Number field 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1

• Label: 44.0.342...125.1
• Degree: $44$
• Signature: $[0, 22]$
• Discriminant: $3.429\times 10^{77}$
• Root discriminant: \(57.83\)
• Ramified primes: $5,23$
• Class number: $45013$ (GRH)
• Class group: $[45013]$ (GRH)
• Galois group: $C_{44}$ (as 44T1)

Normalized defining polynomial

x^44 - x^43 + 11*x^42 - 12*x^41 + 77*x^40 - 74*x^39 + 424*x^38 - 368*x^37 + 2052*x^36 - 1675*x^35 + 8154*x^34 - 6259*x^33 + 27860*x^32 - 18614*x^31 + 82991*x^30 - 50150*x^29 + 218995*x^28 - 122605*x^27 + 487495*x^26 - 239465*x^25 + 942321*x^24 - 367121*x^23 + 1562162*x^22 - 556097*x^21 + 2230394*x^20 - 742219*x^19 + 2512113*x^18 - 607127*x^17 + 2369760*x^16 - 263798*x^15 + 1750478*x^14 - 273124*x^13 + 1076097*x^12 - 237409*x^11 + 402111*x^10 - 41911*x^9 + 123922*x^8 + 19642*x^7 + 17600*x^6 + 1439*x^5 + 2686*x^4 - 361*x^3 + 51*x^2 - 6*x + 1

Invariants

Degree:		$44$
Signature:		$[0, 22]$
Discriminant:		\(342865339180420288801608222738062084913425127327306009945459663867950439453125\) \(\medspace = 5^{33}\cdot 23^{40}\)
Root discriminant:		\(57.83\)
Ramified primes:		\(5\), \(23\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$44$
This field is Galois and abelian over $\Q$.
Conductor:		\(115=5\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(2,·)$, $\chi_{115}(3,·)$, $\chi_{115}(4,·)$, $\chi_{115}(6,·)$, $\chi_{115}(8,·)$, $\chi_{115}(9,·)$, $\chi_{115}(12,·)$, $\chi_{115}(13,·)$, $\chi_{115}(16,·)$, $\chi_{115}(18,·)$, $\chi_{115}(24,·)$, $\chi_{115}(26,·)$, $\chi_{115}(27,·)$, $\chi_{115}(29,·)$, $\chi_{115}(31,·)$, $\chi_{115}(32,·)$, $\chi_{115}(36,·)$, $\chi_{115}(39,·)$, $\chi_{115}(41,·)$, $\chi_{115}(47,·)$, $\chi_{115}(48,·)$, $\chi_{115}(49,·)$, $\chi_{115}(52,·)$, $\chi_{115}(54,·)$, $\chi_{115}(58,·)$, $\chi_{115}(59,·)$, $\chi_{115}(62,·)$, $\chi_{115}(64,·)$, $\chi_{115}(71,·)$, $\chi_{115}(72,·)$, $\chi_{115}(73,·)$, $\chi_{115}(77,·)$, $\chi_{115}(78,·)$, $\chi_{115}(81,·)$, $\chi_{115}(82,·)$, $\chi_{115}(87,·)$, $\chi_{115}(93,·)$, $\chi_{115}(94,·)$, $\chi_{115}(96,·)$, $\chi_{115}(98,·)$, $\chi_{115}(101,·)$, $\chi_{115}(104,·)$, $\chi_{115}(108,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $\frac{1}{91\!\cdots\!21}a^{41}-\frac{91\!\cdots\!50}{91\!\cdots\!21}a^{40}+\frac{22\!\cdots\!07}{91\!\cdots\!21}a^{39}-\frac{32\!\cdots\!98}{91\!\cdots\!21}a^{38}-\frac{21\!\cdots\!81}{91\!\cdots\!21}a^{37}-\frac{14\!\cdots\!13}{91\!\cdots\!21}a^{36}-\frac{57\!\cdots\!32}{91\!\cdots\!21}a^{35}+\frac{41\!\cdots\!08}{91\!\cdots\!21}a^{34}+\frac{41\!\cdots\!13}{91\!\cdots\!21}a^{33}+\frac{12\!\cdots\!02}{91\!\cdots\!21}a^{32}-\frac{25\!\cdots\!84}{91\!\cdots\!21}a^{31}-\frac{23\!\cdots\!58}{91\!\cdots\!21}a^{30}+\frac{40\!\cdots\!59}{91\!\cdots\!21}a^{29}+\frac{20\!\cdots\!39}{91\!\cdots\!21}a^{28}-\frac{16\!\cdots\!56}{91\!\cdots\!21}a^{27}+\frac{13\!\cdots\!12}{91\!\cdots\!21}a^{26}-\frac{41\!\cdots\!29}{91\!\cdots\!21}a^{25}+\frac{17\!\cdots\!65}{91\!\cdots\!21}a^{24}+\frac{39\!\cdots\!90}{91\!\cdots\!21}a^{23}-\frac{31\!\cdots\!57}{91\!\cdots\!21}a^{22}-\frac{10\!\cdots\!61}{91\!\cdots\!21}a^{21}+\frac{42\!\cdots\!85}{91\!\cdots\!21}a^{20}-\frac{30\!\cdots\!43}{91\!\cdots\!21}a^{19}-\frac{12\!\cdots\!02}{91\!\cdots\!21}a^{18}+\frac{41\!\cdots\!80}{91\!\cdots\!21}a^{17}+\frac{22\!\cdots\!83}{91\!\cdots\!21}a^{16}-\frac{11\!\cdots\!91}{91\!\cdots\!21}a^{15}-\frac{45\!\cdots\!82}{91\!\cdots\!21}a^{14}+\frac{19\!\cdots\!42}{91\!\cdots\!21}a^{13}+\frac{71\!\cdots\!75}{91\!\cdots\!21}a^{12}+\frac{44\!\cdots\!82}{91\!\cdots\!21}a^{11}-\frac{38\!\cdots\!32}{91\!\cdots\!21}a^{10}+\frac{24\!\cdots\!71}{91\!\cdots\!21}a^{9}-\frac{14\!\cdots\!53}{91\!\cdots\!21}a^{8}-\frac{97\!\cdots\!15}{91\!\cdots\!21}a^{7}+\frac{33\!\cdots\!73}{91\!\cdots\!21}a^{6}+\frac{31\!\cdots\!36}{91\!\cdots\!21}a^{5}-\frac{90\!\cdots\!30}{91\!\cdots\!21}a^{4}-\frac{44\!\cdots\!72}{91\!\cdots\!21}a^{3}+\frac{29\!\cdots\!60}{91\!\cdots\!21}a^{2}-\frac{22\!\cdots\!84}{91\!\cdots\!21}a-\frac{20\!\cdots\!15}{91\!\cdots\!21}$, $\frac{1}{91\!\cdots\!21}a^{42}+\frac{25\!\cdots\!89}{91\!\cdots\!21}a^{40}-\frac{21\!\cdots\!15}{91\!\cdots\!21}a^{39}+\frac{36\!\cdots\!06}{91\!\cdots\!21}a^{38}-\frac{30\!\cdots\!53}{91\!\cdots\!21}a^{37}+\frac{35\!\cdots\!12}{91\!\cdots\!21}a^{36}-\frac{11\!\cdots\!31}{91\!\cdots\!21}a^{35}+\frac{35\!\cdots\!97}{91\!\cdots\!21}a^{34}-\frac{18\!\cdots\!46}{91\!\cdots\!21}a^{33}+\frac{33\!\cdots\!36}{91\!\cdots\!21}a^{32}-\frac{40\!\cdots\!02}{91\!\cdots\!21}a^{31}+\frac{32\!\cdots\!63}{91\!\cdots\!21}a^{30}+\frac{30\!\cdots\!81}{91\!\cdots\!21}a^{29}-\frac{39\!\cdots\!08}{91\!\cdots\!21}a^{28}+\frac{30\!\cdots\!78}{91\!\cdots\!21}a^{27}+\frac{15\!\cdots\!03}{91\!\cdots\!21}a^{26}+\frac{26\!\cdots\!92}{91\!\cdots\!21}a^{25}+\frac{28\!\cdots\!13}{91\!\cdots\!21}a^{24}+\frac{21\!\cdots\!75}{91\!\cdots\!21}a^{23}+\frac{35\!\cdots\!59}{91\!\cdots\!21}a^{22}-\frac{42\!\cdots\!11}{91\!\cdots\!21}a^{21}-\frac{10\!\cdots\!18}{91\!\cdots\!21}a^{20}+\frac{12\!\cdots\!84}{91\!\cdots\!21}a^{19}+\frac{43\!\cdots\!35}{91\!\cdots\!21}a^{18}-\frac{14\!\cdots\!68}{91\!\cdots\!21}a^{17}-\frac{26\!\cdots\!80}{91\!\cdots\!21}a^{16}+\frac{21\!\cdots\!65}{91\!\cdots\!21}a^{15}+\frac{42\!\cdots\!33}{91\!\cdots\!21}a^{14}+\frac{39\!\cdots\!44}{91\!\cdots\!21}a^{13}+\frac{33\!\cdots\!56}{91\!\cdots\!21}a^{12}+\frac{25\!\cdots\!64}{91\!\cdots\!21}a^{11}-\frac{20\!\cdots\!26}{91\!\cdots\!21}a^{10}+\frac{17\!\cdots\!34}{91\!\cdots\!21}a^{9}-\frac{34\!\cdots\!54}{91\!\cdots\!21}a^{8}-\frac{32\!\cdots\!47}{91\!\cdots\!21}a^{7}+\frac{21\!\cdots\!33}{91\!\cdots\!21}a^{6}-\frac{23\!\cdots\!09}{91\!\cdots\!21}a^{5}-\frac{26\!\cdots\!96}{91\!\cdots\!21}a^{4}-\frac{13\!\cdots\!34}{91\!\cdots\!21}a^{3}-\frac{35\!\cdots\!32}{91\!\cdots\!21}a^{2}-\frac{29\!\cdots\!78}{91\!\cdots\!21}a+\frac{11\!\cdots\!83}{91\!\cdots\!21}$, $\frac{1}{91\!\cdots\!21}a^{43}+\frac{77\!\cdots\!68}{91\!\cdots\!21}a^{40}-\frac{17\!\cdots\!81}{91\!\cdots\!21}a^{39}-\frac{41\!\cdots\!58}{91\!\cdots\!21}a^{38}-\frac{14\!\cdots\!33}{91\!\cdots\!21}a^{37}-\frac{17\!\cdots\!05}{91\!\cdots\!21}a^{36}+\frac{34\!\cdots\!08}{91\!\cdots\!21}a^{35}+\frac{28\!\cdots\!87}{91\!\cdots\!21}a^{34}+\frac{38\!\cdots\!14}{91\!\cdots\!21}a^{33}+\frac{11\!\cdots\!04}{91\!\cdots\!21}a^{32}+\frac{22\!\cdots\!26}{91\!\cdots\!21}a^{31}-\frac{10\!\cdots\!58}{91\!\cdots\!21}a^{30}+\frac{37\!\cdots\!80}{91\!\cdots\!21}a^{29}+\frac{27\!\cdots\!28}{91\!\cdots\!21}a^{28}+\frac{12\!\cdots\!35}{91\!\cdots\!21}a^{27}+\frac{42\!\cdots\!05}{91\!\cdots\!21}a^{26}-\frac{39\!\cdots\!12}{91\!\cdots\!21}a^{25}+\frac{14\!\cdots\!73}{91\!\cdots\!21}a^{24}+\frac{20\!\cdots\!31}{91\!\cdots\!21}a^{23}+\frac{45\!\cdots\!39}{91\!\cdots\!21}a^{22}+\frac{40\!\cdots\!22}{91\!\cdots\!21}a^{21}-\frac{39\!\cdots\!95}{91\!\cdots\!21}a^{20}-\frac{10\!\cdots\!57}{91\!\cdots\!21}a^{19}-\frac{43\!\cdots\!28}{91\!\cdots\!21}a^{18}+\frac{20\!\cdots\!14}{91\!\cdots\!21}a^{17}-\frac{73\!\cdots\!48}{91\!\cdots\!21}a^{16}+\frac{28\!\cdots\!60}{91\!\cdots\!21}a^{15}-\frac{15\!\cdots\!32}{91\!\cdots\!21}a^{14}-\frac{16\!\cdots\!12}{91\!\cdots\!21}a^{13}+\frac{39\!\cdots\!48}{91\!\cdots\!21}a^{12}+\frac{23\!\cdots\!88}{91\!\cdots\!21}a^{11}+\frac{26\!\cdots\!27}{91\!\cdots\!21}a^{10}+\frac{20\!\cdots\!17}{91\!\cdots\!21}a^{9}+\frac{12\!\cdots\!04}{91\!\cdots\!21}a^{8}+\frac{50\!\cdots\!93}{91\!\cdots\!21}a^{7}+\frac{17\!\cdots\!10}{91\!\cdots\!21}a^{6}-\frac{27\!\cdots\!75}{91\!\cdots\!21}a^{5}-\frac{35\!\cdots\!74}{91\!\cdots\!21}a^{4}+\frac{40\!\cdots\!72}{91\!\cdots\!21}a^{3}-\frac{17\!\cdots\!39}{91\!\cdots\!21}a^{2}+\frac{20\!\cdots\!25}{91\!\cdots\!21}a+\frac{43\!\cdots\!39}{91\!\cdots\!21}$

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

Class group and class number

$C_{45013}$, which has order $45013$ (assuming GRH)

Unit group

Rank:		$21$
Torsion generator:		\( -\frac{143504380309978050139881567973335320096946001}{9136101057353736188751549898001468069294153521} a^{43} + \frac{81368225088904635639643242222958289840485360}{9136101057353736188751549898001468069294153521} a^{42} - \frac{1492231294800999393499889646283360262044835668}{9136101057353736188751549898001468069294153521} a^{41} + \frac{1019370527770824485026681601720312810602393430}{9136101057353736188751549898001468069294153521} a^{40} - \frac{10042664933624921435671222612358500651410850334}{9136101057353736188751549898001468069294153521} a^{39} + \frac{5598518894712457990681568250129869301219118290}{9136101057353736188751549898001468069294153521} a^{38} - \frac{54439443092304226137092628289942470954432716948}{9136101057353736188751549898001468069294153521} a^{37} + \frac{25046768213369137622452464058667318173761005651}{9136101057353736188751549898001468069294153521} a^{36} - \frac{261675475689135149812732129394105464547887305550}{9136101057353736188751549898001468069294153521} a^{35} + \frac{106006149934327707187186860862887429199777655855}{9136101057353736188751549898001468069294153521} a^{34} - \frac{1018020153523507803054954131696212731384989055983}{9136101057353736188751549898001468069294153521} a^{33} + \frac{360868913551724277449830863493937698706480336055}{9136101057353736188751549898001468069294153521} a^{32} - \frac{3419087929686705779392212790150592656405732168743}{9136101057353736188751549898001468069294153521} a^{31} + \frac{827520078509455974126373061859624784677633384470}{9136101057353736188751549898001468069294153521} a^{30} - \frac{10105632475745477178767443068810705542601985118826}{9136101057353736188751549898001468069294153521} a^{29} + \frac{1721655411985498108932461991190925395560931023439}{9136101057353736188751549898001468069294153521} a^{28} - \frac{26379968816467594647512817238674814140808086761957}{9136101057353736188751549898001468069294153521} a^{27} + \frac{3165153959303854936220694197994417993417747525060}{9136101057353736188751549898001468069294153521} a^{26} - \frac{57245272100169277205124608505170969150184789408770}{9136101057353736188751549898001468069294153521} a^{25} + \frac{2135469037678029195969847258311039512705177671261}{9136101057353736188751549898001468069294153521} a^{24} - \frac{109043960405876300104464918727943361473069774995405}{9136101057353736188751549898001468069294153521} a^{23} - \frac{9394367245362583061729845311873119449368514197033}{9136101057353736188751549898001468069294153521} a^{22} - \frac{179492447323669580307644301966947668061487024905260}{9136101057353736188751549898001468069294153521} a^{21} - \frac{21789532311278591125661166913986888157234519547176}{9136101057353736188751549898001468069294153521} a^{20} - \frac{249045211393859148621624433309241448155897060631260}{9136101057353736188751549898001468069294153521} a^{19} - \frac{38346669529210729755020489975763052316588856466547}{9136101057353736188751549898001468069294153521} a^{18} - \frac{262393676597008755852667438862819053492774408920205}{9136101057353736188751549898001468069294153521} a^{17} - \frac{76776809216200678040852496849808737961296716745800}{9136101057353736188751549898001468069294153521} a^{16} - \frac{244152822345396638525731568775136429978233150069260}{9136101057353736188751549898001468069294153521} a^{15} - \frac{112902416032775381176555799479769868186552088794493}{9136101057353736188751549898001468069294153521} a^{14} - \frac{179290617172262217286376867432743846823222431058505}{9136101057353736188751549898001468069294153521} a^{13} - \frac{65470244859845637792486912705750698987242722132550}{9136101057353736188751549898001468069294153521} a^{12} - \frac{95430799894192649077541424315906600459814682525175}{9136101057353736188751549898001468069294153521} a^{11} - \frac{31748909586203858691990761692951136137711540127758}{9136101057353736188751549898001468069294153521} a^{10} - \frac{17759400710147720196586546868252742926126088090524}{9136101057353736188751549898001468069294153521} a^{9} - \frac{20183709730699723087162238404965531540012392192071}{9136101057353736188751549898001468069294153521} a^{8} - \frac{6296016141516024306135728642564513083607941961208}{9136101057353736188751549898001468069294153521} a^{7} - \frac{10049103659438267859668632986449870027282711753670}{9136101057353736188751549898001468069294153521} a^{6} - \frac{858010346659704501968645248274200275556980986171}{9136101057353736188751549898001468069294153521} a^{5} - \frac{298361613164131089770410931908806186529504065289}{9136101057353736188751549898001468069294153521} a^{4} + \frac{39651548858446898930005679224874141617425029253}{9136101057353736188751549898001468069294153521} a^{3} - \frac{5840288335645027161315799389478167811794293879}{9136101057353736188751549898001468069294153521} a^{2} + \frac{67219294366979430088793002262755859468098366401}{9136101057353736188751549898001468069294153521} a - \frac{143806742197913588744033711164446545109789525}{9136101057353736188751549898001468069294153521} \)  (order $10$)
Fundamental units:		$\frac{30\!\cdots\!95}{91\!\cdots\!21}a^{43}-\frac{43\!\cdots\!56}{91\!\cdots\!21}a^{42}+\frac{35\!\cdots\!10}{91\!\cdots\!21}a^{41}-\frac{50\!\cdots\!60}{91\!\cdots\!21}a^{40}+\frac{25\!\cdots\!78}{91\!\cdots\!21}a^{39}-\frac{32\!\cdots\!05}{91\!\cdots\!21}a^{38}+\frac{13\!\cdots\!66}{91\!\cdots\!21}a^{37}-\frac{16\!\cdots\!95}{91\!\cdots\!21}a^{36}+\frac{67\!\cdots\!12}{91\!\cdots\!21}a^{35}-\frac{76\!\cdots\!27}{91\!\cdots\!21}a^{34}+\frac{26\!\cdots\!64}{91\!\cdots\!21}a^{33}-\frac{29\!\cdots\!85}{91\!\cdots\!21}a^{32}+\frac{92\!\cdots\!75}{91\!\cdots\!21}a^{31}-\frac{90\!\cdots\!38}{91\!\cdots\!21}a^{30}+\frac{27\!\cdots\!87}{91\!\cdots\!21}a^{29}-\frac{25\!\cdots\!31}{91\!\cdots\!21}a^{28}+\frac{72\!\cdots\!47}{91\!\cdots\!21}a^{27}-\frac{64\!\cdots\!10}{91\!\cdots\!21}a^{26}+\frac{16\!\cdots\!80}{91\!\cdots\!21}a^{25}-\frac{13\!\cdots\!70}{91\!\cdots\!21}a^{24}+\frac{31\!\cdots\!80}{91\!\cdots\!21}a^{23}-\frac{22\!\cdots\!29}{91\!\cdots\!21}a^{22}+\frac{50\!\cdots\!80}{91\!\cdots\!21}a^{21}-\frac{35\!\cdots\!48}{91\!\cdots\!21}a^{20}+\frac{72\!\cdots\!40}{91\!\cdots\!21}a^{19}-\frac{49\!\cdots\!91}{91\!\cdots\!21}a^{18}+\frac{82\!\cdots\!49}{91\!\cdots\!21}a^{17}-\frac{48\!\cdots\!62}{91\!\cdots\!21}a^{16}+\frac{75\!\cdots\!42}{91\!\cdots\!21}a^{15}-\frac{36\!\cdots\!49}{91\!\cdots\!21}a^{14}+\frac{52\!\cdots\!34}{91\!\cdots\!21}a^{13}-\frac{29\!\cdots\!50}{91\!\cdots\!21}a^{12}+\frac{33\!\cdots\!43}{91\!\cdots\!21}a^{11}-\frac{20\!\cdots\!44}{91\!\cdots\!21}a^{10}+\frac{13\!\cdots\!01}{91\!\cdots\!21}a^{9}-\frac{57\!\cdots\!51}{91\!\cdots\!21}a^{8}+\frac{35\!\cdots\!14}{91\!\cdots\!21}a^{7}-\frac{84\!\cdots\!78}{91\!\cdots\!21}a^{6}+\frac{45\!\cdots\!83}{91\!\cdots\!21}a^{5}-\frac{21\!\cdots\!06}{91\!\cdots\!21}a^{4}+\frac{29\!\cdots\!98}{91\!\cdots\!21}a^{3}-\frac{52\!\cdots\!07}{91\!\cdots\!21}a^{2}+\frac{50\!\cdots\!85}{91\!\cdots\!21}a-\frac{98\!\cdots\!51}{91\!\cdots\!21}$, $\frac{43\!\cdots\!25}{91\!\cdots\!21}a^{43}-\frac{87\!\cdots\!65}{91\!\cdots\!21}a^{42}+\frac{50\!\cdots\!00}{91\!\cdots\!21}a^{41}-\frac{97\!\cdots\!30}{91\!\cdots\!21}a^{40}+\frac{36\!\cdots\!75}{91\!\cdots\!21}a^{39}-\frac{62\!\cdots\!60}{91\!\cdots\!21}a^{38}+\frac{20\!\cdots\!50}{91\!\cdots\!21}a^{37}-\frac{32\!\cdots\!45}{91\!\cdots\!21}a^{36}+\frac{97\!\cdots\!46}{91\!\cdots\!21}a^{35}-\frac{15\!\cdots\!50}{91\!\cdots\!21}a^{34}+\frac{38\!\cdots\!50}{91\!\cdots\!21}a^{33}-\frac{58\!\cdots\!20}{91\!\cdots\!21}a^{32}+\frac{13\!\cdots\!00}{91\!\cdots\!21}a^{31}-\frac{18\!\cdots\!95}{91\!\cdots\!21}a^{30}+\frac{38\!\cdots\!50}{91\!\cdots\!21}a^{29}-\frac{52\!\cdots\!90}{91\!\cdots\!21}a^{28}+\frac{10\!\cdots\!60}{91\!\cdots\!21}a^{27}-\frac{13\!\cdots\!80}{91\!\cdots\!21}a^{26}+\frac{22\!\cdots\!08}{91\!\cdots\!21}a^{25}-\frac{27\!\cdots\!25}{91\!\cdots\!21}a^{24}+\frac{41\!\cdots\!65}{91\!\cdots\!21}a^{23}-\frac{49\!\cdots\!75}{91\!\cdots\!21}a^{22}+\frac{65\!\cdots\!30}{91\!\cdots\!21}a^{21}-\frac{78\!\cdots\!00}{91\!\cdots\!21}a^{20}+\frac{90\!\cdots\!60}{91\!\cdots\!21}a^{19}-\frac{10\!\cdots\!75}{91\!\cdots\!21}a^{18}+\frac{98\!\cdots\!45}{91\!\cdots\!21}a^{17}-\frac{10\!\cdots\!50}{91\!\cdots\!21}a^{16}+\frac{80\!\cdots\!94}{91\!\cdots\!21}a^{15}-\frac{85\!\cdots\!00}{91\!\cdots\!21}a^{14}+\frac{42\!\cdots\!30}{91\!\cdots\!21}a^{13}-\frac{66\!\cdots\!75}{91\!\cdots\!21}a^{12}+\frac{27\!\cdots\!75}{91\!\cdots\!21}a^{11}-\frac{39\!\cdots\!90}{91\!\cdots\!21}a^{10}+\frac{79\!\cdots\!80}{91\!\cdots\!21}a^{9}-\frac{72\!\cdots\!85}{91\!\cdots\!21}a^{8}-\frac{71\!\cdots\!65}{91\!\cdots\!21}a^{7}-\frac{10\!\cdots\!45}{91\!\cdots\!21}a^{6}-\frac{22\!\cdots\!77}{91\!\cdots\!21}a^{5}-\frac{19\!\cdots\!65}{91\!\cdots\!21}a^{4}+\frac{26\!\cdots\!65}{91\!\cdots\!21}a^{3}-\frac{37\!\cdots\!05}{91\!\cdots\!21}a^{2}+\frac{45\!\cdots\!65}{91\!\cdots\!21}a+\frac{12\!\cdots\!80}{91\!\cdots\!21}$, $\frac{19\!\cdots\!00}{91\!\cdots\!21}a^{43}-\frac{38\!\cdots\!10}{91\!\cdots\!21}a^{42}+\frac{22\!\cdots\!50}{91\!\cdots\!21}a^{41}-\frac{42\!\cdots\!41}{91\!\cdots\!21}a^{40}+\frac{16\!\cdots\!00}{91\!\cdots\!21}a^{39}-\frac{27\!\cdots\!40}{91\!\cdots\!21}a^{38}+\frac{88\!\cdots\!00}{91\!\cdots\!21}a^{37}-\frac{14\!\cdots\!30}{91\!\cdots\!21}a^{36}+\frac{42\!\cdots\!50}{91\!\cdots\!21}a^{35}-\frac{66\!\cdots\!50}{91\!\cdots\!21}a^{34}+\frac{16\!\cdots\!50}{91\!\cdots\!21}a^{33}-\frac{25\!\cdots\!80}{91\!\cdots\!21}a^{32}+\frac{57\!\cdots\!50}{91\!\cdots\!21}a^{31}-\frac{81\!\cdots\!17}{91\!\cdots\!21}a^{30}+\frac{16\!\cdots\!50}{91\!\cdots\!21}a^{29}-\frac{23\!\cdots\!60}{91\!\cdots\!21}a^{28}+\frac{44\!\cdots\!40}{91\!\cdots\!21}a^{27}-\frac{58\!\cdots\!20}{91\!\cdots\!21}a^{26}+\frac{97\!\cdots\!15}{91\!\cdots\!21}a^{25}-\frac{12\!\cdots\!00}{91\!\cdots\!21}a^{24}+\frac{18\!\cdots\!10}{91\!\cdots\!21}a^{23}-\frac{21\!\cdots\!50}{91\!\cdots\!21}a^{22}+\frac{28\!\cdots\!70}{91\!\cdots\!21}a^{21}-\frac{34\!\cdots\!20}{91\!\cdots\!21}a^{20}+\frac{39\!\cdots\!40}{91\!\cdots\!21}a^{19}-\frac{47\!\cdots\!00}{91\!\cdots\!21}a^{18}+\frac{42\!\cdots\!30}{91\!\cdots\!21}a^{17}-\frac{46\!\cdots\!00}{91\!\cdots\!21}a^{16}+\frac{34\!\cdots\!60}{91\!\cdots\!21}a^{15}-\frac{37\!\cdots\!00}{91\!\cdots\!21}a^{14}+\frac{18\!\cdots\!70}{91\!\cdots\!21}a^{13}-\frac{29\!\cdots\!00}{91\!\cdots\!21}a^{12}+\frac{11\!\cdots\!00}{91\!\cdots\!21}a^{11}-\frac{17\!\cdots\!89}{91\!\cdots\!21}a^{10}+\frac{34\!\cdots\!70}{91\!\cdots\!21}a^{9}-\frac{31\!\cdots\!40}{91\!\cdots\!21}a^{8}-\frac{31\!\cdots\!60}{91\!\cdots\!21}a^{7}-\frac{46\!\cdots\!30}{91\!\cdots\!21}a^{6}-\frac{88\!\cdots\!45}{91\!\cdots\!21}a^{5}-\frac{86\!\cdots\!10}{91\!\cdots\!21}a^{4}+\frac{11\!\cdots\!60}{91\!\cdots\!21}a^{3}-\frac{16\!\cdots\!70}{91\!\cdots\!21}a^{2}+\frac{19\!\cdots\!10}{91\!\cdots\!21}a+\frac{16\!\cdots\!38}{91\!\cdots\!21}$, $\frac{71\!\cdots\!10}{91\!\cdots\!21}a^{43}-\frac{14\!\cdots\!66}{91\!\cdots\!21}a^{42}+\frac{82\!\cdots\!40}{91\!\cdots\!21}a^{41}-\frac{15\!\cdots\!02}{91\!\cdots\!21}a^{40}+\frac{59\!\cdots\!50}{91\!\cdots\!21}a^{39}-\frac{10\!\cdots\!04}{91\!\cdots\!21}a^{38}+\frac{32\!\cdots\!44}{91\!\cdots\!21}a^{37}-\frac{52\!\cdots\!18}{91\!\cdots\!21}a^{36}+\frac{15\!\cdots\!88}{91\!\cdots\!21}a^{35}-\frac{24\!\cdots\!80}{91\!\cdots\!21}a^{34}+\frac{63\!\cdots\!16}{91\!\cdots\!21}a^{33}-\frac{94\!\cdots\!64}{91\!\cdots\!21}a^{32}+\frac{21\!\cdots\!32}{91\!\cdots\!21}a^{31}-\frac{30\!\cdots\!19}{91\!\cdots\!21}a^{30}+\frac{63\!\cdots\!20}{91\!\cdots\!21}a^{29}-\frac{85\!\cdots\!60}{91\!\cdots\!21}a^{28}+\frac{16\!\cdots\!40}{91\!\cdots\!21}a^{27}-\frac{21\!\cdots\!20}{91\!\cdots\!21}a^{26}+\frac{36\!\cdots\!59}{91\!\cdots\!21}a^{25}-\frac{45\!\cdots\!50}{91\!\cdots\!21}a^{24}+\frac{67\!\cdots\!06}{91\!\cdots\!21}a^{23}-\frac{79\!\cdots\!10}{91\!\cdots\!21}a^{22}+\frac{10\!\cdots\!52}{91\!\cdots\!21}a^{21}-\frac{12\!\cdots\!20}{91\!\cdots\!21}a^{20}+\frac{14\!\cdots\!04}{91\!\cdots\!21}a^{19}-\frac{17\!\cdots\!74}{91\!\cdots\!21}a^{18}+\frac{15\!\cdots\!78}{91\!\cdots\!21}a^{17}-\frac{17\!\cdots\!08}{91\!\cdots\!21}a^{16}+\frac{13\!\cdots\!10}{91\!\cdots\!21}a^{15}-\frac{13\!\cdots\!36}{91\!\cdots\!21}a^{14}+\frac{68\!\cdots\!68}{91\!\cdots\!21}a^{13}-\frac{10\!\cdots\!42}{91\!\cdots\!21}a^{12}+\frac{44\!\cdots\!82}{91\!\cdots\!21}a^{11}-\frac{63\!\cdots\!65}{91\!\cdots\!21}a^{10}+\frac{12\!\cdots\!76}{91\!\cdots\!21}a^{9}-\frac{11\!\cdots\!26}{91\!\cdots\!21}a^{8}-\frac{11\!\cdots\!78}{91\!\cdots\!21}a^{7}-\frac{17\!\cdots\!26}{91\!\cdots\!21}a^{6}-\frac{42\!\cdots\!89}{91\!\cdots\!21}a^{5}-\frac{32\!\cdots\!50}{91\!\cdots\!21}a^{4}+\frac{43\!\cdots\!90}{91\!\cdots\!21}a^{3}-\frac{60\!\cdots\!30}{91\!\cdots\!21}a^{2}+\frac{73\!\cdots\!70}{91\!\cdots\!21}a+\frac{12\!\cdots\!96}{91\!\cdots\!21}$, $\frac{24\!\cdots\!50}{91\!\cdots\!21}a^{43}-\frac{49\!\cdots\!50}{91\!\cdots\!21}a^{42}+\frac{28\!\cdots\!00}{91\!\cdots\!21}a^{41}-\frac{55\!\cdots\!75}{91\!\cdots\!21}a^{40}+\frac{20\!\cdots\!50}{91\!\cdots\!21}a^{39}-\frac{35\!\cdots\!00}{91\!\cdots\!21}a^{38}+\frac{11\!\cdots\!00}{91\!\cdots\!21}a^{37}-\frac{18\!\cdots\!50}{91\!\cdots\!21}a^{36}+\frac{55\!\cdots\!75}{91\!\cdots\!21}a^{35}-\frac{86\!\cdots\!00}{91\!\cdots\!21}a^{34}+\frac{22\!\cdots\!00}{91\!\cdots\!21}a^{33}-\frac{33\!\cdots\!00}{91\!\cdots\!21}a^{32}+\frac{75\!\cdots\!00}{91\!\cdots\!21}a^{31}-\frac{10\!\cdots\!00}{91\!\cdots\!21}a^{30}+\frac{21\!\cdots\!00}{91\!\cdots\!21}a^{29}-\frac{29\!\cdots\!00}{91\!\cdots\!21}a^{28}+\frac{57\!\cdots\!00}{91\!\cdots\!21}a^{27}-\frac{75\!\cdots\!00}{91\!\cdots\!21}a^{26}+\frac{12\!\cdots\!29}{91\!\cdots\!21}a^{25}-\frac{15\!\cdots\!50}{91\!\cdots\!21}a^{24}+\frac{23\!\cdots\!50}{91\!\cdots\!21}a^{23}-\frac{27\!\cdots\!50}{91\!\cdots\!21}a^{22}+\frac{37\!\cdots\!00}{91\!\cdots\!21}a^{21}-\frac{44\!\cdots\!00}{91\!\cdots\!21}a^{20}+\frac{51\!\cdots\!00}{91\!\cdots\!21}a^{19}-\frac{61\!\cdots\!50}{91\!\cdots\!21}a^{18}+\frac{55\!\cdots\!50}{91\!\cdots\!21}a^{17}-\frac{60\!\cdots\!00}{91\!\cdots\!21}a^{16}+\frac{45\!\cdots\!55}{91\!\cdots\!21}a^{15}-\frac{48\!\cdots\!00}{91\!\cdots\!21}a^{14}+\frac{23\!\cdots\!00}{91\!\cdots\!21}a^{13}-\frac{37\!\cdots\!50}{91\!\cdots\!21}a^{12}+\frac{15\!\cdots\!50}{91\!\cdots\!21}a^{11}-\frac{22\!\cdots\!00}{91\!\cdots\!21}a^{10}+\frac{44\!\cdots\!00}{91\!\cdots\!21}a^{9}-\frac{41\!\cdots\!50}{91\!\cdots\!21}a^{8}-\frac{40\!\cdots\!50}{91\!\cdots\!21}a^{7}-\frac{59\!\cdots\!50}{91\!\cdots\!21}a^{6}-\frac{11\!\cdots\!05}{91\!\cdots\!21}a^{5}-\frac{11\!\cdots\!50}{91\!\cdots\!21}a^{4}+\frac{15\!\cdots\!50}{91\!\cdots\!21}a^{3}-\frac{21\!\cdots\!50}{91\!\cdots\!21}a^{2}+\frac{25\!\cdots\!50}{91\!\cdots\!21}a+\frac{18\!\cdots\!50}{91\!\cdots\!21}$, $\frac{49\!\cdots\!60}{91\!\cdots\!21}a^{43}-\frac{10\!\cdots\!90}{91\!\cdots\!21}a^{42}+\frac{58\!\cdots\!30}{91\!\cdots\!21}a^{41}-\frac{11\!\cdots\!80}{91\!\cdots\!21}a^{40}+\frac{42\!\cdots\!20}{91\!\cdots\!21}a^{39}-\frac{72\!\cdots\!20}{91\!\cdots\!21}a^{38}+\frac{23\!\cdots\!24}{91\!\cdots\!21}a^{37}-\frac{37\!\cdots\!70}{91\!\cdots\!21}a^{36}+\frac{11\!\cdots\!28}{91\!\cdots\!21}a^{35}-\frac{17\!\cdots\!30}{91\!\cdots\!21}a^{34}+\frac{44\!\cdots\!66}{91\!\cdots\!21}a^{33}-\frac{67\!\cdots\!36}{91\!\cdots\!21}a^{32}+\frac{15\!\cdots\!62}{91\!\cdots\!21}a^{31}-\frac{21\!\cdots\!52}{91\!\cdots\!21}a^{30}+\frac{44\!\cdots\!50}{91\!\cdots\!21}a^{29}-\frac{60\!\cdots\!04}{91\!\cdots\!21}a^{28}+\frac{11\!\cdots\!56}{91\!\cdots\!21}a^{27}-\frac{15\!\cdots\!28}{91\!\cdots\!21}a^{26}+\frac{25\!\cdots\!87}{91\!\cdots\!21}a^{25}-\frac{32\!\cdots\!80}{91\!\cdots\!21}a^{24}+\frac{48\!\cdots\!70}{91\!\cdots\!21}a^{23}-\frac{56\!\cdots\!30}{91\!\cdots\!21}a^{22}+\frac{75\!\cdots\!10}{91\!\cdots\!21}a^{21}-\frac{91\!\cdots\!81}{91\!\cdots\!21}a^{20}+\frac{10\!\cdots\!80}{91\!\cdots\!21}a^{19}-\frac{12\!\cdots\!84}{91\!\cdots\!21}a^{18}+\frac{11\!\cdots\!50}{91\!\cdots\!21}a^{17}-\frac{12\!\cdots\!68}{91\!\cdots\!21}a^{16}+\frac{92\!\cdots\!74}{91\!\cdots\!21}a^{15}-\frac{99\!\cdots\!56}{91\!\cdots\!21}a^{14}+\frac{48\!\cdots\!66}{91\!\cdots\!21}a^{13}-\frac{76\!\cdots\!72}{91\!\cdots\!21}a^{12}+\frac{31\!\cdots\!92}{91\!\cdots\!21}a^{11}-\frac{46\!\cdots\!36}{91\!\cdots\!21}a^{10}+\frac{91\!\cdots\!54}{91\!\cdots\!21}a^{9}-\frac{83\!\cdots\!12}{91\!\cdots\!21}a^{8}-\frac{83\!\cdots\!92}{91\!\cdots\!21}a^{7}-\frac{12\!\cdots\!58}{91\!\cdots\!21}a^{6}-\frac{15\!\cdots\!91}{91\!\cdots\!21}a^{5}-\frac{22\!\cdots\!74}{91\!\cdots\!21}a^{4}+\frac{30\!\cdots\!24}{91\!\cdots\!21}a^{3}-\frac{43\!\cdots\!98}{91\!\cdots\!21}a^{2}+\frac{52\!\cdots\!94}{91\!\cdots\!21}a-\frac{47\!\cdots\!81}{91\!\cdots\!21}$, $\frac{83\!\cdots\!50}{91\!\cdots\!21}a^{43}-\frac{16\!\cdots\!45}{91\!\cdots\!21}a^{42}+\frac{96\!\cdots\!25}{91\!\cdots\!21}a^{41}-\frac{18\!\cdots\!94}{91\!\cdots\!21}a^{40}+\frac{70\!\cdots\!50}{91\!\cdots\!21}a^{39}-\frac{11\!\cdots\!80}{91\!\cdots\!21}a^{38}+\frac{38\!\cdots\!60}{91\!\cdots\!21}a^{37}-\frac{62\!\cdots\!35}{91\!\cdots\!21}a^{36}+\frac{18\!\cdots\!16}{91\!\cdots\!21}a^{35}-\frac{29\!\cdots\!25}{91\!\cdots\!21}a^{34}+\frac{74\!\cdots\!65}{91\!\cdots\!21}a^{33}-\frac{11\!\cdots\!50}{91\!\cdots\!21}a^{32}+\frac{25\!\cdots\!05}{91\!\cdots\!21}a^{31}-\frac{35\!\cdots\!92}{91\!\cdots\!21}a^{30}+\frac{73\!\cdots\!25}{91\!\cdots\!21}a^{29}-\frac{10\!\cdots\!30}{91\!\cdots\!21}a^{28}+\frac{19\!\cdots\!70}{91\!\cdots\!21}a^{27}-\frac{25\!\cdots\!10}{91\!\cdots\!21}a^{26}+\frac{42\!\cdots\!99}{91\!\cdots\!21}a^{25}-\frac{53\!\cdots\!00}{91\!\cdots\!21}a^{24}+\frac{79\!\cdots\!45}{91\!\cdots\!21}a^{23}-\frac{93\!\cdots\!75}{91\!\cdots\!21}a^{22}+\frac{12\!\cdots\!65}{91\!\cdots\!21}a^{21}-\frac{15\!\cdots\!45}{91\!\cdots\!21}a^{20}+\frac{17\!\cdots\!30}{91\!\cdots\!21}a^{19}-\frac{20\!\cdots\!10}{91\!\cdots\!21}a^{18}+\frac{18\!\cdots\!85}{91\!\cdots\!21}a^{17}-\frac{20\!\cdots\!70}{91\!\cdots\!21}a^{16}+\frac{15\!\cdots\!29}{91\!\cdots\!21}a^{15}-\frac{16\!\cdots\!90}{91\!\cdots\!21}a^{14}+\frac{80\!\cdots\!05}{91\!\cdots\!21}a^{13}-\frac{12\!\cdots\!80}{91\!\cdots\!21}a^{12}+\frac{51\!\cdots\!30}{91\!\cdots\!21}a^{11}-\frac{74\!\cdots\!31}{91\!\cdots\!21}a^{10}+\frac{15\!\cdots\!75}{91\!\cdots\!21}a^{9}-\frac{13\!\cdots\!60}{91\!\cdots\!21}a^{8}-\frac{13\!\cdots\!50}{91\!\cdots\!21}a^{7}-\frac{20\!\cdots\!05}{91\!\cdots\!21}a^{6}-\frac{46\!\cdots\!67}{91\!\cdots\!21}a^{5}-\frac{37\!\cdots\!55}{91\!\cdots\!21}a^{4}+\frac{50\!\cdots\!30}{91\!\cdots\!21}a^{3}-\frac{71\!\cdots\!35}{91\!\cdots\!21}a^{2}+\frac{86\!\cdots\!55}{91\!\cdots\!21}a-\frac{10\!\cdots\!63}{91\!\cdots\!21}$, $\frac{70\!\cdots\!25}{91\!\cdots\!21}a^{43}-\frac{14\!\cdots\!85}{91\!\cdots\!21}a^{42}+\frac{81\!\cdots\!00}{91\!\cdots\!21}a^{41}-\frac{15\!\cdots\!95}{91\!\cdots\!21}a^{40}+\frac{59\!\cdots\!75}{91\!\cdots\!21}a^{39}-\frac{10\!\cdots\!40}{91\!\cdots\!21}a^{38}+\frac{32\!\cdots\!10}{91\!\cdots\!21}a^{37}-\frac{52\!\cdots\!05}{91\!\cdots\!21}a^{36}+\frac{15\!\cdots\!20}{91\!\cdots\!21}a^{35}-\frac{24\!\cdots\!50}{91\!\cdots\!21}a^{34}+\frac{62\!\cdots\!90}{91\!\cdots\!21}a^{33}-\frac{94\!\cdots\!20}{91\!\cdots\!21}a^{32}+\frac{21\!\cdots\!80}{91\!\cdots\!21}a^{31}-\frac{29\!\cdots\!00}{91\!\cdots\!21}a^{30}+\frac{62\!\cdots\!50}{91\!\cdots\!21}a^{29}-\frac{84\!\cdots\!70}{91\!\cdots\!21}a^{28}+\frac{16\!\cdots\!80}{91\!\cdots\!21}a^{27}-\frac{21\!\cdots\!40}{91\!\cdots\!21}a^{26}+\frac{35\!\cdots\!35}{91\!\cdots\!21}a^{25}-\frac{44\!\cdots\!25}{91\!\cdots\!21}a^{24}+\frac{67\!\cdots\!85}{91\!\cdots\!21}a^{23}-\frac{79\!\cdots\!75}{91\!\cdots\!21}a^{22}+\frac{10\!\cdots\!70}{91\!\cdots\!21}a^{21}-\frac{12\!\cdots\!95}{91\!\cdots\!21}a^{20}+\frac{14\!\cdots\!40}{91\!\cdots\!21}a^{19}-\frac{17\!\cdots\!35}{91\!\cdots\!21}a^{18}+\frac{15\!\cdots\!05}{91\!\cdots\!21}a^{17}-\frac{17\!\cdots\!70}{91\!\cdots\!21}a^{16}+\frac{12\!\cdots\!60}{91\!\cdots\!21}a^{15}-\frac{13\!\cdots\!40}{91\!\cdots\!21}a^{14}+\frac{68\!\cdots\!10}{91\!\cdots\!21}a^{13}-\frac{10\!\cdots\!55}{91\!\cdots\!21}a^{12}+\frac{43\!\cdots\!55}{91\!\cdots\!21}a^{11}-\frac{63\!\cdots\!36}{91\!\cdots\!21}a^{10}+\frac{12\!\cdots\!80}{91\!\cdots\!21}a^{9}-\frac{11\!\cdots\!45}{91\!\cdots\!21}a^{8}-\frac{11\!\cdots\!65}{91\!\cdots\!21}a^{7}-\frac{17\!\cdots\!25}{91\!\cdots\!21}a^{6}-\frac{37\!\cdots\!11}{91\!\cdots\!21}a^{5}-\frac{31\!\cdots\!45}{91\!\cdots\!21}a^{4}+\frac{43\!\cdots\!45}{91\!\cdots\!21}a^{3}-\frac{60\!\cdots\!65}{91\!\cdots\!21}a^{2}+\frac{73\!\cdots\!45}{91\!\cdots\!21}a+\frac{10\!\cdots\!61}{91\!\cdots\!21}$, $\frac{40\!\cdots\!25}{91\!\cdots\!21}a^{43}-\frac{73\!\cdots\!10}{91\!\cdots\!21}a^{42}+\frac{44\!\cdots\!75}{91\!\cdots\!21}a^{41}-\frac{82\!\cdots\!74}{91\!\cdots\!21}a^{40}+\frac{31\!\cdots\!75}{91\!\cdots\!21}a^{39}-\frac{52\!\cdots\!40}{91\!\cdots\!21}a^{38}+\frac{17\!\cdots\!10}{91\!\cdots\!21}a^{37}-\frac{27\!\cdots\!80}{91\!\cdots\!21}a^{36}+\frac{84\!\cdots\!95}{91\!\cdots\!21}a^{35}-\frac{12\!\cdots\!25}{91\!\cdots\!21}a^{34}+\frac{33\!\cdots\!65}{91\!\cdots\!21}a^{33}-\frac{48\!\cdots\!70}{91\!\cdots\!21}a^{32}+\frac{11\!\cdots\!55}{91\!\cdots\!21}a^{31}-\frac{15\!\cdots\!32}{91\!\cdots\!21}a^{30}+\frac{33\!\cdots\!25}{91\!\cdots\!21}a^{29}-\frac{43\!\cdots\!20}{91\!\cdots\!21}a^{28}+\frac{85\!\cdots\!30}{91\!\cdots\!21}a^{27}-\frac{11\!\cdots\!90}{91\!\cdots\!21}a^{26}+\frac{18\!\cdots\!45}{91\!\cdots\!21}a^{25}-\frac{23\!\cdots\!75}{91\!\cdots\!21}a^{24}+\frac{35\!\cdots\!10}{91\!\cdots\!21}a^{23}-\frac{40\!\cdots\!50}{91\!\cdots\!21}a^{22}+\frac{55\!\cdots\!45}{91\!\cdots\!21}a^{21}-\frac{66\!\cdots\!45}{91\!\cdots\!21}a^{20}+\frac{76\!\cdots\!90}{91\!\cdots\!21}a^{19}-\frac{89\!\cdots\!35}{91\!\cdots\!21}a^{18}+\frac{82\!\cdots\!30}{91\!\cdots\!21}a^{17}-\frac{88\!\cdots\!20}{91\!\cdots\!21}a^{16}+\frac{70\!\cdots\!80}{91\!\cdots\!21}a^{15}-\frac{71\!\cdots\!90}{91\!\cdots\!21}a^{14}+\frac{35\!\cdots\!85}{91\!\cdots\!21}a^{13}-\frac{55\!\cdots\!05}{91\!\cdots\!21}a^{12}+\frac{22\!\cdots\!55}{91\!\cdots\!21}a^{11}-\frac{26\!\cdots\!75}{91\!\cdots\!21}a^{10}+\frac{66\!\cdots\!05}{91\!\cdots\!21}a^{9}-\frac{59\!\cdots\!95}{91\!\cdots\!21}a^{8}-\frac{58\!\cdots\!65}{91\!\cdots\!21}a^{7}-\frac{87\!\cdots\!50}{91\!\cdots\!21}a^{6}-\frac{66\!\cdots\!40}{91\!\cdots\!21}a^{5}-\frac{16\!\cdots\!20}{91\!\cdots\!21}a^{4}+\frac{22\!\cdots\!45}{91\!\cdots\!21}a^{3}-\frac{31\!\cdots\!40}{91\!\cdots\!21}a^{2}+\frac{37\!\cdots\!20}{91\!\cdots\!21}a-\frac{97\!\cdots\!54}{91\!\cdots\!21}$, $\frac{71\!\cdots\!10}{91\!\cdots\!21}a^{43}-\frac{14\!\cdots\!66}{91\!\cdots\!21}a^{42}+\frac{82\!\cdots\!40}{91\!\cdots\!21}a^{41}-\frac{15\!\cdots\!02}{91\!\cdots\!21}a^{40}+\frac{59\!\cdots\!50}{91\!\cdots\!21}a^{39}-\frac{10\!\cdots\!04}{91\!\cdots\!21}a^{38}+\frac{32\!\cdots\!44}{91\!\cdots\!21}a^{37}-\frac{52\!\cdots\!18}{91\!\cdots\!21}a^{36}+\frac{15\!\cdots\!88}{91\!\cdots\!21}a^{35}-\frac{24\!\cdots\!80}{91\!\cdots\!21}a^{34}+\frac{63\!\cdots\!16}{91\!\cdots\!21}a^{33}-\frac{94\!\cdots\!64}{91\!\cdots\!21}a^{32}+\frac{21\!\cdots\!32}{91\!\cdots\!21}a^{31}-\frac{30\!\cdots\!19}{91\!\cdots\!21}a^{30}+\frac{63\!\cdots\!20}{91\!\cdots\!21}a^{29}-\frac{85\!\cdots\!60}{91\!\cdots\!21}a^{28}+\frac{16\!\cdots\!40}{91\!\cdots\!21}a^{27}-\frac{21\!\cdots\!20}{91\!\cdots\!21}a^{26}+\frac{36\!\cdots\!59}{91\!\cdots\!21}a^{25}-\frac{45\!\cdots\!50}{91\!\cdots\!21}a^{24}+\frac{67\!\cdots\!06}{91\!\cdots\!21}a^{23}-\frac{79\!\cdots\!10}{91\!\cdots\!21}a^{22}+\frac{10\!\cdots\!52}{91\!\cdots\!21}a^{21}-\frac{12\!\cdots\!20}{91\!\cdots\!21}a^{20}+\frac{14\!\cdots\!04}{91\!\cdots\!21}a^{19}-\frac{17\!\cdots\!74}{91\!\cdots\!21}a^{18}+\frac{15\!\cdots\!78}{91\!\cdots\!21}a^{17}-\frac{17\!\cdots\!08}{91\!\cdots\!21}a^{16}+\frac{13\!\cdots\!10}{91\!\cdots\!21}a^{15}-\frac{13\!\cdots\!36}{91\!\cdots\!21}a^{14}+\frac{68\!\cdots\!68}{91\!\cdots\!21}a^{13}-\frac{10\!\cdots\!42}{91\!\cdots\!21}a^{12}+\frac{44\!\cdots\!82}{91\!\cdots\!21}a^{11}-\frac{63\!\cdots\!65}{91\!\cdots\!21}a^{10}+\frac{12\!\cdots\!76}{91\!\cdots\!21}a^{9}-\frac{11\!\cdots\!26}{91\!\cdots\!21}a^{8}-\frac{11\!\cdots\!78}{91\!\cdots\!21}a^{7}-\frac{17\!\cdots\!26}{91\!\cdots\!21}a^{6}-\frac{42\!\cdots\!89}{91\!\cdots\!21}a^{5}-\frac{32\!\cdots\!50}{91\!\cdots\!21}a^{4}+\frac{43\!\cdots\!90}{91\!\cdots\!21}a^{3}-\frac{60\!\cdots\!30}{91\!\cdots\!21}a^{2}+\frac{73\!\cdots\!70}{91\!\cdots\!21}a-\frac{79\!\cdots\!25}{91\!\cdots\!21}$, $\frac{20\!\cdots\!75}{91\!\cdots\!21}a^{43}-\frac{41\!\cdots\!85}{91\!\cdots\!21}a^{42}+\frac{23\!\cdots\!50}{91\!\cdots\!21}a^{41}-\frac{46\!\cdots\!70}{91\!\cdots\!21}a^{40}+\frac{17\!\cdots\!25}{91\!\cdots\!21}a^{39}-\frac{29\!\cdots\!40}{91\!\cdots\!21}a^{38}+\frac{95\!\cdots\!90}{91\!\cdots\!21}a^{37}-\frac{15\!\cdots\!05}{91\!\cdots\!21}a^{36}+\frac{45\!\cdots\!50}{91\!\cdots\!21}a^{35}-\frac{72\!\cdots\!00}{91\!\cdots\!21}a^{34}+\frac{18\!\cdots\!60}{91\!\cdots\!21}a^{33}-\frac{27\!\cdots\!40}{91\!\cdots\!21}a^{32}+\frac{62\!\cdots\!70}{91\!\cdots\!21}a^{31}-\frac{87\!\cdots\!65}{91\!\cdots\!21}a^{30}+\frac{18\!\cdots\!00}{91\!\cdots\!21}a^{29}-\frac{24\!\cdots\!50}{91\!\cdots\!21}a^{28}+\frac{47\!\cdots\!00}{91\!\cdots\!21}a^{27}-\frac{63\!\cdots\!00}{91\!\cdots\!21}a^{26}+\frac{10\!\cdots\!70}{91\!\cdots\!21}a^{25}-\frac{13\!\cdots\!75}{91\!\cdots\!21}a^{24}+\frac{19\!\cdots\!85}{91\!\cdots\!21}a^{23}-\frac{23\!\cdots\!75}{91\!\cdots\!21}a^{22}+\frac{30\!\cdots\!20}{91\!\cdots\!21}a^{21}-\frac{37\!\cdots\!39}{91\!\cdots\!21}a^{20}+\frac{42\!\cdots\!40}{91\!\cdots\!21}a^{19}-\frac{51\!\cdots\!65}{91\!\cdots\!21}a^{18}+\frac{46\!\cdots\!05}{91\!\cdots\!21}a^{17}-\frac{50\!\cdots\!30}{91\!\cdots\!21}a^{16}+\frac{37\!\cdots\!50}{91\!\cdots\!21}a^{15}-\frac{40\!\cdots\!60}{91\!\cdots\!21}a^{14}+\frac{19\!\cdots\!80}{91\!\cdots\!21}a^{13}-\frac{31\!\cdots\!45}{91\!\cdots\!21}a^{12}+\frac{12\!\cdots\!45}{91\!\cdots\!21}a^{11}-\frac{18\!\cdots\!04}{91\!\cdots\!21}a^{10}+\frac{37\!\cdots\!10}{91\!\cdots\!21}a^{9}-\frac{34\!\cdots\!35}{91\!\cdots\!21}a^{8}-\frac{33\!\cdots\!55}{91\!\cdots\!21}a^{7}-\frac{49\!\cdots\!85}{91\!\cdots\!21}a^{6}-\frac{12\!\cdots\!45}{91\!\cdots\!21}a^{5}-\frac{93\!\cdots\!25}{91\!\cdots\!21}a^{4}+\frac{12\!\cdots\!75}{91\!\cdots\!21}a^{3}-\frac{17\!\cdots\!25}{91\!\cdots\!21}a^{2}+\frac{21\!\cdots\!25}{91\!\cdots\!21}a+\frac{36\!\cdots\!92}{91\!\cdots\!21}$, $\frac{13\!\cdots\!93}{91\!\cdots\!21}a^{43}-\frac{14\!\cdots\!01}{91\!\cdots\!21}a^{42}+\frac{15\!\cdots\!76}{91\!\cdots\!21}a^{41}-\frac{17\!\cdots\!73}{91\!\cdots\!21}a^{40}+\frac{10\!\cdots\!06}{91\!\cdots\!21}a^{39}-\frac{10\!\cdots\!76}{91\!\cdots\!21}a^{38}+\frac{59\!\cdots\!19}{91\!\cdots\!21}a^{37}-\frac{54\!\cdots\!96}{91\!\cdots\!21}a^{36}+\frac{28\!\cdots\!46}{91\!\cdots\!21}a^{35}-\frac{25\!\cdots\!51}{91\!\cdots\!21}a^{34}+\frac{11\!\cdots\!04}{91\!\cdots\!21}a^{33}-\frac{93\!\cdots\!50}{91\!\cdots\!21}a^{32}+\frac{39\!\cdots\!82}{91\!\cdots\!21}a^{31}-\frac{28\!\cdots\!35}{91\!\cdots\!21}a^{30}+\frac{11\!\cdots\!78}{91\!\cdots\!21}a^{29}-\frac{76\!\cdots\!84}{91\!\cdots\!21}a^{28}+\frac{30\!\cdots\!44}{91\!\cdots\!21}a^{27}-\frac{18\!\cdots\!75}{91\!\cdots\!21}a^{26}+\frac{68\!\cdots\!08}{91\!\cdots\!21}a^{25}-\frac{37\!\cdots\!71}{91\!\cdots\!21}a^{24}+\frac{13\!\cdots\!84}{91\!\cdots\!21}a^{23}-\frac{58\!\cdots\!06}{91\!\cdots\!21}a^{22}+\frac{22\!\cdots\!13}{91\!\cdots\!21}a^{21}-\frac{89\!\cdots\!86}{91\!\cdots\!21}a^{20}+\frac{31\!\cdots\!26}{91\!\cdots\!21}a^{19}-\frac{12\!\cdots\!62}{91\!\cdots\!21}a^{18}+\frac{35\!\cdots\!01}{91\!\cdots\!21}a^{17}-\frac{10\!\cdots\!66}{91\!\cdots\!21}a^{16}+\frac{33\!\cdots\!41}{91\!\cdots\!21}a^{15}-\frac{54\!\cdots\!71}{91\!\cdots\!21}a^{14}+\frac{24\!\cdots\!14}{91\!\cdots\!21}a^{13}-\frac{51\!\cdots\!81}{91\!\cdots\!21}a^{12}+\frac{15\!\cdots\!17}{91\!\cdots\!21}a^{11}-\frac{41\!\cdots\!07}{91\!\cdots\!21}a^{10}+\frac{57\!\cdots\!53}{91\!\cdots\!21}a^{9}-\frac{86\!\cdots\!88}{91\!\cdots\!21}a^{8}+\frac{17\!\cdots\!85}{91\!\cdots\!21}a^{7}+\frac{19\!\cdots\!83}{91\!\cdots\!21}a^{6}+\frac{21\!\cdots\!90}{91\!\cdots\!21}a^{5}+\frac{91\!\cdots\!12}{91\!\cdots\!21}a^{4}+\frac{34\!\cdots\!70}{91\!\cdots\!21}a^{3}-\frac{66\!\cdots\!90}{91\!\cdots\!21}a^{2}+\frac{65\!\cdots\!73}{91\!\cdots\!21}a+\frac{13\!\cdots\!28}{91\!\cdots\!21}$, $\frac{15\!\cdots\!35}{91\!\cdots\!21}a^{43}-\frac{16\!\cdots\!84}{91\!\cdots\!21}a^{42}+\frac{16\!\cdots\!20}{91\!\cdots\!21}a^{41}-\frac{19\!\cdots\!64}{91\!\cdots\!21}a^{40}+\frac{11\!\cdots\!68}{91\!\cdots\!21}a^{39}-\frac{12\!\cdots\!43}{91\!\cdots\!21}a^{38}+\frac{65\!\cdots\!21}{91\!\cdots\!21}a^{37}-\frac{61\!\cdots\!35}{91\!\cdots\!21}a^{36}+\frac{31\!\cdots\!62}{91\!\cdots\!21}a^{35}-\frac{28\!\cdots\!39}{91\!\cdots\!21}a^{34}+\frac{12\!\cdots\!84}{91\!\cdots\!21}a^{33}-\frac{10\!\cdots\!94}{91\!\cdots\!21}a^{32}+\frac{43\!\cdots\!95}{91\!\cdots\!21}a^{31}-\frac{31\!\cdots\!14}{91\!\cdots\!21}a^{30}+\frac{12\!\cdots\!26}{91\!\cdots\!21}a^{29}-\frac{86\!\cdots\!23}{91\!\cdots\!21}a^{28}+\frac{33\!\cdots\!07}{91\!\cdots\!21}a^{27}-\frac{21\!\cdots\!30}{91\!\cdots\!21}a^{26}+\frac{75\!\cdots\!56}{91\!\cdots\!21}a^{25}-\frac{42\!\cdots\!13}{91\!\cdots\!21}a^{24}+\frac{14\!\cdots\!82}{91\!\cdots\!21}a^{23}-\frac{67\!\cdots\!29}{91\!\cdots\!21}a^{22}+\frac{24\!\cdots\!74}{91\!\cdots\!21}a^{21}-\frac{10\!\cdots\!08}{91\!\cdots\!21}a^{20}+\frac{34\!\cdots\!28}{91\!\cdots\!21}a^{19}-\frac{14\!\cdots\!71}{91\!\cdots\!21}a^{18}+\frac{38\!\cdots\!29}{91\!\cdots\!21}a^{17}-\frac{12\!\cdots\!92}{91\!\cdots\!21}a^{16}+\frac{36\!\cdots\!14}{91\!\cdots\!21}a^{15}-\frac{68\!\cdots\!19}{91\!\cdots\!21}a^{14}+\frac{26\!\cdots\!94}{91\!\cdots\!21}a^{13}-\frac{62\!\cdots\!85}{91\!\cdots\!21}a^{12}+\frac{16\!\cdots\!65}{91\!\cdots\!21}a^{11}-\frac{49\!\cdots\!18}{91\!\cdots\!21}a^{10}+\frac{62\!\cdots\!90}{91\!\cdots\!21}a^{9}-\frac{10\!\cdots\!39}{91\!\cdots\!21}a^{8}+\frac{18\!\cdots\!84}{91\!\cdots\!21}a^{7}+\frac{15\!\cdots\!56}{91\!\cdots\!21}a^{6}+\frac{22\!\cdots\!49}{91\!\cdots\!21}a^{5}+\frac{73\!\cdots\!71}{91\!\cdots\!21}a^{4}+\frac{35\!\cdots\!04}{91\!\cdots\!21}a^{3}-\frac{89\!\cdots\!29}{91\!\cdots\!21}a^{2}+\frac{67\!\cdots\!15}{91\!\cdots\!21}a+\frac{10\!\cdots\!61}{91\!\cdots\!21}$, $\frac{14\!\cdots\!48}{91\!\cdots\!21}a^{43}-\frac{15\!\cdots\!70}{91\!\cdots\!21}a^{42}+\frac{16\!\cdots\!46}{91\!\cdots\!21}a^{41}-\frac{18\!\cdots\!38}{91\!\cdots\!21}a^{40}+\frac{11\!\cdots\!10}{91\!\cdots\!21}a^{39}-\frac{11\!\cdots\!36}{91\!\cdots\!21}a^{38}+\frac{63\!\cdots\!90}{91\!\cdots\!21}a^{37}-\frac{59\!\cdots\!46}{91\!\cdots\!21}a^{36}+\frac{30\!\cdots\!42}{91\!\cdots\!21}a^{35}-\frac{27\!\cdots\!12}{91\!\cdots\!21}a^{34}+\frac{12\!\cdots\!96}{91\!\cdots\!21}a^{33}-\frac{10\!\cdots\!31}{91\!\cdots\!21}a^{32}+\frac{41\!\cdots\!62}{91\!\cdots\!21}a^{31}-\frac{30\!\cdots\!14}{91\!\cdots\!21}a^{30}+\frac{12\!\cdots\!19}{91\!\cdots\!21}a^{29}-\frac{83\!\cdots\!42}{91\!\cdots\!21}a^{28}+\frac{32\!\cdots\!54}{91\!\cdots\!21}a^{27}-\frac{20\!\cdots\!30}{91\!\cdots\!21}a^{26}+\frac{73\!\cdots\!28}{91\!\cdots\!21}a^{25}-\frac{40\!\cdots\!51}{91\!\cdots\!21}a^{24}+\frac{14\!\cdots\!54}{91\!\cdots\!21}a^{23}-\frac{64\!\cdots\!50}{91\!\cdots\!21}a^{22}+\frac{23\!\cdots\!88}{91\!\cdots\!21}a^{21}-\frac{99\!\cdots\!00}{91\!\cdots\!21}a^{20}+\frac{33\!\cdots\!16}{91\!\cdots\!21}a^{19}-\frac{13\!\cdots\!50}{91\!\cdots\!21}a^{18}+\frac{37\!\cdots\!31}{91\!\cdots\!21}a^{17}-\frac{11\!\cdots\!62}{91\!\cdots\!21}a^{16}+\frac{35\!\cdots\!22}{91\!\cdots\!21}a^{15}-\frac{64\!\cdots\!98}{91\!\cdots\!21}a^{14}+\frac{25\!\cdots\!16}{91\!\cdots\!21}a^{13}-\frac{59\!\cdots\!29}{91\!\cdots\!21}a^{12}+\frac{16\!\cdots\!26}{91\!\cdots\!21}a^{11}-\frac{46\!\cdots\!24}{91\!\cdots\!21}a^{10}+\frac{60\!\cdots\!51}{91\!\cdots\!21}a^{9}-\frac{10\!\cdots\!26}{91\!\cdots\!21}a^{8}+\frac{18\!\cdots\!45}{91\!\cdots\!21}a^{7}+\frac{16\!\cdots\!04}{91\!\cdots\!21}a^{6}+\frac{21\!\cdots\!04}{91\!\cdots\!21}a^{5}+\frac{34\!\cdots\!59}{91\!\cdots\!21}a^{4}+\frac{35\!\cdots\!54}{91\!\cdots\!21}a^{3}-\frac{74\!\cdots\!79}{91\!\cdots\!21}a^{2}+\frac{66\!\cdots\!98}{91\!\cdots\!21}a+\frac{11\!\cdots\!93}{91\!\cdots\!21}$, $\frac{12\!\cdots\!69}{91\!\cdots\!21}a^{43}-\frac{12\!\cdots\!57}{91\!\cdots\!21}a^{42}+\frac{13\!\cdots\!68}{91\!\cdots\!21}a^{41}-\frac{14\!\cdots\!71}{91\!\cdots\!21}a^{40}+\frac{92\!\cdots\!06}{91\!\cdots\!21}a^{39}-\frac{88\!\cdots\!82}{91\!\cdots\!21}a^{38}+\frac{50\!\cdots\!17}{91\!\cdots\!21}a^{37}-\frac{44\!\cdots\!78}{91\!\cdots\!21}a^{36}+\frac{24\!\cdots\!10}{91\!\cdots\!21}a^{35}-\frac{20\!\cdots\!91}{91\!\cdots\!21}a^{34}+\frac{98\!\cdots\!56}{91\!\cdots\!21}a^{33}-\frac{75\!\cdots\!33}{91\!\cdots\!21}a^{32}+\frac{33\!\cdots\!56}{91\!\cdots\!21}a^{31}-\frac{22\!\cdots\!81}{91\!\cdots\!21}a^{30}+\frac{99\!\cdots\!78}{91\!\cdots\!21}a^{29}-\frac{60\!\cdots\!54}{91\!\cdots\!21}a^{28}+\frac{26\!\cdots\!11}{91\!\cdots\!21}a^{27}-\frac{14\!\cdots\!45}{91\!\cdots\!21}a^{26}+\frac{58\!\cdots\!52}{91\!\cdots\!21}a^{25}-\frac{28\!\cdots\!47}{91\!\cdots\!21}a^{24}+\frac{11\!\cdots\!28}{91\!\cdots\!21}a^{23}-\frac{44\!\cdots\!03}{91\!\cdots\!21}a^{22}+\frac{18\!\cdots\!31}{91\!\cdots\!21}a^{21}-\frac{66\!\cdots\!26}{91\!\cdots\!21}a^{20}+\frac{26\!\cdots\!42}{91\!\cdots\!21}a^{19}-\frac{88\!\cdots\!72}{91\!\cdots\!21}a^{18}+\frac{30\!\cdots\!84}{91\!\cdots\!21}a^{17}-\frac{72\!\cdots\!10}{91\!\cdots\!21}a^{16}+\frac{28\!\cdots\!31}{91\!\cdots\!21}a^{15}-\frac{31\!\cdots\!67}{91\!\cdots\!21}a^{14}+\frac{21\!\cdots\!26}{91\!\cdots\!21}a^{13}-\frac{32\!\cdots\!77}{91\!\cdots\!21}a^{12}+\frac{13\!\cdots\!65}{91\!\cdots\!21}a^{11}-\frac{28\!\cdots\!37}{91\!\cdots\!21}a^{10}+\frac{48\!\cdots\!87}{91\!\cdots\!21}a^{9}-\frac{49\!\cdots\!94}{91\!\cdots\!21}a^{8}+\frac{15\!\cdots\!65}{91\!\cdots\!21}a^{7}+\frac{24\!\cdots\!13}{91\!\cdots\!21}a^{6}+\frac{21\!\cdots\!36}{91\!\cdots\!21}a^{5}+\frac{22\!\cdots\!46}{91\!\cdots\!21}a^{4}+\frac{32\!\cdots\!86}{91\!\cdots\!21}a^{3}-\frac{27\!\cdots\!23}{91\!\cdots\!21}a^{2}+\frac{62\!\cdots\!69}{91\!\cdots\!21}a+\frac{18\!\cdots\!22}{91\!\cdots\!21}$, $\frac{22\!\cdots\!00}{91\!\cdots\!21}a^{43}-\frac{38\!\cdots\!81}{91\!\cdots\!21}a^{42}+\frac{26\!\cdots\!16}{91\!\cdots\!21}a^{41}-\frac{44\!\cdots\!60}{91\!\cdots\!21}a^{40}+\frac{19\!\cdots\!98}{91\!\cdots\!21}a^{39}-\frac{29\!\cdots\!45}{91\!\cdots\!21}a^{38}+\frac{10\!\cdots\!31}{91\!\cdots\!21}a^{37}-\frac{15\!\cdots\!79}{91\!\cdots\!21}a^{36}+\frac{52\!\cdots\!37}{91\!\cdots\!21}a^{35}-\frac{70\!\cdots\!62}{91\!\cdots\!21}a^{34}+\frac{20\!\cdots\!64}{91\!\cdots\!21}a^{33}-\frac{27\!\cdots\!80}{91\!\cdots\!21}a^{32}+\frac{72\!\cdots\!83}{91\!\cdots\!21}a^{31}-\frac{86\!\cdots\!63}{91\!\cdots\!21}a^{30}+\frac{21\!\cdots\!57}{91\!\cdots\!21}a^{29}-\frac{24\!\cdots\!11}{91\!\cdots\!21}a^{28}+\frac{56\!\cdots\!87}{91\!\cdots\!21}a^{27}-\frac{62\!\cdots\!61}{91\!\cdots\!21}a^{26}+\frac{12\!\cdots\!30}{91\!\cdots\!21}a^{25}-\frac{13\!\cdots\!75}{91\!\cdots\!21}a^{24}+\frac{24\!\cdots\!55}{91\!\cdots\!21}a^{23}-\frac{23\!\cdots\!64}{91\!\cdots\!21}a^{22}+\frac{40\!\cdots\!85}{91\!\cdots\!21}a^{21}-\frac{37\!\cdots\!18}{91\!\cdots\!21}a^{20}+\frac{58\!\cdots\!80}{91\!\cdots\!21}a^{19}-\frac{51\!\cdots\!56}{91\!\cdots\!21}a^{18}+\frac{67\!\cdots\!29}{91\!\cdots\!21}a^{17}-\frac{52\!\cdots\!33}{91\!\cdots\!21}a^{16}+\frac{61\!\cdots\!52}{91\!\cdots\!21}a^{15}-\frac{42\!\cdots\!44}{91\!\cdots\!21}a^{14}+\frac{42\!\cdots\!54}{91\!\cdots\!21}a^{13}-\frac{33\!\cdots\!70}{91\!\cdots\!21}a^{12}+\frac{27\!\cdots\!62}{91\!\cdots\!21}a^{11}-\frac{21\!\cdots\!19}{91\!\cdots\!21}a^{10}+\frac{12\!\cdots\!91}{91\!\cdots\!21}a^{9}-\frac{69\!\cdots\!01}{91\!\cdots\!21}a^{8}+\frac{31\!\cdots\!09}{91\!\cdots\!21}a^{7}-\frac{14\!\cdots\!45}{91\!\cdots\!21}a^{6}-\frac{39\!\cdots\!22}{91\!\cdots\!21}a^{5}-\frac{23\!\cdots\!76}{91\!\cdots\!21}a^{4}+\frac{31\!\cdots\!13}{91\!\cdots\!21}a^{3}-\frac{53\!\cdots\!02}{91\!\cdots\!21}a^{2}-\frac{50\!\cdots\!78}{91\!\cdots\!21}a-\frac{99\!\cdots\!01}{91\!\cdots\!21}$, $\frac{17\!\cdots\!52}{91\!\cdots\!21}a^{43}-\frac{19\!\cdots\!29}{91\!\cdots\!21}a^{42}+\frac{19\!\cdots\!74}{91\!\cdots\!21}a^{41}-\frac{23\!\cdots\!80}{91\!\cdots\!21}a^{40}+\frac{13\!\cdots\!94}{91\!\cdots\!21}a^{39}-\frac{14\!\cdots\!90}{91\!\cdots\!21}a^{38}+\frac{75\!\cdots\!73}{91\!\cdots\!21}a^{37}-\frac{73\!\cdots\!64}{91\!\cdots\!21}a^{36}+\frac{36\!\cdots\!94}{91\!\cdots\!21}a^{35}-\frac{33\!\cdots\!28}{91\!\cdots\!21}a^{34}+\frac{14\!\cdots\!76}{91\!\cdots\!21}a^{33}-\frac{12\!\cdots\!72}{91\!\cdots\!21}a^{32}+\frac{49\!\cdots\!68}{91\!\cdots\!21}a^{31}-\frac{38\!\cdots\!52}{91\!\cdots\!21}a^{30}+\frac{14\!\cdots\!55}{91\!\cdots\!21}a^{29}-\frac{10\!\cdots\!40}{91\!\cdots\!21}a^{28}+\frac{39\!\cdots\!57}{91\!\cdots\!21}a^{27}-\frac{26\!\cdots\!40}{91\!\cdots\!21}a^{26}+\frac{87\!\cdots\!04}{91\!\cdots\!21}a^{25}-\frac{52\!\cdots\!55}{91\!\cdots\!21}a^{24}+\frac{16\!\cdots\!30}{91\!\cdots\!21}a^{23}-\frac{83\!\cdots\!82}{91\!\cdots\!21}a^{22}+\frac{27\!\cdots\!70}{91\!\cdots\!21}a^{21}-\frac{13\!\cdots\!04}{91\!\cdots\!21}a^{20}+\frac{39\!\cdots\!40}{91\!\cdots\!21}a^{19}-\frac{17\!\cdots\!88}{91\!\cdots\!21}a^{18}+\frac{44\!\cdots\!09}{91\!\cdots\!21}a^{17}-\frac{15\!\cdots\!34}{91\!\cdots\!21}a^{16}+\frac{41\!\cdots\!08}{91\!\cdots\!21}a^{15}-\frac{95\!\cdots\!94}{91\!\cdots\!21}a^{14}+\frac{30\!\cdots\!96}{91\!\cdots\!21}a^{13}-\frac{84\!\cdots\!17}{91\!\cdots\!21}a^{12}+\frac{18\!\cdots\!42}{91\!\cdots\!21}a^{11}-\frac{63\!\cdots\!26}{91\!\cdots\!21}a^{10}+\frac{72\!\cdots\!25}{91\!\cdots\!21}a^{9}-\frac{15\!\cdots\!68}{91\!\cdots\!21}a^{8}+\frac{21\!\cdots\!93}{91\!\cdots\!21}a^{7}+\frac{95\!\cdots\!90}{91\!\cdots\!21}a^{6}+\frac{22\!\cdots\!02}{91\!\cdots\!21}a^{5}-\frac{15\!\cdots\!63}{91\!\cdots\!21}a^{4}+\frac{37\!\cdots\!02}{91\!\cdots\!21}a^{3}-\frac{11\!\cdots\!64}{91\!\cdots\!21}a^{2}+\frac{71\!\cdots\!02}{91\!\cdots\!21}a+\frac{52\!\cdots\!39}{91\!\cdots\!21}$, $\frac{21\!\cdots\!03}{91\!\cdots\!21}a^{43}-\frac{18\!\cdots\!14}{91\!\cdots\!21}a^{42}+\frac{23\!\cdots\!40}{91\!\cdots\!21}a^{41}-\frac{22\!\cdots\!87}{91\!\cdots\!21}a^{40}+\frac{16\!\cdots\!44}{91\!\cdots\!21}a^{39}-\frac{13\!\cdots\!00}{91\!\cdots\!21}a^{38}+\frac{88\!\cdots\!08}{91\!\cdots\!21}a^{37}-\frac{65\!\cdots\!08}{91\!\cdots\!21}a^{36}+\frac{42\!\cdots\!34}{91\!\cdots\!21}a^{35}-\frac{29\!\cdots\!36}{91\!\cdots\!21}a^{34}+\frac{17\!\cdots\!30}{91\!\cdots\!21}a^{33}-\frac{10\!\cdots\!38}{91\!\cdots\!21}a^{32}+\frac{58\!\cdots\!67}{91\!\cdots\!21}a^{31}-\frac{31\!\cdots\!03}{91\!\cdots\!21}a^{30}+\frac{17\!\cdots\!70}{91\!\cdots\!21}a^{29}-\frac{81\!\cdots\!49}{91\!\cdots\!21}a^{28}+\frac{45\!\cdots\!46}{91\!\cdots\!21}a^{27}-\frac{19\!\cdots\!45}{91\!\cdots\!21}a^{26}+\frac{10\!\cdots\!89}{91\!\cdots\!21}a^{25}-\frac{36\!\cdots\!66}{91\!\cdots\!21}a^{24}+\frac{19\!\cdots\!69}{91\!\cdots\!21}a^{23}-\frac{49\!\cdots\!12}{91\!\cdots\!21}a^{22}+\frac{32\!\cdots\!72}{91\!\cdots\!21}a^{21}-\frac{70\!\cdots\!24}{91\!\cdots\!21}a^{20}+\frac{46\!\cdots\!40}{91\!\cdots\!21}a^{19}-\frac{90\!\cdots\!33}{91\!\cdots\!21}a^{18}+\frac{52\!\cdots\!20}{91\!\cdots\!21}a^{17}-\frac{52\!\cdots\!54}{91\!\cdots\!21}a^{16}+\frac{49\!\cdots\!56}{91\!\cdots\!21}a^{15}+\frac{16\!\cdots\!22}{91\!\cdots\!21}a^{14}+\frac{37\!\cdots\!56}{91\!\cdots\!21}a^{13}-\frac{32\!\cdots\!92}{91\!\cdots\!21}a^{12}+\frac{22\!\cdots\!01}{91\!\cdots\!21}a^{11}-\frac{17\!\cdots\!47}{91\!\cdots\!21}a^{10}+\frac{83\!\cdots\!10}{91\!\cdots\!21}a^{9}+\frac{31\!\cdots\!46}{91\!\cdots\!21}a^{8}+\frac{26\!\cdots\!04}{91\!\cdots\!21}a^{7}+\frac{80\!\cdots\!60}{91\!\cdots\!21}a^{6}+\frac{49\!\cdots\!55}{91\!\cdots\!21}a^{5}+\frac{10\!\cdots\!16}{91\!\cdots\!21}a^{4}+\frac{71\!\cdots\!69}{91\!\cdots\!21}a^{3}+\frac{19\!\cdots\!92}{91\!\cdots\!21}a^{2}+\frac{10\!\cdots\!64}{91\!\cdots\!21}a-\frac{10\!\cdots\!24}{91\!\cdots\!21}$, $\frac{23\!\cdots\!33}{91\!\cdots\!21}a^{43}-\frac{22\!\cdots\!53}{91\!\cdots\!21}a^{42}+\frac{25\!\cdots\!96}{91\!\cdots\!21}a^{41}-\frac{26\!\cdots\!86}{91\!\cdots\!21}a^{40}+\frac{17\!\cdots\!43}{91\!\cdots\!21}a^{39}-\frac{16\!\cdots\!08}{91\!\cdots\!21}a^{38}+\frac{97\!\cdots\!35}{91\!\cdots\!21}a^{37}-\frac{81\!\cdots\!79}{91\!\cdots\!21}a^{36}+\frac{47\!\cdots\!27}{91\!\cdots\!21}a^{35}-\frac{37\!\cdots\!82}{91\!\cdots\!21}a^{34}+\frac{18\!\cdots\!36}{91\!\cdots\!21}a^{33}-\frac{13\!\cdots\!89}{91\!\cdots\!21}a^{32}+\frac{63\!\cdots\!34}{91\!\cdots\!21}a^{31}-\frac{40\!\cdots\!30}{91\!\cdots\!21}a^{30}+\frac{18\!\cdots\!82}{91\!\cdots\!21}a^{29}-\frac{10\!\cdots\!48}{91\!\cdots\!21}a^{28}+\frac{50\!\cdots\!60}{91\!\cdots\!21}a^{27}-\frac{26\!\cdots\!50}{91\!\cdots\!21}a^{26}+\frac{11\!\cdots\!41}{91\!\cdots\!21}a^{25}-\frac{51\!\cdots\!33}{91\!\cdots\!21}a^{24}+\frac{21\!\cdots\!83}{91\!\cdots\!21}a^{23}-\frac{77\!\cdots\!79}{91\!\cdots\!21}a^{22}+\frac{35\!\cdots\!16}{91\!\cdots\!21}a^{21}-\frac{11\!\cdots\!78}{91\!\cdots\!21}a^{20}+\frac{50\!\cdots\!52}{91\!\cdots\!21}a^{19}-\frac{15\!\cdots\!45}{91\!\cdots\!21}a^{18}+\frac{56\!\cdots\!82}{91\!\cdots\!21}a^{17}-\frac{12\!\cdots\!62}{91\!\cdots\!21}a^{16}+\frac{53\!\cdots\!76}{91\!\cdots\!21}a^{15}-\frac{43\!\cdots\!48}{91\!\cdots\!21}a^{14}+\frac{39\!\cdots\!16}{91\!\cdots\!21}a^{13}-\frac{50\!\cdots\!82}{91\!\cdots\!21}a^{12}+\frac{24\!\cdots\!59}{91\!\cdots\!21}a^{11}-\frac{47\!\cdots\!94}{91\!\cdots\!21}a^{10}+\frac{87\!\cdots\!14}{91\!\cdots\!21}a^{9}-\frac{64\!\cdots\!60}{91\!\cdots\!21}a^{8}+\frac{27\!\cdots\!99}{91\!\cdots\!21}a^{7}+\frac{54\!\cdots\!01}{91\!\cdots\!21}a^{6}+\frac{38\!\cdots\!60}{91\!\cdots\!21}a^{5}+\frac{35\!\cdots\!81}{91\!\cdots\!21}a^{4}+\frac{56\!\cdots\!07}{91\!\cdots\!21}a^{3}-\frac{76\!\cdots\!55}{91\!\cdots\!21}a^{2}-\frac{63\!\cdots\!67}{91\!\cdots\!21}a-\frac{11\!\cdots\!02}{91\!\cdots\!21}$, $\frac{11\!\cdots\!55}{91\!\cdots\!21}a^{43}-\frac{11\!\cdots\!63}{91\!\cdots\!21}a^{42}+\frac{12\!\cdots\!40}{91\!\cdots\!21}a^{41}-\frac{13\!\cdots\!49}{91\!\cdots\!21}a^{40}+\frac{88\!\cdots\!10}{91\!\cdots\!21}a^{39}-\frac{84\!\cdots\!78}{91\!\cdots\!21}a^{38}+\frac{48\!\cdots\!91}{91\!\cdots\!21}a^{37}-\frac{41\!\cdots\!70}{91\!\cdots\!21}a^{36}+\frac{23\!\cdots\!90}{91\!\cdots\!21}a^{35}-\frac{19\!\cdots\!17}{91\!\cdots\!21}a^{34}+\frac{94\!\cdots\!60}{91\!\cdots\!21}a^{33}-\frac{70\!\cdots\!63}{91\!\cdots\!21}a^{32}+\frac{32\!\cdots\!80}{91\!\cdots\!21}a^{31}-\frac{21\!\cdots\!11}{91\!\cdots\!21}a^{30}+\frac{95\!\cdots\!44}{91\!\cdots\!21}a^{29}-\frac{56\!\cdots\!82}{91\!\cdots\!21}a^{28}+\frac{25\!\cdots\!35}{91\!\cdots\!21}a^{27}-\frac{13\!\cdots\!95}{91\!\cdots\!21}a^{26}+\frac{56\!\cdots\!36}{91\!\cdots\!21}a^{25}-\frac{26\!\cdots\!83}{91\!\cdots\!21}a^{24}+\frac{10\!\cdots\!12}{91\!\cdots\!21}a^{23}-\frac{40\!\cdots\!20}{91\!\cdots\!21}a^{22}+\frac{18\!\cdots\!89}{91\!\cdots\!21}a^{21}-\frac{61\!\cdots\!30}{91\!\cdots\!21}a^{20}+\frac{25\!\cdots\!78}{91\!\cdots\!21}a^{19}-\frac{81\!\cdots\!20}{91\!\cdots\!21}a^{18}+\frac{29\!\cdots\!95}{91\!\cdots\!21}a^{17}-\frac{65\!\cdots\!70}{91\!\cdots\!21}a^{16}+\frac{27\!\cdots\!77}{91\!\cdots\!21}a^{15}-\frac{26\!\cdots\!45}{91\!\cdots\!21}a^{14}+\frac{20\!\cdots\!50}{91\!\cdots\!21}a^{13}-\frac{27\!\cdots\!50}{91\!\cdots\!21}a^{12}+\frac{12\!\cdots\!87}{91\!\cdots\!21}a^{11}-\frac{25\!\cdots\!59}{91\!\cdots\!21}a^{10}+\frac{46\!\cdots\!39}{91\!\cdots\!21}a^{9}-\frac{40\!\cdots\!18}{91\!\cdots\!21}a^{8}+\frac{14\!\cdots\!10}{91\!\cdots\!21}a^{7}+\frac{25\!\cdots\!39}{91\!\cdots\!21}a^{6}+\frac{21\!\cdots\!76}{91\!\cdots\!21}a^{5}+\frac{25\!\cdots\!72}{91\!\cdots\!21}a^{4}+\frac{32\!\cdots\!66}{91\!\cdots\!21}a^{3}-\frac{36\!\cdots\!94}{91\!\cdots\!21}a^{2}+\frac{61\!\cdots\!95}{91\!\cdots\!21}a+\frac{19\!\cdots\!76}{91\!\cdots\!21}$, $\frac{29\!\cdots\!61}{91\!\cdots\!21}a^{43}+\frac{75\!\cdots\!37}{91\!\cdots\!21}a^{42}+\frac{28\!\cdots\!92}{91\!\cdots\!21}a^{41}+\frac{47\!\cdots\!88}{91\!\cdots\!21}a^{40}+\frac{18\!\cdots\!60}{91\!\cdots\!21}a^{39}+\frac{58\!\cdots\!11}{91\!\cdots\!21}a^{38}+\frac{10\!\cdots\!35}{91\!\cdots\!21}a^{37}+\frac{43\!\cdots\!63}{91\!\cdots\!21}a^{36}+\frac{48\!\cdots\!74}{91\!\cdots\!21}a^{35}+\frac{24\!\cdots\!43}{91\!\cdots\!21}a^{34}+\frac{18\!\cdots\!72}{91\!\cdots\!21}a^{33}+\frac{10\!\cdots\!75}{91\!\cdots\!21}a^{32}+\frac{63\!\cdots\!49}{91\!\cdots\!21}a^{31}+\frac{43\!\cdots\!38}{91\!\cdots\!21}a^{30}+\frac{19\!\cdots\!44}{91\!\cdots\!21}a^{29}+\frac{14\!\cdots\!23}{91\!\cdots\!21}a^{28}+\frac{50\!\cdots\!60}{91\!\cdots\!21}a^{27}+\frac{41\!\cdots\!10}{91\!\cdots\!21}a^{26}+\frac{11\!\cdots\!00}{91\!\cdots\!21}a^{25}+\frac{10\!\cdots\!11}{91\!\cdots\!21}a^{24}+\frac{21\!\cdots\!26}{91\!\cdots\!21}a^{23}+\frac{22\!\cdots\!15}{91\!\cdots\!21}a^{22}+\frac{38\!\cdots\!22}{91\!\cdots\!21}a^{21}+\frac{38\!\cdots\!80}{91\!\cdots\!21}a^{20}+\frac{54\!\cdots\!24}{91\!\cdots\!21}a^{19}+\frac{56\!\cdots\!95}{91\!\cdots\!21}a^{18}+\frac{60\!\cdots\!20}{91\!\cdots\!21}a^{17}+\frac{69\!\cdots\!16}{91\!\cdots\!21}a^{16}+\frac{63\!\cdots\!42}{91\!\cdots\!21}a^{15}+\frac{75\!\cdots\!79}{91\!\cdots\!21}a^{14}+\frac{56\!\cdots\!22}{91\!\cdots\!21}a^{13}+\frac{54\!\cdots\!28}{91\!\cdots\!21}a^{12}+\frac{32\!\cdots\!75}{91\!\cdots\!21}a^{11}+\frac{30\!\cdots\!86}{91\!\cdots\!21}a^{10}+\frac{10\!\cdots\!68}{91\!\cdots\!21}a^{9}+\frac{12\!\cdots\!71}{91\!\cdots\!21}a^{8}+\frac{46\!\cdots\!01}{91\!\cdots\!21}a^{7}+\frac{49\!\cdots\!44}{91\!\cdots\!21}a^{6}+\frac{20\!\cdots\!27}{91\!\cdots\!21}a^{5}+\frac{85\!\cdots\!51}{91\!\cdots\!21}a^{4}+\frac{24\!\cdots\!12}{91\!\cdots\!21}a^{3}+\frac{11\!\cdots\!83}{91\!\cdots\!21}a^{2}+\frac{47\!\cdots\!21}{91\!\cdots\!21}a+\frac{39\!\cdots\!65}{91\!\cdots\!21}$ (assuming GRH)
Regulator:		\( 36326958556899850 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{22}\cdot 36326958556899850 \cdot 45013}{10\cdot\sqrt{342865339180420288801608222738062084913425127327306009945459663867950439453125}}\cr\approx \mathstrut & 0.101382339536813 \end{aligned}\] (assuming GRH)

Galois group

$C_{44}$ (as 44T1):

A cyclic group of order 44

The 44 conjugacy class representatives for $C_{44}$

Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$44$	$44$	R	$44$	${\href{/padicField/11.11.0.1}{11} }^{4}$	$44$	$44$	$22^{2}$	R	$22^{2}$	${\href{/padicField/31.11.0.1}{11} }^{4}$	$44$	${\href{/padicField/41.11.0.1}{11} }^{4}$	$44$	${\href{/padicField/47.4.0.1}{4} }^{11}$	$44$	$22^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(5\)		Deg $44$	$4$	$11$	$33$
\(23\)		Deg $44$	$11$	$4$	$40$