!\cdots\!51}a^{16}+\frac{90311123518870}{47\!\cdots\!51}a^{15}-\frac{291198152775432}{47\!\cdots\!51}a^{14}-\frac{22\!\cdots\!22}{47\!\cdots\!51}a^{13}-\frac{14\!\cdots\!58}{47\!\cdots\!51}a^{12}-\frac{13\!\cdots\!18}{47\!\cdots\!51}a^{11}+\frac{918325217722504}{47\!\cdots\!51}a^{10}-\frac{1765444680248}{16806901544171}a^{9}+\frac{12\!\cdots\!39}{47\!\cdots\!51}a^{8}+\frac{944060707910600}{47\!\cdots\!51}a^{7}-\frac{19\!\cdots\!57}{47\!\cdots\!51}a^{6}+\frac{139172850885016}{47\!\cdots\!51}a^{5}-\frac{16\!\cdots\!18}{47\!\cdots\!51}a^{4}-\frac{870540702822637}{47\!\cdots\!51}a^{3}-\frac{18\!\cdots\!52}{47\!\cdots\!51}a^{2}+\frac{202513801551297}{47\!\cdots\!51}a+\frac{705493826352278}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{37}-\frac{804333750000}{47\!\cdots\!51}a^{23}+\frac{580199818578990}{47\!\cdots\!51}a^{22}+\frac{77417123437500}{47\!\cdots\!51}a^{21}+\frac{10\!\cdots\!98}{47\!\cdots\!51}a^{20}+\frac{14\!\cdots\!51}{47\!\cdots\!51}a^{19}+\frac{11\!\cdots\!28}{47\!\cdots\!51}a^{18}-\frac{54277114004867}{47\!\cdots\!51}a^{17}-\frac{19\!\cdots\!42}{47\!\cdots\!51}a^{16}+\frac{21\!\cdots\!13}{47\!\cdots\!51}a^{15}-\frac{17\!\cdots\!89}{47\!\cdots\!51}a^{14}+\frac{201129606444340}{47\!\cdots\!51}a^{13}-\frac{17\!\cdots\!72}{47\!\cdots\!51}a^{12}+\frac{18\!\cdots\!34}{47\!\cdots\!51}a^{11}-\frac{11\!\cdots\!61}{47\!\cdots\!51}a^{10}-\frac{16\!\cdots\!08}{47\!\cdots\!51}a^{9}-\frac{595979938587184}{47\!\cdots\!51}a^{8}+\frac{18\!\cdots\!33}{47\!\cdots\!51}a^{7}-\frac{19\!\cdots\!22}{47\!\cdots\!51}a^{6}-\frac{17\!\cdots\!00}{47\!\cdots\!51}a^{5}+\frac{20\!\cdots\!11}{47\!\cdots\!51}a^{4}-\frac{14\!\cdots\!85}{47\!\cdots\!51}a^{3}-\frac{19\!\cdots\!16}{47\!\cdots\!51}a^{2}-\frac{893265771375208}{47\!\cdots\!51}a-\frac{13\!\cdots\!39}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{38}-\frac{898406128210098}{47\!\cdots\!51}a^{23}-\frac{19102926562500}{47\!\cdots\!51}a^{22}+\frac{10\!\cdots\!82}{47\!\cdots\!51}a^{21}+\frac{17\!\cdots\!00}{47\!\cdots\!51}a^{20}+\frac{56202117025128}{47\!\cdots\!51}a^{19}-\frac{17\!\cdots\!35}{47\!\cdots\!51}a^{18}-\frac{295636658477248}{47\!\cdots\!51}a^{17}+\frac{14\!\cdots\!44}{47\!\cdots\!51}a^{16}-\frac{560718160905875}{47\!\cdots\!51}a^{15}-\frac{10\!\cdots\!93}{47\!\cdots\!51}a^{14}-\frac{11\!\cdots\!49}{47\!\cdots\!51}a^{13}+\frac{490995227337166}{47\!\cdots\!51}a^{12}+\frac{571680170917438}{47\!\cdots\!51}a^{11}-\frac{529197379642341}{47\!\cdots\!51}a^{10}-\frac{258717737535371}{47\!\cdots\!51}a^{9}+\frac{803805340180766}{47\!\cdots\!51}a^{8}+\frac{17\!\cdots\!90}{47\!\cdots\!51}a^{7}-\frac{23\!\cdots\!32}{47\!\cdots\!51}a^{6}-\frac{10\!\cdots\!75}{47\!\cdots\!51}a^{5}-\frac{67148976930772}{47\!\cdots\!51}a^{4}-\frac{21\!\cdots\!85}{47\!\cdots\!51}a^{3}-\frac{16\!\cdots\!10}{47\!\cdots\!51}a^{2}-\frac{19\!\cdots\!03}{47\!\cdots\!51}a-\frac{19\!\cdots\!60}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{39}-\frac{24032714062500}{47\!\cdots\!51}a^{23}+\frac{19\!\cdots\!95}{47\!\cdots\!51}a^{22}+\frac{23\!\cdots\!00}{47\!\cdots\!51}a^{21}+\frac{21\!\cdots\!30}{47\!\cdots\!51}a^{20}-\frac{17\!\cdots\!29}{47\!\cdots\!51}a^{19}+\frac{12\!\cdots\!96}{47\!\cdots\!51}a^{18}+\frac{17\!\cdots\!23}{47\!\cdots\!51}a^{17}-\frac{14\!\cdots\!23}{47\!\cdots\!51}a^{16}-\frac{13\!\cdots\!82}{47\!\cdots\!51}a^{15}+\frac{16\!\cdots\!03}{47\!\cdots\!51}a^{14}-\frac{23\!\cdots\!97}{47\!\cdots\!51}a^{13}-\frac{23\!\cdots\!68}{47\!\cdots\!51}a^{12}+\frac{314509831925504}{47\!\cdots\!51}a^{11}+\frac{10\!\cdots\!68}{47\!\cdots\!51}a^{10}-\frac{893695418706875}{47\!\cdots\!51}a^{9}+\frac{699541625294356}{47\!\cdots\!51}a^{8}-\frac{19\!\cdots\!60}{47\!\cdots\!51}a^{7}+\frac{10\!\cdots\!64}{47\!\cdots\!51}a^{6}+\frac{890825896455125}{47\!\cdots\!51}a^{5}+\frac{640445000637385}{47\!\cdots\!51}a^{4}-\frac{13\!\cdots\!19}{47\!\cdots\!51}a^{3}-\frac{20\!\cdots\!22}{47\!\cdots\!51}a^{2}-\frac{17\!\cdots\!42}{47\!\cdots\!51}a-\frac{13\!\cdots\!33}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{40}+\frac{77610659462373}{47\!\cdots\!51}a^{23}-\frac{534060312500000}{47\!\cdots\!51}a^{22}-\frac{244353522620835}{47\!\cdots\!51}a^{21}-\frac{14\!\cdots\!61}{47\!\cdots\!51}a^{20}-\frac{688667510325980}{47\!\cdots\!51}a^{19}-\frac{22\!\cdots\!11}{47\!\cdots\!51}a^{18}+\frac{13\!\cdots\!82}{47\!\cdots\!51}a^{17}-\frac{659663606624923}{47\!\cdots\!51}a^{16}+\frac{501661553756600}{47\!\cdots\!51}a^{15}+\frac{474978065748026}{47\!\cdots\!51}a^{14}+\frac{18\!\cdots\!65}{47\!\cdots\!51}a^{13}-\frac{786274579813760}{47\!\cdots\!51}a^{12}-\frac{17\!\cdots\!93}{47\!\cdots\!51}a^{11}+\frac{766564298518037}{47\!\cdots\!51}a^{10}+\frac{15\!\cdots\!59}{47\!\cdots\!51}a^{9}+\frac{449039610343698}{47\!\cdots\!51}a^{8}-\frac{13\!\cdots\!45}{47\!\cdots\!51}a^{7}+\frac{402914777454639}{47\!\cdots\!51}a^{6}+\frac{27818439158349}{47\!\cdots\!51}a^{5}-\frac{115674122758191}{47\!\cdots\!51}a^{4}-\frac{15\!\cdots\!02}{47\!\cdots\!51}a^{3}+\frac{17\!\cdots\!95}{47\!\cdots\!51}a^{2}-\frac{619721671823351}{47\!\cdots\!51}a+\frac{10\!\cdots\!72}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{41}-\frac{684264775390625}{47\!\cdots\!51}a^{23}-\frac{376553054960177}{47\!\cdots\!51}a^{22}+\frac{16\!\cdots\!61}{47\!\cdots\!51}a^{21}+\frac{15\!\cdots\!24}{47\!\cdots\!51}a^{20}-\frac{16\!\cdots\!28}{47\!\cdots\!51}a^{19}-\frac{17\!\cdots\!91}{47\!\cdots\!51}a^{18}-\frac{820715755062860}{47\!\cdots\!51}a^{17}+\frac{11\!\cdots\!15}{47\!\cdots\!51}a^{16}+\frac{16\!\cdots\!43}{47\!\cdots\!51}a^{15}+\frac{18\!\cdots\!22}{47\!\cdots\!51}a^{14}+\frac{58269218361028}{47\!\cdots\!51}a^{13}-\frac{19\!\cdots\!26}{47\!\cdots\!51}a^{12}+\frac{15\!\cdots\!95}{47\!\cdots\!51}a^{11}+\frac{11375341457268}{47\!\cdots\!51}a^{10}-\frac{21\!\cdots\!33}{47\!\cdots\!51}a^{9}-\frac{14\!\cdots\!52}{47\!\cdots\!51}a^{8}-\frac{13\!\cdots\!70}{47\!\cdots\!51}a^{7}-\frac{161030866736781}{47\!\cdots\!51}a^{6}+\frac{11\!\cdots\!07}{47\!\cdots\!51}a^{5}-\frac{695810124151884}{47\!\cdots\!51}a^{4}-\frac{371669570489346}{47\!\cdots\!51}a^{3}+\frac{69136533080513}{47\!\cdots\!51}a^{2}+\frac{12\!\cdots\!79}{47\!\cdots\!51}a+\frac{20\!\cdots\!21}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{42}-\frac{183001145176809}{47\!\cdots\!51}a^{23}-\frac{201342281466972}{47\!\cdots\!51}a^{22}-\frac{18\!\cdots\!92}{47\!\cdots\!51}a^{21}+\frac{20\!\cdots\!60}{47\!\cdots\!51}a^{20}+\frac{13\!\cdots\!24}{47\!\cdots\!51}a^{19}+\frac{11\!\cdots\!19}{47\!\cdots\!51}a^{18}+\frac{745315517896831}{47\!\cdots\!51}a^{17}+\frac{20\!\cdots\!09}{47\!\cdots\!51}a^{16}-\frac{151137112321611}{47\!\cdots\!51}a^{15}-\frac{16\!\cdots\!97}{47\!\cdots\!51}a^{14}-\frac{21\!\cdots\!96}{47\!\cdots\!51}a^{13}+\frac{11\!\cdots\!96}{47\!\cdots\!51}a^{12}-\frac{318182115325453}{47\!\cdots\!51}a^{11}-\frac{15\!\cdots\!04}{47\!\cdots\!51}a^{10}-\frac{12\!\cdots\!34}{47\!\cdots\!51}a^{9}-\frac{144435043856008}{47\!\cdots\!51}a^{8}-\frac{715052220598035}{47\!\cdots\!51}a^{7}+\frac{11\!\cdots\!53}{47\!\cdots\!51}a^{6}+\frac{227558798950206}{47\!\cdots\!51}a^{5}-\frac{386594344869681}{47\!\cdots\!51}a^{4}-\frac{17\!\cdots\!68}{47\!\cdots\!51}a^{3}+\frac{641780684603296}{47\!\cdots\!51}a^{2}+\frac{98912469540823}{47\!\cdots\!51}a-\frac{809575526159184}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{43}+\frac{166984845413829}{47\!\cdots\!51}a^{23}-\frac{184884437518517}{47\!\cdots\!51}a^{22}+\frac{21\!\cdots\!04}{47\!\cdots\!51}a^{21}+\frac{18\!\cdots\!80}{47\!\cdots\!51}a^{20}-\frac{14\!\cdots\!30}{47\!\cdots\!51}a^{19}-\frac{665291815293707}{47\!\cdots\!51}a^{18}-\frac{950122580720886}{47\!\cdots\!51}a^{17}-\frac{329023035046865}{47\!\cdots\!51}a^{16}-\frac{11\!\cdots\!03}{47\!\cdots\!51}a^{15}+\frac{849827030573631}{47\!\cdots\!51}a^{14}+\frac{18\!\cdots\!34}{47\!\cdots\!51}a^{13}-\frac{16\!\cdots\!56}{47\!\cdots\!51}a^{12}+\frac{20\!\cdots\!75}{47\!\cdots\!51}a^{11}+\frac{22\!\cdots\!44}{47\!\cdots\!51}a^{10}+\frac{921633323926325}{47\!\cdots\!51}a^{9}+\frac{851196183637975}{47\!\cdots\!51}a^{8}-\frac{20\!\cdots\!46}{47\!\cdots\!51}a^{7}+\frac{20\!\cdots\!03}{47\!\cdots\!51}a^{6}+\frac{221008160129933}{47\!\cdots\!51}a^{5}+\frac{700044202523554}{47\!\cdots\!51}a^{4}-\frac{20\!\cdots\!46}{47\!\cdots\!51}a^{3}-\frac{22\!\cdots\!40}{47\!\cdots\!51}a^{2}-\frac{637299013622020}{47\!\cdots\!51}a+\frac{357061544119819}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{44}-\frac{17\!\cdots\!85}{47\!\cdots\!51}a^{23}-\frac{13\!\cdots\!71}{47\!\cdots\!51}a^{22}-\frac{13\!\cdots\!89}{47\!\cdots\!51}a^{21}-\frac{299552715601357}{47\!\cdots\!51}a^{20}-\frac{22\!\cdots\!42}{47\!\cdots\!51}a^{19}-\frac{11\!\cdots\!04}{47\!\cdots\!51}a^{18}+\frac{267100055006198}{47\!\cdots\!51}a^{17}-\frac{103547696080122}{47\!\cdots\!51}a^{16}-\frac{372160123604447}{47\!\cdots\!51}a^{15}+\frac{17\!\cdots\!69}{47\!\cdots\!51}a^{14}+\frac{15\!\cdots\!35}{47\!\cdots\!51}a^{13}+\frac{200461966043186}{47\!\cdots\!51}a^{12}+\frac{13\!\cdots\!12}{47\!\cdots\!51}a^{11}+\frac{18\!\cdots\!70}{47\!\cdots\!51}a^{10}+\frac{20\!\cdots\!44}{47\!\cdots\!51}a^{9}+\frac{22\!\cdots\!31}{47\!\cdots\!51}a^{8}-\frac{15\!\cdots\!04}{47\!\cdots\!51}a^{7}+\frac{23\!\cdots\!83}{47\!\cdots\!51}a^{6}+\frac{19\!\cdots\!48}{47\!\cdots\!51}a^{5}-\frac{15\!\cdots\!49}{47\!\cdots\!51}a^{4}+\frac{482442528155550}{47\!\cdots\!51}a^{3}+\frac{12\!\cdots\!31}{47\!\cdots\!51}a^{2}-\frac{11\!\cdots\!38}{47\!\cdots\!51}a-\frac{11\!\cdots\!35}{47\!\cdots\!51}$, $\frac{1}{47\!\cdots\!51}a^{45}+\frac{253064142796605}{47\!\cdots\!51}a^{23}+\frac{167828006090006}{47\!\cdots\!51}a^{22}+\frac{457765271191943}{47\!\cdots\!51}a^{21}-\frac{14\!\cdots\!37}{47\!\cdots\!51}a^{20}-\frac{12\!\cdots\!96}{47\!\cdots\!51}a^{19}-\frac{20\!\cdots\!25}{47\!\cdots\!51}a^{18}+\frac{22\!\cdots\!50}{47\!\cdots\!51}a^{17}+\frac{512397791872490}{47\!\cdots\!51}a^{16}+\frac{14\!\cdots\!08}{47\!\cdots\!51}a^{15}+\frac{23\!\cdots\!07}{47\!\cdots\!51}a^{14}-\frac{701130635511820}{47\!\cdots\!51}a^{13}+\frac{779053928192401}{47\!\cdots\!51}a^{12}+\frac{12\!\cdots\!29}{47\!\cdots\!51}a^{11}+\frac{596620585173846}{47\!\cdots\!51}a^{10}-\frac{630149928542343}{47\!\cdots\!51}a^{9}-\frac{720148692293576}{47\!\cdots\!51}a^{8}+\frac{13\!\cdots\!03}{47\!\cdots\!51}a^{7}+\frac{447492526988377}{47\!\cdots\!51}a^{6}-\frac{15\!\cdots\!66}{47\!\cdots\!51}a^{5}+\frac{17\!\cdots\!51}{47\!\cdots\!51}a^{4}-\frac{167460286869887}{47\!\cdots\!51}a^{3}+\frac{18\!\cdots\!41}{47\!\cdots\!51}a^{2}+\frac{66560790233342}{47\!\cdots\!51}a-\frac{14\!\cdots\!12}{47\!\cdots\!51}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$45$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

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
\(19\)		Deg $46$	$2$	$23$	$23$
\(47\)		Deg $46$	$46$	$1$	$45$