Normalized defining polynomial

\( x^{46} - x^{45} + 189 x^{44} - 189 x^{43} + 16733 x^{42} - 16733 x^{41} + 922141 x^{40} + \cdots + 15\!\cdots\!41 \) 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:		$46$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 23]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(-350\!\cdots\!791\) -35053291047098822445710921889723167962319821671472348329989416363445958210915442479511871796900053441791 \(\medspace = -\,17^{23}\cdot 47^{45}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(178.23\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$17^{1/2}47^{45/46}\approx 178.22653556251296$
Ramified primes:		\(17\), \(47\) 17, 47	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{-799}) \)
$\card{ \Gal(K/\Q) }$:		$46$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
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.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{38\!\cdots\!49}a^{24}-\frac{18\!\cdots\!93}{38\!\cdots\!49}a^{23}+\frac{96}{38\!\cdots\!49}a^{22}+\frac{15\!\cdots\!49}{38\!\cdots\!49}a^{21}+\frac{4032}{38\!\cdots\!49}a^{20}+\frac{24\!\cdots\!76}{38\!\cdots\!49}a^{19}+\frac{97280}{38\!\cdots\!49}a^{18}+\frac{88\!\cdots\!54}{38\!\cdots\!49}a^{17}+\frac{1488384}{38\!\cdots\!49}a^{16}-\frac{42\!\cdots\!63}{38\!\cdots\!49}a^{15}+\frac{15040512}{38\!\cdots\!49}a^{14}-\frac{14\!\cdots\!22}{38\!\cdots\!49}a^{13}+\frac{101384192}{38\!\cdots\!49}a^{12}-\frac{11\!\cdots\!63}{38\!\cdots\!49}a^{11}+\frac{449839104}{38\!\cdots\!49}a^{10}+\frac{26\!\cdots\!12}{38\!\cdots\!49}a^{9}+\frac{1265172480}{38\!\cdots\!49}a^{8}-\frac{13\!\cdots\!61}{38\!\cdots\!49}a^{7}+\frac{2099249152}{38\!\cdots\!49}a^{6}-\frac{14\!\cdots\!31}{38\!\cdots\!49}a^{5}+\frac{1799356416}{38\!\cdots\!49}a^{4}-\frac{30\!\cdots\!50}{38\!\cdots\!49}a^{3}+\frac{603979776}{38\!\cdots\!49}a^{2}-\frac{56\!\cdots\!00}{38\!\cdots\!49}a+\frac{33554432}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{25}+\frac{100}{38\!\cdots\!49}a^{23}-\frac{23\!\cdots\!26}{38\!\cdots\!49}a^{22}+\frac{4400}{38\!\cdots\!49}a^{21}-\frac{13\!\cdots\!43}{38\!\cdots\!49}a^{20}+\frac{112000}{38\!\cdots\!49}a^{19}+\frac{86\!\cdots\!52}{38\!\cdots\!49}a^{18}+\frac{1824000}{38\!\cdots\!49}a^{17}+\frac{68\!\cdots\!16}{38\!\cdots\!49}a^{16}+\frac{19845120}{38\!\cdots\!49}a^{15}-\frac{10\!\cdots\!84}{38\!\cdots\!49}a^{14}+\frac{146227200}{38\!\cdots\!49}a^{13}+\frac{13\!\cdots\!06}{38\!\cdots\!49}a^{12}+\frac{724172800}{38\!\cdots\!49}a^{11}-\frac{14\!\cdots\!65}{38\!\cdots\!49}a^{10}+\frac{2342912000}{38\!\cdots\!49}a^{9}-\frac{10\!\cdots\!87}{38\!\cdots\!49}a^{8}+\frac{4685824000}{38\!\cdots\!49}a^{7}+\frac{17\!\cdots\!75}{38\!\cdots\!49}a^{6}+\frac{5248122880}{38\!\cdots\!49}a^{5}+\frac{19\!\cdots\!91}{38\!\cdots\!49}a^{4}+\frac{2726297600}{38\!\cdots\!49}a^{3}-\frac{15\!\cdots\!93}{38\!\cdots\!49}a^{2}+\frac{419430400}{38\!\cdots\!49}a-\frac{11\!\cdots\!41}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{26}+\frac{15\!\cdots\!22}{38\!\cdots\!49}a^{23}-\frac{5200}{38\!\cdots\!49}a^{22}+\frac{18\!\cdots\!66}{38\!\cdots\!49}a^{21}-\frac{291200}{38\!\cdots\!49}a^{20}-\frac{20\!\cdots\!54}{38\!\cdots\!49}a^{19}-\frac{7904000}{38\!\cdots\!49}a^{18}+\frac{20\!\cdots\!43}{38\!\cdots\!49}a^{17}-\frac{128993280}{38\!\cdots\!49}a^{16}-\frac{89\!\cdots\!23}{38\!\cdots\!49}a^{15}-\frac{1357824000}{38\!\cdots\!49}a^{14}-\frac{55\!\cdots\!07}{38\!\cdots\!49}a^{13}-\frac{9414246400}{38\!\cdots\!49}a^{12}+\frac{49\!\cdots\!14}{38\!\cdots\!49}a^{11}-\frac{42640998400}{38\!\cdots\!49}a^{10}-\frac{43\!\cdots\!44}{38\!\cdots\!49}a^{9}-\frac{121831424000}{38\!\cdots\!49}a^{8}+\frac{13\!\cdots\!79}{38\!\cdots\!49}a^{7}-\frac{204676792320}{38\!\cdots\!49}a^{6}-\frac{15\!\cdots\!20}{38\!\cdots\!49}a^{5}-\frac{177209344000}{38\!\cdots\!49}a^{4}-\frac{13\!\cdots\!85}{38\!\cdots\!49}a^{3}-\frac{59978547200}{38\!\cdots\!49}a^{2}+\frac{64\!\cdots\!10}{38\!\cdots\!49}a-\frac{3355443200}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{27}-\frac{5616}{38\!\cdots\!49}a^{23}-\frac{16\!\cdots\!35}{38\!\cdots\!49}a^{22}-\frac{329472}{38\!\cdots\!49}a^{21}+\frac{68\!\cdots\!66}{38\!\cdots\!49}a^{20}-\frac{9434880}{38\!\cdots\!49}a^{19}-\frac{74\!\cdots\!98}{38\!\cdots\!49}a^{18}-\frac{163897344}{38\!\cdots\!49}a^{17}-\frac{11\!\cdots\!69}{38\!\cdots\!49}a^{16}-\frac{1857503232}{38\!\cdots\!49}a^{15}-\frac{16\!\cdots\!67}{38\!\cdots\!49}a^{14}-\frac{14077919232}{38\!\cdots\!49}a^{13}-\frac{12\!\cdots\!79}{38\!\cdots\!49}a^{12}-\frac{71171702784}{38\!\cdots\!49}a^{11}-\frac{52\!\cdots\!75}{38\!\cdots\!49}a^{10}-\frac{233916334080}{38\!\cdots\!49}a^{9}+\frac{15\!\cdots\!07}{38\!\cdots\!49}a^{8}-\frac{473680576512}{38\!\cdots\!49}a^{7}+\frac{62\!\cdots\!68}{38\!\cdots\!49}a^{6}-\frac{535881056256}{38\!\cdots\!49}a^{5}-\frac{16\!\cdots\!09}{38\!\cdots\!49}a^{4}-\frac{280699600896}{38\!\cdots\!49}a^{3}+\frac{13\!\cdots\!22}{38\!\cdots\!49}a^{2}-\frac{43486543872}{38\!\cdots\!49}a+\frac{46\!\cdots\!45}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{28}+\frac{52\!\cdots\!04}{38\!\cdots\!49}a^{23}+\frac{209664}{38\!\cdots\!49}a^{22}+\frac{15\!\cdots\!55}{38\!\cdots\!49}a^{21}+\frac{13208832}{38\!\cdots\!49}a^{20}+\frac{13\!\cdots\!70}{38\!\cdots\!49}a^{19}+\frac{382427136}{38\!\cdots\!49}a^{18}+\frac{15\!\cdots\!93}{38\!\cdots\!49}a^{17}+\frac{6501261312}{38\!\cdots\!49}a^{16}-\frac{75\!\cdots\!46}{38\!\cdots\!49}a^{15}+\frac{70389596160}{38\!\cdots\!49}a^{14}-\frac{28\!\cdots\!81}{38\!\cdots\!49}a^{13}+\frac{498201919488}{38\!\cdots\!49}a^{12}-\frac{63\!\cdots\!21}{38\!\cdots\!49}a^{11}+\frac{2292380073984}{38\!\cdots\!49}a^{10}-\frac{10\!\cdots\!66}{38\!\cdots\!49}a^{9}+\frac{6631528071168}{38\!\cdots\!49}a^{8}+\frac{12\!\cdots\!41}{38\!\cdots\!49}a^{7}+\frac{11253502181376}{38\!\cdots\!49}a^{6}-\frac{87\!\cdots\!45}{38\!\cdots\!49}a^{5}+\frac{9824486031360}{38\!\cdots\!49}a^{4}-\frac{16\!\cdots\!30}{38\!\cdots\!49}a^{3}+\frac{3348463878144}{38\!\cdots\!49}a^{2}-\frac{78\!\cdots\!37}{38\!\cdots\!49}a+\frac{188441690112}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{29}+\frac{233856}{38\!\cdots\!49}a^{23}+\frac{56\!\cdots\!08}{38\!\cdots\!49}a^{22}+\frac{15434496}{38\!\cdots\!49}a^{21}+\frac{50\!\cdots\!86}{38\!\cdots\!49}a^{20}+\frac{471453696}{38\!\cdots\!49}a^{19}-\frac{16\!\cdots\!96}{38\!\cdots\!49}a^{18}+\frac{8531066880}{38\!\cdots\!49}a^{17}-\frac{18\!\cdots\!55}{38\!\cdots\!49}a^{16}+\frac{99447865344}{38\!\cdots\!49}a^{15}+\frac{53\!\cdots\!25}{38\!\cdots\!49}a^{14}+\frac{769412431872}{38\!\cdots\!49}a^{13}+\frac{12\!\cdots\!13}{38\!\cdots\!49}a^{12}+\frac{3951550267392}{38\!\cdots\!49}a^{11}+\frac{12\!\cdots\!49}{38\!\cdots\!49}a^{10}+\frac{13149696688128}{38\!\cdots\!49}a^{9}-\frac{72\!\cdots\!74}{38\!\cdots\!49}a^{8}+\frac{26897106862080}{38\!\cdots\!49}a^{7}-\frac{16\!\cdots\!22}{38\!\cdots\!49}a^{6}+\frac{30682625605632}{38\!\cdots\!49}a^{5}-\frac{15\!\cdots\!68}{38\!\cdots\!49}a^{4}+\frac{16184242077696}{38\!\cdots\!49}a^{3}+\frac{13\!\cdots\!61}{38\!\cdots\!49}a^{2}+\frac{2522219544576}{38\!\cdots\!49}a+\frac{54\!\cdots\!22}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{30}+\frac{18\!\cdots\!31}{38\!\cdots\!49}a^{23}-\frac{7015680}{38\!\cdots\!49}a^{22}-\frac{10\!\cdots\!17}{38\!\cdots\!49}a^{21}-\frac{471453696}{38\!\cdots\!49}a^{20}+\frac{18\!\cdots\!17}{38\!\cdots\!49}a^{19}-\frac{14218444800}{38\!\cdots\!49}a^{18}+\frac{17\!\cdots\!41}{38\!\cdots\!49}a^{17}-\frac{248619663360}{38\!\cdots\!49}a^{16}+\frac{76\!\cdots\!52}{38\!\cdots\!49}a^{15}-\frac{2747901542400}{38\!\cdots\!49}a^{14}+\frac{80\!\cdots\!99}{38\!\cdots\!49}a^{13}-\frac{19757751336960}{38\!\cdots\!49}a^{12}-\frac{17\!\cdots\!13}{38\!\cdots\!49}a^{11}-\frac{92047876816896}{38\!\cdots\!49}a^{10}-\frac{14\!\cdots\!99}{38\!\cdots\!49}a^{9}-\frac{268971068620800}{38\!\cdots\!49}a^{8}+\frac{56\!\cdots\!07}{38\!\cdots\!49}a^{7}-\frac{460239384084480}{38\!\cdots\!49}a^{6}+\frac{15\!\cdots\!21}{38\!\cdots\!49}a^{5}-\frac{404606051942400}{38\!\cdots\!49}a^{4}+\frac{15\!\cdots\!66}{38\!\cdots\!49}a^{3}-\frac{138722074951680}{38\!\cdots\!49}a^{2}+\frac{12\!\cdots\!50}{38\!\cdots\!49}a-\frac{7846905249792}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{31}-\frac{8055040}{38\!\cdots\!49}a^{23}-\frac{15\!\cdots\!39}{38\!\cdots\!49}a^{22}-\frac{567074816}{38\!\cdots\!49}a^{21}-\frac{18\!\cdots\!62}{38\!\cdots\!49}a^{20}-\frac{18043289600}{38\!\cdots\!49}a^{19}+\frac{71\!\cdots\!54}{38\!\cdots\!49}a^{18}-\frac{335826124800}{38\!\cdots\!49}a^{17}+\frac{17\!\cdots\!62}{38\!\cdots\!49}a^{16}-\frac{3996330885120}{38\!\cdots\!49}a^{15}-\frac{56\!\cdots\!88}{38\!\cdots\!49}a^{14}-\frac{31409758535680}{38\!\cdots\!49}a^{13}+\frac{43\!\cdots\!05}{38\!\cdots\!49}a^{12}-\frac{163330744385536}{38\!\cdots\!49}a^{11}+\frac{80\!\cdots\!58}{38\!\cdots\!49}a^{10}-\frac{549010905497600}{38\!\cdots\!49}a^{9}+\frac{76\!\cdots\!47}{38\!\cdots\!49}a^{8}-\frac{11\!\cdots\!00}{38\!\cdots\!49}a^{7}+\frac{15\!\cdots\!23}{38\!\cdots\!49}a^{6}-\frac{13\!\cdots\!60}{38\!\cdots\!49}a^{5}-\frac{11\!\cdots\!53}{38\!\cdots\!49}a^{4}-\frac{690185138339840}{38\!\cdots\!49}a^{3}+\frac{14\!\cdots\!43}{38\!\cdots\!49}a^{2}-\frac{108112916774912}{38\!\cdots\!49}a-\frac{63\!\cdots\!53}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{32}-\frac{12\!\cdots\!65}{38\!\cdots\!49}a^{23}+\frac{206209024}{38\!\cdots\!49}a^{22}-\frac{11\!\cdots\!95}{38\!\cdots\!49}a^{21}+\frac{14434631680}{38\!\cdots\!49}a^{20}+\frac{24\!\cdots\!42}{38\!\cdots\!49}a^{19}+\frac{447768166400}{38\!\cdots\!49}a^{18}-\frac{12\!\cdots\!54}{38\!\cdots\!49}a^{17}+\frac{7992661770240}{38\!\cdots\!49}a^{16}+\frac{13\!\cdots\!85}{38\!\cdots\!49}a^{15}+\frac{89742167244800}{38\!\cdots\!49}a^{14}-\frac{11\!\cdots\!92}{38\!\cdots\!49}a^{13}+\frac{653322977542144}{38\!\cdots\!49}a^{12}+\frac{96\!\cdots\!59}{38\!\cdots\!49}a^{11}+\frac{30\!\cdots\!60}{38\!\cdots\!49}a^{10}-\frac{10\!\cdots\!02}{38\!\cdots\!49}a^{9}+\frac{90\!\cdots\!00}{38\!\cdots\!49}a^{8}-\frac{13\!\cdots\!95}{38\!\cdots\!49}a^{7}+\frac{15\!\cdots\!20}{38\!\cdots\!49}a^{6}+\frac{13\!\cdots\!38}{38\!\cdots\!49}a^{5}+\frac{13\!\cdots\!00}{38\!\cdots\!49}a^{4}+\frac{32\!\cdots\!42}{38\!\cdots\!49}a^{3}+\frac{47\!\cdots\!28}{38\!\cdots\!49}a^{2}-\frac{82\!\cdots\!35}{38\!\cdots\!49}a+\frac{270282291937280}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{33}+\frac{243032064}{38\!\cdots\!49}a^{23}-\frac{46\!\cdots\!25}{38\!\cdots\!49}a^{22}+\frac{17822351360}{38\!\cdots\!49}a^{21}+\frac{15\!\cdots\!39}{38\!\cdots\!49}a^{20}+\frac{583276953600}{38\!\cdots\!49}a^{19}+\frac{23\!\cdots\!73}{38\!\cdots\!49}a^{18}+\frac{11082262118400}{38\!\cdots\!49}a^{17}+\frac{15\!\cdots\!79}{38\!\cdots\!49}a^{16}+\frac{133972235386880}{38\!\cdots\!49}a^{15}+\frac{63\!\cdots\!74}{38\!\cdots\!49}a^{14}+\frac{10\!\cdots\!24}{38\!\cdots\!49}a^{13}+\frac{70\!\cdots\!73}{38\!\cdots\!49}a^{12}+\frac{55\!\cdots\!20}{38\!\cdots\!49}a^{11}-\frac{12\!\cdots\!42}{38\!\cdots\!49}a^{10}+\frac{18\!\cdots\!00}{38\!\cdots\!49}a^{9}-\frac{56\!\cdots\!57}{38\!\cdots\!49}a^{8}+\frac{39\!\cdots\!00}{38\!\cdots\!49}a^{7}-\frac{56\!\cdots\!97}{38\!\cdots\!49}a^{6}+\frac{45\!\cdots\!40}{38\!\cdots\!49}a^{5}+\frac{29\!\cdots\!33}{38\!\cdots\!49}a^{4}+\frac{24\!\cdots\!68}{38\!\cdots\!49}a^{3}+\frac{31\!\cdots\!90}{38\!\cdots\!49}a^{2}+\frac{38\!\cdots\!60}{38\!\cdots\!49}a+\frac{15\!\cdots\!26}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{34}+\frac{68\!\cdots\!99}{38\!\cdots\!49}a^{23}-\frac{5508726784}{38\!\cdots\!49}a^{22}-\frac{59\!\cdots\!25}{38\!\cdots\!49}a^{21}-\frac{396628328448}{38\!\cdots\!49}a^{20}+\frac{83\!\cdots\!40}{38\!\cdots\!49}a^{19}-\frac{12559897067520}{38\!\cdots\!49}a^{18}-\frac{16\!\cdots\!09}{38\!\cdots\!49}a^{17}-\frac{227752800157696}{38\!\cdots\!49}a^{16}-\frac{54\!\cdots\!66}{38\!\cdots\!49}a^{15}-\frac{25\!\cdots\!44}{38\!\cdots\!49}a^{14}+\frac{80\!\cdots\!48}{38\!\cdots\!49}a^{13}-\frac{19\!\cdots\!68}{38\!\cdots\!49}a^{12}-\frac{12\!\cdots\!82}{38\!\cdots\!49}a^{11}-\frac{90\!\cdots\!56}{38\!\cdots\!49}a^{10}+\frac{10\!\cdots\!14}{38\!\cdots\!49}a^{9}-\frac{26\!\cdots\!20}{38\!\cdots\!49}a^{8}+\frac{10\!\cdots\!04}{38\!\cdots\!49}a^{7}-\frac{46\!\cdots\!88}{38\!\cdots\!49}a^{6}-\frac{14\!\cdots\!48}{38\!\cdots\!49}a^{5}-\frac{41\!\cdots\!56}{38\!\cdots\!49}a^{4}+\frac{14\!\cdots\!13}{38\!\cdots\!49}a^{3}-\frac{14\!\cdots\!04}{38\!\cdots\!49}a^{2}-\frac{16\!\cdots\!19}{38\!\cdots\!49}a-\frac{81\!\cdots\!48}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{35}-\frac{6648463360}{38\!\cdots\!49}a^{23}-\frac{61\!\cdots\!96}{38\!\cdots\!49}a^{22}-\frac{501484093440}{38\!\cdots\!49}a^{21}+\frac{10\!\cdots\!58}{38\!\cdots\!49}a^{20}-\frac{16754127667200}{38\!\cdots\!49}a^{19}+\frac{13\!\cdots\!39}{38\!\cdots\!49}a^{18}-\frac{323381257830400}{38\!\cdots\!49}a^{17}+\frac{13\!\cdots\!80}{38\!\cdots\!49}a^{16}-\frac{39\!\cdots\!96}{38\!\cdots\!49}a^{15}-\frac{54\!\cdots\!66}{38\!\cdots\!49}a^{14}-\frac{31\!\cdots\!20}{38\!\cdots\!49}a^{13}-\frac{28\!\cdots\!25}{38\!\cdots\!49}a^{12}-\frac{16\!\cdots\!80}{38\!\cdots\!49}a^{11}-\frac{86\!\cdots\!96}{38\!\cdots\!49}a^{10}-\frac{57\!\cdots\!00}{38\!\cdots\!49}a^{9}+\frac{19\!\cdots\!44}{38\!\cdots\!49}a^{8}-\frac{12\!\cdots\!00}{38\!\cdots\!49}a^{7}-\frac{37\!\cdots\!13}{38\!\cdots\!49}a^{6}-\frac{13\!\cdots\!72}{38\!\cdots\!49}a^{5}-\frac{60\!\cdots\!78}{38\!\cdots\!49}a^{4}-\frac{74\!\cdots\!60}{38\!\cdots\!49}a^{3}+\frac{12\!\cdots\!50}{38\!\cdots\!49}a^{2}-\frac{11\!\cdots\!40}{38\!\cdots\!49}a+\frac{16\!\cdots\!34}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{36}+\frac{15\!\cdots\!73}{38\!\cdots\!49}a^{23}+\frac{136768389120}{38\!\cdots\!49}a^{22}+\frac{17\!\cdots\!24}{38\!\cdots\!49}a^{21}+\frac{10052476600320}{38\!\cdots\!49}a^{20}+\frac{17\!\cdots\!46}{38\!\cdots\!49}a^{19}+\frac{323381257830400}{38\!\cdots\!49}a^{18}+\frac{10\!\cdots\!62}{38\!\cdots\!49}a^{17}+\frac{59\!\cdots\!44}{38\!\cdots\!49}a^{16}-\frac{98\!\cdots\!88}{38\!\cdots\!49}a^{15}+\frac{68\!\cdots\!00}{38\!\cdots\!49}a^{14}-\frac{18\!\cdots\!31}{38\!\cdots\!49}a^{13}+\frac{50\!\cdots\!40}{38\!\cdots\!49}a^{12}-\frac{12\!\cdots\!79}{38\!\cdots\!49}a^{11}-\frac{14\!\cdots\!09}{38\!\cdots\!49}a^{10}-\frac{17\!\cdots\!22}{38\!\cdots\!49}a^{9}-\frac{47\!\cdots\!98}{38\!\cdots\!49}a^{8}+\frac{12\!\cdots\!87}{38\!\cdots\!49}a^{7}+\frac{10\!\cdots\!01}{38\!\cdots\!49}a^{6}+\frac{21\!\cdots\!39}{38\!\cdots\!49}a^{5}-\frac{30\!\cdots\!47}{38\!\cdots\!49}a^{4}+\frac{14\!\cdots\!49}{38\!\cdots\!49}a^{3}+\frac{56\!\cdots\!71}{38\!\cdots\!49}a^{2}+\frac{16\!\cdots\!52}{38\!\cdots\!49}a+\frac{22\!\cdots\!20}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{37}+\frac{168681013248}{38\!\cdots\!49}a^{23}-\frac{16\!\cdots\!37}{38\!\cdots\!49}a^{22}+\frac{12988438020096}{38\!\cdots\!49}a^{21}-\frac{10\!\cdots\!52}{38\!\cdots\!49}a^{20}+\frac{440819714621440}{38\!\cdots\!49}a^{19}+\frac{49\!\cdots\!96}{38\!\cdots\!49}a^{18}+\frac{86\!\cdots\!56}{38\!\cdots\!49}a^{17}-\frac{59\!\cdots\!90}{38\!\cdots\!49}a^{16}+\frac{10\!\cdots\!56}{38\!\cdots\!49}a^{15}+\frac{14\!\cdots\!20}{38\!\cdots\!49}a^{14}+\frac{86\!\cdots\!96}{38\!\cdots\!49}a^{13}-\frac{66\!\cdots\!24}{38\!\cdots\!49}a^{12}+\frac{76\!\cdots\!63}{38\!\cdots\!49}a^{11}-\frac{34\!\cdots\!34}{38\!\cdots\!49}a^{10}+\frac{44\!\cdots\!44}{38\!\cdots\!49}a^{9}+\frac{11\!\cdots\!73}{38\!\cdots\!49}a^{8}-\frac{13\!\cdots\!57}{38\!\cdots\!49}a^{7}+\frac{14\!\cdots\!51}{38\!\cdots\!49}a^{6}+\frac{32\!\cdots\!58}{38\!\cdots\!49}a^{5}-\frac{82\!\cdots\!97}{38\!\cdots\!49}a^{4}+\frac{16\!\cdots\!43}{38\!\cdots\!49}a^{3}-\frac{10\!\cdots\!00}{38\!\cdots\!49}a^{2}-\frac{53\!\cdots\!53}{38\!\cdots\!49}a-\frac{14\!\cdots\!64}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{38}+\frac{16\!\cdots\!75}{38\!\cdots\!49}a^{23}-\frac{3204939251712}{38\!\cdots\!49}a^{22}-\frac{20\!\cdots\!19}{38\!\cdots\!49}a^{21}-\frac{239302130794496}{38\!\cdots\!49}a^{20}-\frac{18\!\cdots\!25}{38\!\cdots\!49}a^{19}-\frac{77\!\cdots\!84}{38\!\cdots\!49}a^{18}+\frac{13\!\cdots\!47}{38\!\cdots\!49}a^{17}-\frac{14\!\cdots\!76}{38\!\cdots\!49}a^{16}+\frac{13\!\cdots\!10}{38\!\cdots\!49}a^{15}-\frac{16\!\cdots\!80}{38\!\cdots\!49}a^{14}-\frac{16\!\cdots\!37}{38\!\cdots\!49}a^{13}-\frac{97\!\cdots\!57}{38\!\cdots\!49}a^{12}-\frac{74\!\cdots\!65}{38\!\cdots\!49}a^{11}+\frac{13\!\cdots\!32}{38\!\cdots\!49}a^{10}+\frac{98\!\cdots\!83}{38\!\cdots\!49}a^{9}+\frac{30\!\cdots\!47}{38\!\cdots\!49}a^{8}+\frac{97\!\cdots\!15}{38\!\cdots\!49}a^{7}-\frac{42\!\cdots\!30}{38\!\cdots\!49}a^{6}-\frac{14\!\cdots\!45}{38\!\cdots\!49}a^{5}+\frac{15\!\cdots\!46}{38\!\cdots\!49}a^{4}-\frac{49\!\cdots\!49}{38\!\cdots\!49}a^{3}+\frac{12\!\cdots\!22}{38\!\cdots\!49}a^{2}+\frac{64\!\cdots\!36}{38\!\cdots\!49}a-\frac{18\!\cdots\!87}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{39}-\frac{4032020348928}{38\!\cdots\!49}a^{23}+\frac{17\!\cdots\!88}{38\!\cdots\!49}a^{22}-\frac{315393591738368}{38\!\cdots\!49}a^{21}+\frac{15\!\cdots\!40}{38\!\cdots\!49}a^{20}-\frac{10\!\cdots\!64}{38\!\cdots\!49}a^{19}-\frac{34\!\cdots\!48}{38\!\cdots\!49}a^{18}-\frac{21\!\cdots\!40}{38\!\cdots\!49}a^{17}-\frac{24\!\cdots\!84}{38\!\cdots\!49}a^{16}+\frac{11\!\cdots\!37}{38\!\cdots\!49}a^{15}-\frac{40\!\cdots\!54}{38\!\cdots\!49}a^{14}+\frac{12\!\cdots\!58}{38\!\cdots\!49}a^{13}-\frac{16\!\cdots\!42}{38\!\cdots\!49}a^{12}-\frac{15\!\cdots\!66}{38\!\cdots\!49}a^{11}-\frac{15\!\cdots\!88}{38\!\cdots\!49}a^{10}+\frac{23\!\cdots\!09}{38\!\cdots\!49}a^{9}-\frac{57\!\cdots\!08}{38\!\cdots\!49}a^{8}-\frac{13\!\cdots\!11}{38\!\cdots\!49}a^{7}+\frac{92\!\cdots\!70}{38\!\cdots\!49}a^{6}-\frac{10\!\cdots\!45}{38\!\cdots\!49}a^{5}+\frac{98\!\cdots\!14}{38\!\cdots\!49}a^{4}+\frac{31\!\cdots\!12}{38\!\cdots\!49}a^{3}-\frac{11\!\cdots\!94}{38\!\cdots\!49}a^{2}-\frac{94\!\cdots\!30}{38\!\cdots\!49}a-\frac{18\!\cdots\!29}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{40}+\frac{13\!\cdots\!42}{38\!\cdots\!49}a^{23}+\frac{71680361758720}{38\!\cdots\!49}a^{22}-\frac{15\!\cdots\!53}{38\!\cdots\!49}a^{21}+\frac{54\!\cdots\!32}{38\!\cdots\!49}a^{20}-\frac{17\!\cdots\!51}{38\!\cdots\!49}a^{19}+\frac{17\!\cdots\!00}{38\!\cdots\!49}a^{18}-\frac{11\!\cdots\!03}{38\!\cdots\!49}a^{17}-\frac{50\!\cdots\!09}{38\!\cdots\!49}a^{16}-\frac{16\!\cdots\!95}{38\!\cdots\!49}a^{15}+\frac{46\!\cdots\!10}{38\!\cdots\!49}a^{14}+\frac{43\!\cdots\!55}{38\!\cdots\!49}a^{13}+\frac{80\!\cdots\!16}{38\!\cdots\!49}a^{12}+\frac{11\!\cdots\!08}{38\!\cdots\!49}a^{11}+\frac{10\!\cdots\!93}{38\!\cdots\!49}a^{10}-\frac{96\!\cdots\!78}{38\!\cdots\!49}a^{9}-\frac{85\!\cdots\!43}{38\!\cdots\!49}a^{8}-\frac{17\!\cdots\!18}{38\!\cdots\!49}a^{7}+\frac{17\!\cdots\!64}{38\!\cdots\!49}a^{6}-\frac{27\!\cdots\!70}{38\!\cdots\!49}a^{5}-\frac{76\!\cdots\!01}{38\!\cdots\!49}a^{4}-\frac{13\!\cdots\!92}{38\!\cdots\!49}a^{3}-\frac{81\!\cdots\!68}{38\!\cdots\!49}a^{2}-\frac{11\!\cdots\!47}{38\!\cdots\!49}a+\frac{86\!\cdots\!81}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{41}+\frac{91840463503360}{38\!\cdots\!49}a^{23}+\frac{40\!\cdots\!32}{38\!\cdots\!49}a^{22}+\frac{72\!\cdots\!12}{38\!\cdots\!49}a^{21}-\frac{73\!\cdots\!63}{38\!\cdots\!49}a^{20}+\frac{25\!\cdots\!00}{38\!\cdots\!49}a^{19}+\frac{18\!\cdots\!27}{38\!\cdots\!49}a^{18}+\frac{11\!\cdots\!51}{38\!\cdots\!49}a^{17}-\frac{11\!\cdots\!14}{38\!\cdots\!49}a^{16}+\frac{16\!\cdots\!96}{38\!\cdots\!49}a^{15}+\frac{13\!\cdots\!30}{38\!\cdots\!49}a^{14}-\frac{52\!\cdots\!95}{38\!\cdots\!49}a^{13}+\frac{34\!\cdots\!47}{38\!\cdots\!49}a^{12}+\frac{10\!\cdots\!13}{38\!\cdots\!49}a^{11}-\frac{16\!\cdots\!25}{38\!\cdots\!49}a^{10}-\frac{35\!\cdots\!29}{38\!\cdots\!49}a^{9}-\frac{12\!\cdots\!91}{38\!\cdots\!49}a^{8}-\frac{16\!\cdots\!11}{38\!\cdots\!49}a^{7}+\frac{37\!\cdots\!93}{38\!\cdots\!49}a^{6}+\frac{17\!\cdots\!14}{38\!\cdots\!49}a^{5}-\frac{25\!\cdots\!31}{38\!\cdots\!49}a^{4}-\frac{11\!\cdots\!93}{38\!\cdots\!49}a^{3}+\frac{18\!\cdots\!20}{38\!\cdots\!49}a^{2}-\frac{16\!\cdots\!82}{38\!\cdots\!49}a+\frac{37\!\cdots\!12}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{42}+\frac{19\!\cdots\!16}{38\!\cdots\!49}a^{23}-\frac{15\!\cdots\!48}{38\!\cdots\!49}a^{22}+\frac{15\!\cdots\!18}{38\!\cdots\!49}a^{21}-\frac{11\!\cdots\!20}{38\!\cdots\!49}a^{20}+\frac{41\!\cdots\!91}{38\!\cdots\!49}a^{19}-\frac{67\!\cdots\!51}{38\!\cdots\!49}a^{18}+\frac{13\!\cdots\!96}{38\!\cdots\!49}a^{17}-\frac{62\!\cdots\!29}{38\!\cdots\!49}a^{16}+\frac{16\!\cdots\!27}{38\!\cdots\!49}a^{15}+\frac{86\!\cdots\!25}{38\!\cdots\!49}a^{14}+\frac{36\!\cdots\!53}{38\!\cdots\!49}a^{13}+\frac{16\!\cdots\!69}{38\!\cdots\!49}a^{12}+\frac{58\!\cdots\!44}{38\!\cdots\!49}a^{11}-\frac{90\!\cdots\!25}{38\!\cdots\!49}a^{10}-\frac{40\!\cdots\!00}{38\!\cdots\!49}a^{9}-\frac{12\!\cdots\!63}{38\!\cdots\!49}a^{8}+\frac{39\!\cdots\!42}{38\!\cdots\!49}a^{7}-\frac{99\!\cdots\!51}{38\!\cdots\!49}a^{6}-\frac{69\!\cdots\!99}{38\!\cdots\!49}a^{5}-\frac{78\!\cdots\!28}{38\!\cdots\!49}a^{4}+\frac{12\!\cdots\!59}{38\!\cdots\!49}a^{3}-\frac{13\!\cdots\!84}{38\!\cdots\!49}a^{2}-\frac{11\!\cdots\!26}{38\!\cdots\!49}a-\frac{12\!\cdots\!22}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{43}-\frac{20\!\cdots\!08}{38\!\cdots\!49}a^{23}-\frac{14\!\cdots\!22}{38\!\cdots\!49}a^{22}-\frac{16\!\cdots\!40}{38\!\cdots\!49}a^{21}-\frac{66\!\cdots\!21}{38\!\cdots\!49}a^{20}-\frac{17\!\cdots\!51}{38\!\cdots\!49}a^{19}+\frac{76\!\cdots\!84}{38\!\cdots\!49}a^{18}-\frac{14\!\cdots\!79}{38\!\cdots\!49}a^{17}-\frac{14\!\cdots\!98}{38\!\cdots\!49}a^{16}+\frac{38\!\cdots\!59}{38\!\cdots\!49}a^{15}+\frac{18\!\cdots\!10}{38\!\cdots\!49}a^{14}+\frac{13\!\cdots\!34}{38\!\cdots\!49}a^{13}-\frac{12\!\cdots\!89}{38\!\cdots\!49}a^{12}+\frac{36\!\cdots\!36}{38\!\cdots\!49}a^{11}-\frac{34\!\cdots\!20}{38\!\cdots\!49}a^{10}+\frac{12\!\cdots\!88}{38\!\cdots\!49}a^{9}+\frac{18\!\cdots\!67}{38\!\cdots\!49}a^{8}-\frac{10\!\cdots\!87}{38\!\cdots\!49}a^{7}-\frac{17\!\cdots\!42}{38\!\cdots\!49}a^{6}+\frac{36\!\cdots\!07}{38\!\cdots\!49}a^{5}+\frac{14\!\cdots\!79}{38\!\cdots\!49}a^{4}+\frac{52\!\cdots\!68}{38\!\cdots\!49}a^{3}+\frac{10\!\cdots\!91}{38\!\cdots\!49}a^{2}+\frac{18\!\cdots\!14}{38\!\cdots\!49}a-\frac{94\!\cdots\!46}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{44}-\frac{16\!\cdots\!38}{38\!\cdots\!49}a^{23}+\frac{32\!\cdots\!28}{38\!\cdots\!49}a^{22}-\frac{66\!\cdots\!48}{38\!\cdots\!49}a^{21}-\frac{13\!\cdots\!93}{38\!\cdots\!49}a^{20}+\frac{47\!\cdots\!04}{38\!\cdots\!49}a^{19}-\frac{17\!\cdots\!38}{38\!\cdots\!49}a^{18}-\frac{27\!\cdots\!24}{38\!\cdots\!49}a^{17}+\frac{34\!\cdots\!60}{38\!\cdots\!49}a^{16}+\frac{15\!\cdots\!74}{38\!\cdots\!49}a^{15}+\frac{59\!\cdots\!53}{38\!\cdots\!49}a^{14}-\frac{70\!\cdots\!76}{38\!\cdots\!49}a^{13}-\frac{15\!\cdots\!09}{38\!\cdots\!49}a^{12}+\frac{12\!\cdots\!60}{38\!\cdots\!49}a^{11}-\frac{29\!\cdots\!16}{38\!\cdots\!49}a^{10}+\frac{71\!\cdots\!83}{38\!\cdots\!49}a^{9}+\frac{20\!\cdots\!44}{38\!\cdots\!49}a^{8}-\frac{46\!\cdots\!84}{38\!\cdots\!49}a^{7}+\frac{64\!\cdots\!53}{38\!\cdots\!49}a^{6}+\frac{18\!\cdots\!40}{38\!\cdots\!49}a^{5}+\frac{18\!\cdots\!50}{38\!\cdots\!49}a^{4}+\frac{19\!\cdots\!13}{38\!\cdots\!49}a^{3}+\frac{15\!\cdots\!19}{38\!\cdots\!49}a^{2}-\frac{44\!\cdots\!83}{38\!\cdots\!49}a-\frac{10\!\cdots\!80}{38\!\cdots\!49}$, $\frac{1}{38\!\cdots\!49}a^{45}+\frac{42\!\cdots\!40}{38\!\cdots\!49}a^{23}+\frac{17\!\cdots\!40}{38\!\cdots\!49}a^{22}-\frac{40\!\cdots\!89}{38\!\cdots\!49}a^{21}-\frac{50\!\cdots\!23}{38\!\cdots\!49}a^{20}-\frac{18\!\cdots\!68}{38\!\cdots\!49}a^{19}-\frac{18\!\cdots\!55}{38\!\cdots\!49}a^{18}+\frac{16\!\cdots\!85}{38\!\cdots\!49}a^{17}-\frac{88\!\cdots\!51}{38\!\cdots\!49}a^{16}-\frac{92\!\cdots\!18}{38\!\cdots\!49}a^{15}-\frac{10\!\cdots\!27}{38\!\cdots\!49}a^{14}-\frac{36\!\cdots\!09}{38\!\cdots\!49}a^{13}-\frac{15\!\cdots\!54}{38\!\cdots\!49}a^{12}+\frac{18\!\cdots\!53}{38\!\cdots\!49}a^{11}-\frac{84\!\cdots\!96}{38\!\cdots\!49}a^{10}-\frac{15\!\cdots\!93}{38\!\cdots\!49}a^{9}+\frac{15\!\cdots\!27}{38\!\cdots\!49}a^{8}+\frac{10\!\cdots\!23}{38\!\cdots\!49}a^{7}-\frac{11\!\cdots\!79}{38\!\cdots\!49}a^{6}-\frac{14\!\cdots\!39}{38\!\cdots\!49}a^{5}-\frac{17\!\cdots\!03}{38\!\cdots\!49}a^{4}+\frac{12\!\cdots\!33}{38\!\cdots\!49}a^{3}+\frac{19\!\cdots\!60}{38\!\cdots\!49}a^{2}+\frac{61\!\cdots\!63}{38\!\cdots\!49}a+\frac{34\!\cdots\!94}{38\!\cdots\!49}$ 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, 1/3840885093721692949*a^24 - 1861928776928916393/3840885093721692949*a^23 + 96/3840885093721692949*a^22 + 1542381740015874549/3840885093721692949*a^21 + 4032/3840885093721692949*a^20 + 241108101087894776/3840885093721692949*a^19 + 97280/3840885093721692949*a^18 + 888202592337969354/3840885093721692949*a^17 + 1488384/3840885093721692949*a^16 - 424548735578019863/3840885093721692949*a^15 + 15040512/3840885093721692949*a^14 - 1401864802913723422/3840885093721692949*a^13 + 101384192/3840885093721692949*a^12 - 1120278318247727563/3840885093721692949*a^11 + 449839104/3840885093721692949*a^10 + 262838506403126012/3840885093721692949*a^9 + 1265172480/3840885093721692949*a^8 - 137364603376836161/3840885093721692949*a^7 + 2099249152/3840885093721692949*a^6 - 1463447835743012531/3840885093721692949*a^5 + 1799356416/3840885093721692949*a^4 - 309677884500054950/3840885093721692949*a^3 + 603979776/3840885093721692949*a^2 - 56305069909100900/3840885093721692949*a + 33554432/3840885093721692949, 1/3840885093721692949*a^25 + 100/3840885093721692949*a^23 - 234055079727720326/3840885093721692949*a^22 + 4400/3840885093721692949*a^21 - 1392421547430923943/3840885093721692949*a^20 + 112000/3840885093721692949*a^19 + 860372509728591452/3840885093721692949*a^18 + 1824000/3840885093721692949*a^17 + 686005037987175416/3840885093721692949*a^16 + 19845120/3840885093721692949*a^15 - 1095626410095508984/3840885093721692949*a^14 + 146227200/3840885093721692949*a^13 + 1370760303014498606/3840885093721692949*a^12 + 724172800/3840885093721692949*a^11 - 142588520863643565/3840885093721692949*a^10 + 2342912000/3840885093721692949*a^9 - 1037572799207019787/3840885093721692949*a^8 + 4685824000/3840885093721692949*a^7 + 1765739495307653375/3840885093721692949*a^6 + 5248122880/3840885093721692949*a^5 + 1909499100154559691/3840885093721692949*a^4 + 2726297600/3840885093721692949*a^3 - 1540731416171191893/3840885093721692949*a^2 + 419430400/3840885093721692949*a - 1149380656381697541/3840885093721692949, 1/3840885093721692949*a^26 + 1596338114522657422/3840885093721692949*a^23 - 5200/3840885093721692949*a^22 + 1845693293571032066/3840885093721692949*a^21 - 291200/3840885093721692949*a^20 - 205127036730728454/3840885093721692949*a^19 - 7904000/3840885093721692949*a^18 + 206102959789177843/3840885093721692949*a^17 - 128993280/3840885093721692949*a^16 - 890488883232145123/3840885093721692949*a^15 - 1357824000/3840885093721692949*a^14 - 555507873315798307/3840885093721692949*a^13 - 9414246400/3840885093721692949*a^12 + 499575585980017214/3840885093721692949*a^11 - 42640998400/3840885093721692949*a^10 - 435227783467770344/3840885093721692949*a^9 - 121831424000/3840885093721692949*a^8 + 138659458104497679/3840885093721692949*a^7 - 204676792320/3840885093721692949*a^6 - 1540235980690212220/3840885093721692949*a^5 - 177209344000/3840885093721692949*a^4 - 1300023715939240485/3840885093721692949*a^3 - 59978547200/3840885093721692949*a^2 + 640241240806699510/3840885093721692949*a - 3355443200/3840885093721692949, 1/3840885093721692949*a^27 - 5616/3840885093721692949*a^23 - 1608247045458055435/3840885093721692949*a^22 - 329472/3840885093721692949*a^21 + 683012285471928566/3840885093721692949*a^20 - 9434880/3840885093721692949*a^19 - 740453542557213298/3840885093721692949*a^18 - 163897344/3840885093721692949*a^17 - 1161528522361665669/3840885093721692949*a^16 - 1857503232/3840885093721692949*a^15 - 1677219976636556267/3840885093721692949*a^14 - 14077919232/3840885093721692949*a^13 - 1229139745938076779/3840885093721692949*a^12 - 71171702784/3840885093721692949*a^11 - 520552978662356775/3840885093721692949*a^10 - 233916334080/3840885093721692949*a^9 + 1579069575122964007/3840885093721692949*a^8 - 473680576512/3840885093721692949*a^7 + 622169837549513068/3840885093721692949*a^6 - 535881056256/3840885093721692949*a^5 - 1641324496717586209/3840885093721692949*a^4 - 280699600896/3840885093721692949*a^3 + 1385177503196444822/3840885093721692949*a^2 - 43486543872/3840885093721692949*a + 468150358324062845/3840885093721692949, 1/3840885093721692949*a^28 + 529851925917381604/3840885093721692949*a^23 + 209664/3840885093721692949*a^22 + 1502977872205795755/3840885093721692949*a^21 + 13208832/3840885093721692949*a^20 + 1331089177023930670/3840885093721692949*a^19 + 382427136/3840885093721692949*a^18 + 1515378396916778593/3840885093721692949*a^17 + 6501261312/3840885093721692949*a^16 - 753275781624785546/3840885093721692949*a^15 + 70389596160/3840885093721692949*a^14 - 287430779938269281/3840885093721692949*a^13 + 498201919488/3840885093721692949*a^12 - 633804741767300121/3840885093721692949*a^11 + 2292380073984/3840885093721692949*a^10 - 1060639547773137966/3840885093721692949*a^9 + 6631528071168/3840885093721692949*a^8 + 1200461111297915641/3840885093721692949*a^7 + 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1417024383475675722/3840885093721692949*a^22 - 160837698993520640/3840885093721692949*a^21 - 661530674652167821/3840885093721692949*a^20 - 1788434371051529451/3840885093721692949*a^19 + 762454605796915784/3840885093721692949*a^18 - 1448164191371538079/3840885093721692949*a^17 - 1492043314537364698/3840885093721692949*a^16 + 38263944722939759/3840885093721692949*a^15 + 1881624301498062910/3840885093721692949*a^14 + 1366967843153145134/3840885093721692949*a^13 - 1266634335746435789/3840885093721692949*a^12 + 362786478589765736/3840885093721692949*a^11 - 349678202606961020/3840885093721692949*a^10 + 1292867509031882288/3840885093721692949*a^9 + 183682759689312167/3840885093721692949*a^8 - 1093541637028943087/3840885093721692949*a^7 - 1737537347216041342/3840885093721692949*a^6 + 368417110477345107/3840885093721692949*a^5 + 1469746016052938479/3840885093721692949*a^4 + 529216582292418168/3840885093721692949*a^3 + 1069445433917389491/3840885093721692949*a^2 + 1872654452388707114/3840885093721692949*a - 945204401468412346/3840885093721692949, 1/3840885093721692949*a^44 - 1625845174826941638/3840885093721692949*a^23 + 32167539798704128/3840885093721692949*a^22 - 668618010360167348/3840885093721692949*a^21 - 1363984529221475093/3840885093721692949*a^20 + 478961177476934704/3840885093721692949*a^19 - 1754661995056780238/3840885093721692949*a^18 - 273923292789547424/3840885093721692949*a^17 + 341998170176935560/3840885093721692949*a^16 + 1530611547176869174/3840885093721692949*a^15 + 595397027703409753/3840885093721692949*a^14 - 701868282355884376/3840885093721692949*a^13 - 1566307271641823209/3840885093721692949*a^12 + 1297885844619145060/3840885093721692949*a^11 - 294782237845372216/3840885093721692949*a^10 + 71881355983255283/3840885093721692949*a^9 + 202521090461318044/3840885093721692949*a^8 - 469506678621900684/3840885093721692949*a^7 + 641155145826876653/3840885093721692949*a^6 + 1880158542937895040/3840885093721692949*a^5 + 1860387782226786050/3840885093721692949*a^4 + 1912912387957623213/3840885093721692949*a^3 + 1540560395597941819/3840885093721692949*a^2 - 442675401304579583/3840885093721692949*a - 1085362195855512780/3840885093721692949, 1/3840885093721692949*a^45 + 42574685027696640/3840885093721692949*a^23 + 1777115024158511940/3840885093721692949*a^22 - 406527168154163989/3840885093721692949*a^21 - 504148903224244823/3840885093721692949*a^20 - 1865218489581265568/3840885093721692949*a^19 - 1862590493500949655/3840885093721692949*a^18 + 1662195188740362985/3840885093721692949*a^17 - 882951589493935151/3840885093721692949*a^16 - 922276941593365418/3840885093721692949*a^15 - 1035489129088145927/3840885093721692949*a^14 - 367723943524953009/3840885093721692949*a^13 - 1519985802794128054/3840885093721692949*a^12 + 1841235936775543253/3840885093721692949*a^11 - 843736132992546696/3840885093721692949*a^10 - 1556889542649559693/3840885093721692949*a^9 + 1516370182428166627/3840885093721692949*a^8 + 1027631738826801623/3840885093721692949*a^7 - 1112428041041949879/3840885093721692949*a^6 - 1465195187495006939/3840885093721692949*a^5 - 1749557567945584703/3840885093721692949*a^4 + 1266763995948389033/3840885093721692949*a^3 + 1900255784580953160/3840885093721692949*a^2 + 610713501354428963/3840885093721692949*a + 343545348867068094/3840885093721692949

Class group and class number

not computed

Unit group

Rank:		$22$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		not computed	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		not computed	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

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):

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

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
\(17\) 17		Deg $46$	$2$	$23$	$23$
\(47\) 47		Deg $46$	$46$	$1$	$45$