Number field 40.0.100383397447978918530459891214693626269465146363712847232818603515625.1

• Label: 40.0.100...625.1
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $1.004\times 10^{68}$
• Root discriminant: \(50.12\)
• Ramified primes: $3,5,11$
• Class number: $8420$ (GRH)
• Class group: [2, 4210] (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

x^40 - x^39 + 11*x^38 - 11*x^37 + 77*x^36 - 76*x^35 + 439*x^34 - 429*x^33 + 2233*x^32 - 2167*x^31 + 9197*x^30 - 8833*x^29 + 32603*x^28 - 30778*x^27 + 102597*x^26 - 95699*x^25 + 285912*x^24 - 263693*x^23 + 659494*x^22 - 595562*x^21 + 1330759*x^20 - 1163338*x^19 + 2342802*x^18 - 1938861*x^17 + 3371093*x^16 - 2519131*x^15 + 3317622*x^14 - 1836384*x^13 + 2360500*x^12 - 558855*x^11 + 869342*x^10 - 264572*x^9 + 355025*x^8 - 154704*x^7 + 86679*x^6 - 26432*x^5 + 14707*x^4 - 1397*x^3 + 132*x^2 - 12*x + 1

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(100383397447978918530459891214693626269465146363712847232818603515625\) \(\medspace = 3^{20}\cdot 5^{30}\cdot 11^{36}\)
Root discriminant:		\(50.12\)
Galois root discriminant:		$3^{1/2}5^{3/4}11^{9/10}\approx 50.12351825429183$
Ramified primes:		\(3\), \(5\), \(11\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$40$
This field is Galois and abelian over $\Q$.
Conductor:		\(165=3\cdot 5\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(131,·)$, $\chi_{165}(4,·)$, $\chi_{165}(133,·)$, $\chi_{165}(134,·)$, $\chi_{165}(8,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(148,·)$, $\chi_{165}(149,·)$, $\chi_{165}(157,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(161,·)$, $\chi_{165}(34,·)$, $\chi_{165}(163,·)$, $\chi_{165}(164,·)$, $\chi_{165}(37,·)$, $\chi_{165}(41,·)$, $\chi_{165}(29,·)$, $\chi_{165}(49,·)$, $\chi_{165}(58,·)$, $\chi_{165}(62,·)$, $\chi_{165}(64,·)$, $\chi_{165}(67,·)$, $\chi_{165}(68,·)$, $\chi_{165}(74,·)$, $\chi_{165}(82,·)$, $\chi_{165}(83,·)$, $\chi_{165}(91,·)$, $\chi_{165}(97,·)$, $\chi_{165}(98,·)$, $\chi_{165}(101,·)$, $\chi_{165}(103,·)$, $\chi_{165}(107,·)$, $\chi_{165}(116,·)$, $\chi_{165}(136,·)$, $\chi_{165}(124,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{62\!\cdots\!99}a^{37}+\frac{11\!\cdots\!31}{62\!\cdots\!99}a^{36}-\frac{51\!\cdots\!61}{62\!\cdots\!99}a^{35}-\frac{24\!\cdots\!17}{62\!\cdots\!99}a^{34}+\frac{60\!\cdots\!84}{62\!\cdots\!99}a^{33}-\frac{26\!\cdots\!64}{62\!\cdots\!99}a^{32}+\frac{44\!\cdots\!98}{62\!\cdots\!99}a^{31}+\frac{14\!\cdots\!63}{62\!\cdots\!99}a^{30}-\frac{10\!\cdots\!04}{62\!\cdots\!99}a^{29}-\frac{11\!\cdots\!97}{62\!\cdots\!99}a^{28}-\frac{22\!\cdots\!19}{62\!\cdots\!99}a^{27}-\frac{50\!\cdots\!06}{62\!\cdots\!99}a^{26}-\frac{11\!\cdots\!77}{62\!\cdots\!99}a^{25}+\frac{86\!\cdots\!85}{62\!\cdots\!99}a^{24}-\frac{45\!\cdots\!04}{62\!\cdots\!99}a^{23}+\frac{13\!\cdots\!93}{62\!\cdots\!99}a^{22}-\frac{30\!\cdots\!92}{62\!\cdots\!99}a^{21}+\frac{22\!\cdots\!10}{62\!\cdots\!99}a^{20}-\frac{95\!\cdots\!39}{62\!\cdots\!99}a^{19}+\frac{92\!\cdots\!26}{62\!\cdots\!99}a^{18}-\frac{30\!\cdots\!51}{62\!\cdots\!99}a^{17}+\frac{23\!\cdots\!47}{62\!\cdots\!99}a^{16}+\frac{56\!\cdots\!29}{62\!\cdots\!99}a^{15}+\frac{14\!\cdots\!71}{62\!\cdots\!99}a^{14}+\frac{16\!\cdots\!98}{62\!\cdots\!99}a^{13}+\frac{30\!\cdots\!56}{62\!\cdots\!99}a^{12}+\frac{28\!\cdots\!62}{62\!\cdots\!99}a^{11}-\frac{12\!\cdots\!64}{62\!\cdots\!99}a^{10}+\frac{13\!\cdots\!22}{62\!\cdots\!99}a^{9}+\frac{19\!\cdots\!36}{62\!\cdots\!99}a^{8}+\frac{16\!\cdots\!11}{62\!\cdots\!99}a^{7}-\frac{28\!\cdots\!87}{62\!\cdots\!99}a^{6}-\frac{26\!\cdots\!77}{62\!\cdots\!99}a^{5}-\frac{26\!\cdots\!77}{62\!\cdots\!99}a^{4}+\frac{86\!\cdots\!80}{62\!\cdots\!99}a^{3}+\frac{19\!\cdots\!40}{62\!\cdots\!99}a^{2}-\frac{23\!\cdots\!05}{62\!\cdots\!99}a-\frac{87\!\cdots\!28}{62\!\cdots\!99}$, $\frac{1}{62\!\cdots\!99}a^{38}-\frac{27\!\cdots\!74}{62\!\cdots\!99}a^{36}-\frac{16\!\cdots\!63}{62\!\cdots\!99}a^{35}-\frac{10\!\cdots\!15}{62\!\cdots\!99}a^{34}+\frac{22\!\cdots\!99}{62\!\cdots\!99}a^{33}-\frac{22\!\cdots\!14}{62\!\cdots\!99}a^{32}-\frac{22\!\cdots\!42}{62\!\cdots\!99}a^{31}-\frac{95\!\cdots\!12}{62\!\cdots\!99}a^{30}+\frac{11\!\cdots\!49}{62\!\cdots\!99}a^{29}-\frac{37\!\cdots\!69}{62\!\cdots\!99}a^{28}-\frac{13\!\cdots\!64}{62\!\cdots\!99}a^{27}-\frac{67\!\cdots\!84}{62\!\cdots\!99}a^{26}+\frac{72\!\cdots\!16}{62\!\cdots\!99}a^{25}-\frac{65\!\cdots\!72}{62\!\cdots\!99}a^{24}-\frac{23\!\cdots\!59}{48\!\cdots\!29}a^{23}-\frac{12\!\cdots\!00}{62\!\cdots\!99}a^{22}-\frac{62\!\cdots\!27}{62\!\cdots\!99}a^{21}+\frac{15\!\cdots\!02}{62\!\cdots\!99}a^{20}+\frac{19\!\cdots\!06}{62\!\cdots\!99}a^{19}+\frac{26\!\cdots\!77}{62\!\cdots\!99}a^{18}-\frac{30\!\cdots\!74}{62\!\cdots\!99}a^{17}-\frac{81\!\cdots\!40}{62\!\cdots\!99}a^{16}-\frac{73\!\cdots\!31}{62\!\cdots\!99}a^{15}-\frac{84\!\cdots\!02}{62\!\cdots\!99}a^{14}-\frac{14\!\cdots\!32}{62\!\cdots\!99}a^{13}-\frac{18\!\cdots\!38}{62\!\cdots\!99}a^{12}+\frac{75\!\cdots\!78}{62\!\cdots\!99}a^{11}+\frac{27\!\cdots\!91}{62\!\cdots\!99}a^{10}+\frac{86\!\cdots\!59}{62\!\cdots\!99}a^{9}+\frac{84\!\cdots\!90}{62\!\cdots\!99}a^{8}+\frac{35\!\cdots\!17}{62\!\cdots\!99}a^{7}+\frac{24\!\cdots\!78}{62\!\cdots\!99}a^{6}+\frac{62\!\cdots\!63}{62\!\cdots\!99}a^{5}+\frac{11\!\cdots\!57}{62\!\cdots\!99}a^{4}-\frac{98\!\cdots\!56}{62\!\cdots\!99}a^{3}+\frac{20\!\cdots\!28}{62\!\cdots\!99}a^{2}-\frac{18\!\cdots\!91}{62\!\cdots\!99}a-\frac{28\!\cdots\!24}{62\!\cdots\!99}$, $\frac{1}{62\!\cdots\!99}a^{39}-\frac{71\!\cdots\!95}{62\!\cdots\!99}a^{36}+\frac{13\!\cdots\!69}{62\!\cdots\!99}a^{35}-\frac{50\!\cdots\!59}{62\!\cdots\!99}a^{34}+\frac{23\!\cdots\!38}{62\!\cdots\!99}a^{33}+\frac{91\!\cdots\!87}{62\!\cdots\!99}a^{32}+\frac{16\!\cdots\!03}{62\!\cdots\!99}a^{31}-\frac{14\!\cdots\!44}{62\!\cdots\!99}a^{30}+\frac{20\!\cdots\!33}{62\!\cdots\!99}a^{29}+\frac{11\!\cdots\!62}{62\!\cdots\!99}a^{28}+\frac{31\!\cdots\!18}{62\!\cdots\!99}a^{27}-\frac{21\!\cdots\!61}{62\!\cdots\!99}a^{26}-\frac{74\!\cdots\!10}{62\!\cdots\!99}a^{25}-\frac{23\!\cdots\!00}{62\!\cdots\!99}a^{24}+\frac{16\!\cdots\!24}{62\!\cdots\!99}a^{23}-\frac{10\!\cdots\!97}{62\!\cdots\!99}a^{22}-\frac{30\!\cdots\!38}{62\!\cdots\!99}a^{21}+\frac{28\!\cdots\!50}{62\!\cdots\!99}a^{20}-\frac{35\!\cdots\!02}{62\!\cdots\!99}a^{19}-\frac{24\!\cdots\!07}{62\!\cdots\!99}a^{18}+\frac{27\!\cdots\!26}{62\!\cdots\!99}a^{17}-\frac{13\!\cdots\!62}{62\!\cdots\!99}a^{16}+\frac{14\!\cdots\!32}{62\!\cdots\!99}a^{15}-\frac{32\!\cdots\!03}{62\!\cdots\!99}a^{14}+\frac{29\!\cdots\!30}{62\!\cdots\!99}a^{13}-\frac{18\!\cdots\!17}{62\!\cdots\!99}a^{12}+\frac{27\!\cdots\!52}{62\!\cdots\!99}a^{11}-\frac{26\!\cdots\!02}{62\!\cdots\!99}a^{10}+\frac{26\!\cdots\!08}{62\!\cdots\!99}a^{9}+\frac{41\!\cdots\!27}{62\!\cdots\!99}a^{8}+\frac{23\!\cdots\!54}{62\!\cdots\!99}a^{7}+\frac{29\!\cdots\!76}{62\!\cdots\!99}a^{6}-\frac{70\!\cdots\!93}{62\!\cdots\!99}a^{5}+\frac{12\!\cdots\!88}{62\!\cdots\!99}a^{4}-\frac{30\!\cdots\!20}{62\!\cdots\!99}a^{3}-\frac{50\!\cdots\!09}{62\!\cdots\!99}a^{2}-\frac{19\!\cdots\!44}{62\!\cdots\!99}a+\frac{12\!\cdots\!37}{62\!\cdots\!99}$

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

Class group and class number

$C_{2}\times C_{4210}$, which has order $8420$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( -\frac{59880346098408791098299272789362685357516}{629634570480290540834950639222275992422799} a^{39} + \frac{59923417166230540017835647848862686469485}{629634570480290540834950639222275992422799} a^{38} - \frac{658684870851470608554420434033240777735632}{629634570480290540834950639222275992422799} a^{37} + \frac{659102621621900436516916348162576903091230}{629634570480290540834950639222275992422799} a^{36} - \frac{4610794095960294259880943038232685444149424}{629634570480290540834950639222275992422799} a^{35} + \frac{4553629146411418780622486716850854805405508}{629634570480290540834950639222275992422799} a^{34} - \frac{26287470417488836793001176699332133647536284}{629634570480290540834950639222275992422799} a^{33} + \frac{25703507135233631324083436411982875192969514}{629634570480290540834950639222275992422799} a^{32} - \frac{133712600038636776607848997038983559225363077}{629634570480290540834950639222275992422799} a^{31} + \frac{129834014949123602677921755321547272621534533}{629634570480290540834950639222275992422799} a^{30} - \frac{550717774079340817421172323929002886189816475}{629634570480290540834950639222275992422799} a^{29} + \frac{529203987224656135713637988018602321394953229}{629634570480290540834950639222275992422799} a^{28} - \frac{1952266796876913291595389085116591016476079057}{629634570480290540834950639222275992422799} a^{27} + \frac{1843935227846960699017541795713366892238970523}{629634570480290540834950639222275992422799} a^{26} - \frac{6143477561909504249484659739055067873592457650}{629634570480290540834950639222275992422799} a^{25} + \frac{5733290415949261471238810642039932716657538655}{629634570480290540834950639222275992422799} a^{24} - \frac{17120253785234335381878357574589049629028461427}{629634570480290540834950639222275992422799} a^{23} + \frac{15797333989026247915531403973640215763445528824}{629634570480290540834950639222275992422799} a^{22} - \frac{39489889866686315408936674624883437413642872819}{629634570480290540834950639222275992422799} a^{21} + \frac{35677141842047483236327771644652129762158219934}{629634570480290540834950639222275992422799} a^{20} - \frac{79683818001439454646881866492165583462781850252}{629634570480290540834950639222275992422799} a^{19} + \frac{69687756671595218894154698639152661628431092973}{629634570480290540834950639222275992422799} a^{18} - \frac{140281329543899637550028426487498901407507488892}{629634570480290540834950639222275992422799} a^{17} + \frac{116141065724587239923898970013023413143451445866}{629634570480290540834950639222275992422799} a^{16} - \frac{201846595116391300426723804454894605012413800413}{629634570480290540834950639222275992422799} a^{15} + \frac{150892754544876901405905018985861207897128250023}{629634570480290540834950639222275992422799} a^{14} - \frac{198627743652997438843782079881948397446261793932}{629634570480290540834950639222275992422799} a^{13} + \frac{109978283778736209799638562652239529010803785905}{629634570480290540834950639222275992422799} a^{12} - \frac{141292409726405291153449228629236179962201937439}{629634570480290540834950639222275992422799} a^{11} + \frac{33474063152214351307038059166616931237778466293}{629634570480290540834950639222275992422799} a^{10} - \frac{51996724578570492517080933859754757404097951946}{629634570480290540834950639222275992422799} a^{9} + \frac{15846430208429339102104373485823256018140980561}{629634570480290540834950639222275992422799} a^{8} - \frac{21252836162408427135483941509097837595973280506}{629634570480290540834950639222275992422799} a^{7} + \frac{9261548468520440715248056576602962396089895433}{629634570480290540834950639222275992422799} a^{6} - \frac{39606584930697257627771268775517217532247550}{4806370767025118632327867474979206049029} a^{5} + \frac{1573152445923325012601706492198424947717742599}{629634570480290540834950639222275992422799} a^{4} - \frac{6719164888834533842323566636973089673397143}{4806370767025118632327867474979206049029} a^{3} + \frac{83610104176961074416576288145789713275403364}{629634570480290540834950639222275992422799} a^{2} - \frac{7900150876926620030386637623455353375417814}{629634570480290540834950639222275992422799} a + \frac{718191536651028329767565181901402114269030}{629634570480290540834950639222275992422799} \)  (order $10$)
Fundamental units:		$\frac{59\!\cdots\!16}{62\!\cdots\!99}a^{39}-\frac{59\!\cdots\!85}{62\!\cdots\!99}a^{38}+\frac{65\!\cdots\!32}{62\!\cdots\!99}a^{37}-\frac{65\!\cdots\!30}{62\!\cdots\!99}a^{36}+\frac{46\!\cdots\!24}{62\!\cdots\!99}a^{35}-\frac{45\!\cdots\!08}{62\!\cdots\!99}a^{34}+\frac{26\!\cdots\!84}{62\!\cdots\!99}a^{33}-\frac{25\!\cdots\!14}{62\!\cdots\!99}a^{32}+\frac{13\!\cdots\!77}{62\!\cdots\!99}a^{31}-\frac{12\!\cdots\!33}{62\!\cdots\!99}a^{30}+\frac{55\!\cdots\!75}{62\!\cdots\!99}a^{29}-\frac{52\!\cdots\!29}{62\!\cdots\!99}a^{28}+\frac{19\!\cdots\!57}{62\!\cdots\!99}a^{27}-\frac{18\!\cdots\!23}{62\!\cdots\!99}a^{26}+\frac{61\!\cdots\!50}{62\!\cdots\!99}a^{25}-\frac{57\!\cdots\!55}{62\!\cdots\!99}a^{24}+\frac{17\!\cdots\!27}{62\!\cdots\!99}a^{23}-\frac{15\!\cdots\!24}{62\!\cdots\!99}a^{22}+\frac{39\!\cdots\!19}{62\!\cdots\!99}a^{21}-\frac{35\!\cdots\!34}{62\!\cdots\!99}a^{20}+\frac{79\!\cdots\!52}{62\!\cdots\!99}a^{19}-\frac{69\!\cdots\!73}{62\!\cdots\!99}a^{18}+\frac{14\!\cdots\!92}{62\!\cdots\!99}a^{17}-\frac{11\!\cdots\!66}{62\!\cdots\!99}a^{16}+\frac{20\!\cdots\!13}{62\!\cdots\!99}a^{15}-\frac{15\!\cdots\!23}{62\!\cdots\!99}a^{14}+\frac{19\!\cdots\!32}{62\!\cdots\!99}a^{13}-\frac{10\!\cdots\!05}{62\!\cdots\!99}a^{12}+\frac{14\!\cdots\!39}{62\!\cdots\!99}a^{11}-\frac{33\!\cdots\!93}{62\!\cdots\!99}a^{10}+\frac{51\!\cdots\!46}{62\!\cdots\!99}a^{9}-\frac{15\!\cdots\!61}{62\!\cdots\!99}a^{8}+\frac{21\!\cdots\!06}{62\!\cdots\!99}a^{7}-\frac{92\!\cdots\!33}{62\!\cdots\!99}a^{6}+\frac{39\!\cdots\!50}{48\!\cdots\!29}a^{5}-\frac{15\!\cdots\!99}{62\!\cdots\!99}a^{4}+\frac{67\!\cdots\!43}{48\!\cdots\!29}a^{3}-\frac{83\!\cdots\!64}{62\!\cdots\!99}a^{2}+\frac{79\!\cdots\!14}{62\!\cdots\!99}a-\frac{88\!\cdots\!31}{62\!\cdots\!99}$, $\frac{15\!\cdots\!08}{62\!\cdots\!99}a^{39}+\frac{40\!\cdots\!96}{62\!\cdots\!99}a^{38}+\frac{15\!\cdots\!38}{62\!\cdots\!99}a^{37}+\frac{23\!\cdots\!59}{62\!\cdots\!99}a^{36}+\frac{10\!\cdots\!07}{62\!\cdots\!99}a^{35}+\frac{24\!\cdots\!53}{62\!\cdots\!99}a^{34}+\frac{58\!\cdots\!53}{62\!\cdots\!99}a^{33}+\frac{18\!\cdots\!24}{62\!\cdots\!99}a^{32}+\frac{29\!\cdots\!67}{62\!\cdots\!99}a^{31}+\frac{11\!\cdots\!27}{62\!\cdots\!99}a^{30}+\frac{11\!\cdots\!77}{62\!\cdots\!99}a^{29}+\frac{52\!\cdots\!53}{62\!\cdots\!99}a^{28}+\frac{38\!\cdots\!78}{62\!\cdots\!99}a^{27}+\frac{25\!\cdots\!31}{62\!\cdots\!99}a^{26}+\frac{11\!\cdots\!43}{62\!\cdots\!99}a^{25}+\frac{91\!\cdots\!38}{62\!\cdots\!99}a^{24}+\frac{31\!\cdots\!29}{62\!\cdots\!99}a^{23}+\frac{28\!\cdots\!18}{62\!\cdots\!99}a^{22}+\frac{66\!\cdots\!52}{62\!\cdots\!99}a^{21}+\frac{78\!\cdots\!23}{62\!\cdots\!99}a^{20}+\frac{12\!\cdots\!18}{62\!\cdots\!99}a^{19}+\frac{20\!\cdots\!02}{62\!\cdots\!99}a^{18}+\frac{20\!\cdots\!83}{62\!\cdots\!99}a^{17}+\frac{51\!\cdots\!42}{62\!\cdots\!99}a^{16}+\frac{25\!\cdots\!62}{62\!\cdots\!99}a^{15}+\frac{11\!\cdots\!43}{62\!\cdots\!99}a^{14}+\frac{17\!\cdots\!25}{62\!\cdots\!99}a^{13}+\frac{20\!\cdots\!49}{62\!\cdots\!99}a^{12}+\frac{12\!\cdots\!26}{62\!\cdots\!99}a^{11}+\frac{26\!\cdots\!52}{62\!\cdots\!99}a^{10}+\frac{80\!\cdots\!32}{62\!\cdots\!99}a^{9}+\frac{89\!\cdots\!65}{62\!\cdots\!99}a^{8}+\frac{21\!\cdots\!54}{62\!\cdots\!99}a^{7}+\frac{28\!\cdots\!80}{62\!\cdots\!99}a^{6}-\frac{61\!\cdots\!95}{62\!\cdots\!99}a^{5}+\frac{75\!\cdots\!80}{62\!\cdots\!99}a^{4}-\frac{71\!\cdots\!69}{62\!\cdots\!99}a^{3}+\frac{20\!\cdots\!76}{62\!\cdots\!99}a^{2}-\frac{62\!\cdots\!81}{62\!\cdots\!99}a+\frac{57\!\cdots\!87}{62\!\cdots\!99}$, $\frac{33\!\cdots\!46}{62\!\cdots\!99}a^{39}-\frac{42\!\cdots\!10}{62\!\cdots\!99}a^{38}+\frac{36\!\cdots\!10}{62\!\cdots\!99}a^{37}-\frac{45\!\cdots\!69}{62\!\cdots\!99}a^{36}+\frac{25\!\cdots\!46}{62\!\cdots\!99}a^{35}-\frac{23\!\cdots\!82}{48\!\cdots\!29}a^{34}+\frac{14\!\cdots\!28}{62\!\cdots\!99}a^{33}-\frac{17\!\cdots\!16}{62\!\cdots\!99}a^{32}+\frac{74\!\cdots\!25}{62\!\cdots\!99}a^{31}-\frac{88\!\cdots\!80}{62\!\cdots\!99}a^{30}+\frac{30\!\cdots\!16}{62\!\cdots\!99}a^{29}-\frac{36\!\cdots\!54}{62\!\cdots\!99}a^{28}+\frac{10\!\cdots\!22}{62\!\cdots\!99}a^{27}-\frac{12\!\cdots\!97}{62\!\cdots\!99}a^{26}+\frac{33\!\cdots\!10}{62\!\cdots\!99}a^{25}-\frac{38\!\cdots\!50}{62\!\cdots\!99}a^{24}+\frac{94\!\cdots\!90}{62\!\cdots\!99}a^{23}-\frac{10\!\cdots\!96}{62\!\cdots\!99}a^{22}+\frac{21\!\cdots\!24}{62\!\cdots\!99}a^{21}-\frac{23\!\cdots\!66}{62\!\cdots\!99}a^{20}+\frac{43\!\cdots\!08}{62\!\cdots\!99}a^{19}-\frac{45\!\cdots\!70}{62\!\cdots\!99}a^{18}+\frac{76\!\cdots\!20}{62\!\cdots\!99}a^{17}-\frac{76\!\cdots\!50}{62\!\cdots\!99}a^{16}+\frac{10\!\cdots\!84}{62\!\cdots\!99}a^{15}-\frac{98\!\cdots\!06}{62\!\cdots\!99}a^{14}+\frac{10\!\cdots\!90}{62\!\cdots\!99}a^{13}-\frac{70\!\cdots\!44}{62\!\cdots\!99}a^{12}+\frac{66\!\cdots\!89}{62\!\cdots\!99}a^{11}-\frac{25\!\cdots\!38}{62\!\cdots\!99}a^{10}+\frac{13\!\cdots\!02}{62\!\cdots\!99}a^{9}-\frac{13\!\cdots\!66}{62\!\cdots\!99}a^{8}+\frac{66\!\cdots\!30}{62\!\cdots\!99}a^{7}-\frac{65\!\cdots\!40}{62\!\cdots\!99}a^{6}+\frac{93\!\cdots\!44}{48\!\cdots\!29}a^{5}-\frac{52\!\cdots\!02}{62\!\cdots\!99}a^{4}+\frac{49\!\cdots\!12}{62\!\cdots\!99}a^{3}-\frac{46\!\cdots\!20}{62\!\cdots\!99}a^{2}-\frac{86\!\cdots\!63}{62\!\cdots\!99}a-\frac{33\!\cdots\!72}{62\!\cdots\!99}$, $\frac{22\!\cdots\!51}{62\!\cdots\!99}a^{39}-\frac{28\!\cdots\!35}{62\!\cdots\!99}a^{38}+\frac{24\!\cdots\!85}{62\!\cdots\!99}a^{37}-\frac{30\!\cdots\!19}{62\!\cdots\!99}a^{36}+\frac{17\!\cdots\!26}{62\!\cdots\!99}a^{35}-\frac{16\!\cdots\!17}{48\!\cdots\!29}a^{34}+\frac{97\!\cdots\!68}{62\!\cdots\!99}a^{33}-\frac{11\!\cdots\!96}{62\!\cdots\!99}a^{32}+\frac{49\!\cdots\!65}{62\!\cdots\!99}a^{31}-\frac{59\!\cdots\!80}{62\!\cdots\!99}a^{30}+\frac{20\!\cdots\!96}{62\!\cdots\!99}a^{29}-\frac{24\!\cdots\!74}{62\!\cdots\!99}a^{28}+\frac{72\!\cdots\!82}{62\!\cdots\!99}a^{27}-\frac{83\!\cdots\!76}{62\!\cdots\!99}a^{26}+\frac{22\!\cdots\!10}{62\!\cdots\!99}a^{25}-\frac{25\!\cdots\!50}{62\!\cdots\!99}a^{24}+\frac{63\!\cdots\!40}{62\!\cdots\!99}a^{23}-\frac{70\!\cdots\!76}{62\!\cdots\!99}a^{22}+\frac{14\!\cdots\!14}{62\!\cdots\!99}a^{21}-\frac{15\!\cdots\!96}{62\!\cdots\!99}a^{20}+\frac{29\!\cdots\!98}{62\!\cdots\!99}a^{19}-\frac{30\!\cdots\!70}{62\!\cdots\!99}a^{18}+\frac{51\!\cdots\!20}{62\!\cdots\!99}a^{17}-\frac{51\!\cdots\!90}{62\!\cdots\!99}a^{16}+\frac{72\!\cdots\!79}{62\!\cdots\!99}a^{15}-\frac{66\!\cdots\!11}{62\!\cdots\!99}a^{14}+\frac{69\!\cdots\!15}{62\!\cdots\!99}a^{13}-\frac{47\!\cdots\!39}{62\!\cdots\!99}a^{12}+\frac{44\!\cdots\!75}{62\!\cdots\!99}a^{11}-\frac{17\!\cdots\!03}{62\!\cdots\!99}a^{10}+\frac{90\!\cdots\!12}{62\!\cdots\!99}a^{9}-\frac{90\!\cdots\!46}{62\!\cdots\!99}a^{8}+\frac{44\!\cdots\!80}{62\!\cdots\!99}a^{7}-\frac{46\!\cdots\!79}{62\!\cdots\!99}a^{6}+\frac{62\!\cdots\!89}{48\!\cdots\!29}a^{5}-\frac{34\!\cdots\!37}{62\!\cdots\!99}a^{4}+\frac{33\!\cdots\!47}{62\!\cdots\!99}a^{3}-\frac{31\!\cdots\!70}{62\!\cdots\!99}a^{2}-\frac{60\!\cdots\!20}{62\!\cdots\!99}a-\frac{22\!\cdots\!57}{62\!\cdots\!99}$, $\frac{41\!\cdots\!09}{62\!\cdots\!99}a^{39}-\frac{52\!\cdots\!65}{62\!\cdots\!99}a^{38}+\frac{45\!\cdots\!83}{62\!\cdots\!99}a^{37}-\frac{56\!\cdots\!77}{62\!\cdots\!99}a^{36}+\frac{31\!\cdots\!78}{62\!\cdots\!99}a^{35}-\frac{29\!\cdots\!63}{48\!\cdots\!29}a^{34}+\frac{18\!\cdots\!64}{62\!\cdots\!99}a^{33}-\frac{21\!\cdots\!16}{62\!\cdots\!99}a^{32}+\frac{91\!\cdots\!14}{62\!\cdots\!99}a^{31}-\frac{11\!\cdots\!48}{62\!\cdots\!99}a^{30}+\frac{37\!\cdots\!44}{62\!\cdots\!99}a^{29}-\frac{44\!\cdots\!46}{62\!\cdots\!99}a^{28}+\frac{13\!\cdots\!90}{62\!\cdots\!99}a^{27}-\frac{15\!\cdots\!90}{62\!\cdots\!99}a^{26}+\frac{42\!\cdots\!14}{62\!\cdots\!99}a^{25}-\frac{47\!\cdots\!38}{62\!\cdots\!99}a^{24}+\frac{11\!\cdots\!08}{62\!\cdots\!99}a^{23}-\frac{13\!\cdots\!68}{62\!\cdots\!99}a^{22}+\frac{26\!\cdots\!44}{62\!\cdots\!99}a^{21}-\frac{29\!\cdots\!76}{62\!\cdots\!99}a^{20}+\frac{54\!\cdots\!14}{62\!\cdots\!99}a^{19}-\frac{56\!\cdots\!62}{62\!\cdots\!99}a^{18}+\frac{95\!\cdots\!16}{62\!\cdots\!99}a^{17}-\frac{94\!\cdots\!90}{62\!\cdots\!99}a^{16}+\frac{13\!\cdots\!09}{62\!\cdots\!99}a^{15}-\frac{12\!\cdots\!45}{62\!\cdots\!99}a^{14}+\frac{12\!\cdots\!53}{62\!\cdots\!99}a^{13}-\frac{87\!\cdots\!21}{62\!\cdots\!99}a^{12}+\frac{82\!\cdots\!85}{62\!\cdots\!99}a^{11}-\frac{32\!\cdots\!77}{62\!\cdots\!99}a^{10}+\frac{16\!\cdots\!92}{62\!\cdots\!99}a^{9}-\frac{16\!\cdots\!26}{62\!\cdots\!99}a^{8}+\frac{81\!\cdots\!20}{62\!\cdots\!99}a^{7}-\frac{82\!\cdots\!25}{62\!\cdots\!99}a^{6}+\frac{11\!\cdots\!31}{48\!\cdots\!29}a^{5}-\frac{64\!\cdots\!43}{62\!\cdots\!99}a^{4}+\frac{61\!\cdots\!93}{62\!\cdots\!99}a^{3}-\frac{57\!\cdots\!70}{62\!\cdots\!99}a^{2}-\frac{12\!\cdots\!32}{62\!\cdots\!99}a-\frac{41\!\cdots\!55}{62\!\cdots\!99}$, $\frac{22\!\cdots\!95}{62\!\cdots\!99}a^{39}-\frac{17\!\cdots\!91}{62\!\cdots\!99}a^{38}+\frac{24\!\cdots\!06}{62\!\cdots\!99}a^{37}-\frac{18\!\cdots\!50}{62\!\cdots\!99}a^{36}+\frac{16\!\cdots\!80}{62\!\cdots\!99}a^{35}-\frac{13\!\cdots\!61}{62\!\cdots\!99}a^{34}+\frac{94\!\cdots\!58}{62\!\cdots\!99}a^{33}-\frac{73\!\cdots\!95}{62\!\cdots\!99}a^{32}+\frac{47\!\cdots\!38}{62\!\cdots\!99}a^{31}-\frac{37\!\cdots\!36}{62\!\cdots\!99}a^{30}+\frac{19\!\cdots\!22}{62\!\cdots\!99}a^{29}-\frac{15\!\cdots\!38}{62\!\cdots\!99}a^{28}+\frac{68\!\cdots\!54}{62\!\cdots\!99}a^{27}-\frac{52\!\cdots\!92}{62\!\cdots\!99}a^{26}+\frac{21\!\cdots\!47}{62\!\cdots\!99}a^{25}-\frac{16\!\cdots\!03}{62\!\cdots\!99}a^{24}+\frac{59\!\cdots\!44}{62\!\cdots\!99}a^{23}-\frac{44\!\cdots\!41}{62\!\cdots\!99}a^{22}+\frac{13\!\cdots\!31}{62\!\cdots\!99}a^{21}-\frac{99\!\cdots\!31}{62\!\cdots\!99}a^{20}+\frac{26\!\cdots\!98}{62\!\cdots\!99}a^{19}-\frac{19\!\cdots\!15}{62\!\cdots\!99}a^{18}+\frac{46\!\cdots\!40}{62\!\cdots\!99}a^{17}-\frac{31\!\cdots\!19}{62\!\cdots\!99}a^{16}+\frac{66\!\cdots\!68}{62\!\cdots\!99}a^{15}-\frac{39\!\cdots\!39}{62\!\cdots\!99}a^{14}+\frac{62\!\cdots\!21}{62\!\cdots\!99}a^{13}-\frac{24\!\cdots\!51}{62\!\cdots\!99}a^{12}+\frac{44\!\cdots\!77}{62\!\cdots\!99}a^{11}-\frac{78\!\cdots\!87}{62\!\cdots\!99}a^{10}+\frac{17\!\cdots\!49}{62\!\cdots\!99}a^{9}-\frac{15\!\cdots\!76}{62\!\cdots\!99}a^{8}+\frac{69\!\cdots\!66}{62\!\cdots\!99}a^{7}-\frac{16\!\cdots\!02}{62\!\cdots\!99}a^{6}+\frac{12\!\cdots\!92}{62\!\cdots\!99}a^{5}-\frac{18\!\cdots\!40}{62\!\cdots\!99}a^{4}+\frac{22\!\cdots\!03}{62\!\cdots\!99}a^{3}+\frac{31\!\cdots\!99}{62\!\cdots\!99}a^{2}+\frac{20\!\cdots\!04}{62\!\cdots\!99}a-\frac{57\!\cdots\!94}{62\!\cdots\!99}$, $\frac{12\!\cdots\!31}{62\!\cdots\!99}a^{39}-\frac{11\!\cdots\!90}{62\!\cdots\!99}a^{38}+\frac{13\!\cdots\!88}{62\!\cdots\!99}a^{37}-\frac{12\!\cdots\!05}{62\!\cdots\!99}a^{36}+\frac{94\!\cdots\!49}{62\!\cdots\!99}a^{35}-\frac{89\!\cdots\!09}{62\!\cdots\!99}a^{34}+\frac{53\!\cdots\!39}{62\!\cdots\!99}a^{33}-\frac{50\!\cdots\!81}{62\!\cdots\!99}a^{32}+\frac{27\!\cdots\!00}{62\!\cdots\!99}a^{31}-\frac{25\!\cdots\!89}{62\!\cdots\!99}a^{30}+\frac{11\!\cdots\!98}{62\!\cdots\!99}a^{29}-\frac{10\!\cdots\!59}{62\!\cdots\!99}a^{28}+\frac{39\!\cdots\!13}{62\!\cdots\!99}a^{27}-\frac{36\!\cdots\!77}{62\!\cdots\!99}a^{26}+\frac{12\!\cdots\!57}{62\!\cdots\!99}a^{25}-\frac{11\!\cdots\!21}{62\!\cdots\!99}a^{24}+\frac{34\!\cdots\!59}{62\!\cdots\!99}a^{23}-\frac{30\!\cdots\!29}{62\!\cdots\!99}a^{22}+\frac{79\!\cdots\!04}{62\!\cdots\!99}a^{21}-\frac{69\!\cdots\!85}{62\!\cdots\!99}a^{20}+\frac{16\!\cdots\!69}{62\!\cdots\!99}a^{19}-\frac{13\!\cdots\!44}{62\!\cdots\!99}a^{18}+\frac{28\!\cdots\!05}{62\!\cdots\!99}a^{17}-\frac{17\!\cdots\!36}{48\!\cdots\!29}a^{16}+\frac{40\!\cdots\!48}{62\!\cdots\!99}a^{15}-\frac{29\!\cdots\!31}{62\!\cdots\!99}a^{14}+\frac{39\!\cdots\!27}{62\!\cdots\!99}a^{13}-\frac{20\!\cdots\!11}{62\!\cdots\!99}a^{12}+\frac{28\!\cdots\!14}{62\!\cdots\!99}a^{11}-\frac{55\!\cdots\!42}{62\!\cdots\!99}a^{10}+\frac{10\!\cdots\!56}{62\!\cdots\!99}a^{9}-\frac{27\!\cdots\!89}{62\!\cdots\!99}a^{8}+\frac{42\!\cdots\!53}{62\!\cdots\!99}a^{7}-\frac{17\!\cdots\!44}{62\!\cdots\!99}a^{6}+\frac{97\!\cdots\!65}{62\!\cdots\!99}a^{5}-\frac{27\!\cdots\!49}{62\!\cdots\!99}a^{4}+\frac{16\!\cdots\!89}{62\!\cdots\!99}a^{3}-\frac{88\!\cdots\!86}{62\!\cdots\!99}a^{2}+\frac{83\!\cdots\!71}{62\!\cdots\!99}a-\frac{13\!\cdots\!79}{62\!\cdots\!99}$, $\frac{61\!\cdots\!24}{62\!\cdots\!99}a^{39}-\frac{59\!\cdots\!89}{62\!\cdots\!99}a^{38}+\frac{67\!\cdots\!70}{62\!\cdots\!99}a^{37}-\frac{65\!\cdots\!71}{62\!\cdots\!99}a^{36}+\frac{47\!\cdots\!31}{62\!\cdots\!99}a^{35}-\frac{45\!\cdots\!55}{62\!\cdots\!99}a^{34}+\frac{26\!\cdots\!37}{62\!\cdots\!99}a^{33}-\frac{25\!\cdots\!90}{62\!\cdots\!99}a^{32}+\frac{13\!\cdots\!44}{62\!\cdots\!99}a^{31}-\frac{12\!\cdots\!06}{62\!\cdots\!99}a^{30}+\frac{56\!\cdots\!52}{62\!\cdots\!99}a^{29}-\frac{52\!\cdots\!76}{62\!\cdots\!99}a^{28}+\frac{19\!\cdots\!35}{62\!\cdots\!99}a^{27}-\frac{18\!\cdots\!92}{62\!\cdots\!99}a^{26}+\frac{62\!\cdots\!93}{62\!\cdots\!99}a^{25}-\frac{57\!\cdots\!17}{62\!\cdots\!99}a^{24}+\frac{17\!\cdots\!56}{62\!\cdots\!99}a^{23}-\frac{15\!\cdots\!06}{62\!\cdots\!99}a^{22}+\frac{40\!\cdots\!71}{62\!\cdots\!99}a^{21}-\frac{35\!\cdots\!11}{62\!\cdots\!99}a^{20}+\frac{80\!\cdots\!70}{62\!\cdots\!99}a^{19}-\frac{69\!\cdots\!71}{62\!\cdots\!99}a^{18}+\frac{14\!\cdots\!75}{62\!\cdots\!99}a^{17}-\frac{11\!\cdots\!24}{62\!\cdots\!99}a^{16}+\frac{20\!\cdots\!75}{62\!\cdots\!99}a^{15}-\frac{14\!\cdots\!80}{62\!\cdots\!99}a^{14}+\frac{20\!\cdots\!57}{62\!\cdots\!99}a^{13}-\frac{10\!\cdots\!56}{62\!\cdots\!99}a^{12}+\frac{14\!\cdots\!65}{62\!\cdots\!99}a^{11}-\frac{30\!\cdots\!41}{62\!\cdots\!99}a^{10}+\frac{52\!\cdots\!78}{62\!\cdots\!99}a^{9}-\frac{14\!\cdots\!96}{62\!\cdots\!99}a^{8}+\frac{21\!\cdots\!60}{62\!\cdots\!99}a^{7}-\frac{89\!\cdots\!53}{62\!\cdots\!99}a^{6}+\frac{51\!\cdots\!55}{62\!\cdots\!99}a^{5}-\frac{14\!\cdots\!19}{62\!\cdots\!99}a^{4}+\frac{87\!\cdots\!64}{62\!\cdots\!99}a^{3}-\frac{63\!\cdots\!88}{62\!\cdots\!99}a^{2}+\frac{78\!\cdots\!33}{62\!\cdots\!99}a-\frac{82\!\cdots\!44}{62\!\cdots\!99}$, $\frac{50\!\cdots\!59}{62\!\cdots\!99}a^{39}+\frac{15\!\cdots\!27}{62\!\cdots\!99}a^{38}+\frac{50\!\cdots\!65}{62\!\cdots\!99}a^{37}+\frac{16\!\cdots\!23}{62\!\cdots\!99}a^{36}+\frac{32\!\cdots\!76}{62\!\cdots\!99}a^{35}+\frac{11\!\cdots\!50}{62\!\cdots\!99}a^{34}+\frac{17\!\cdots\!29}{62\!\cdots\!99}a^{33}+\frac{66\!\cdots\!87}{62\!\cdots\!99}a^{32}+\frac{88\!\cdots\!81}{62\!\cdots\!99}a^{31}+\frac{33\!\cdots\!31}{62\!\cdots\!99}a^{30}+\frac{34\!\cdots\!75}{62\!\cdots\!99}a^{29}+\frac{14\!\cdots\!16}{62\!\cdots\!99}a^{28}+\frac{11\!\cdots\!46}{62\!\cdots\!99}a^{27}+\frac{51\!\cdots\!92}{62\!\cdots\!99}a^{26}+\frac{34\!\cdots\!10}{62\!\cdots\!99}a^{25}+\frac{16\!\cdots\!83}{62\!\cdots\!99}a^{24}+\frac{90\!\cdots\!14}{62\!\cdots\!99}a^{23}+\frac{47\!\cdots\!61}{62\!\cdots\!99}a^{22}+\frac{18\!\cdots\!31}{62\!\cdots\!99}a^{21}+\frac{11\!\cdots\!84}{62\!\cdots\!99}a^{20}+\frac{33\!\cdots\!64}{62\!\cdots\!99}a^{19}+\frac{23\!\cdots\!81}{62\!\cdots\!99}a^{18}+\frac{53\!\cdots\!29}{62\!\cdots\!99}a^{17}+\frac{46\!\cdots\!61}{62\!\cdots\!99}a^{16}+\frac{62\!\cdots\!82}{62\!\cdots\!99}a^{15}+\frac{78\!\cdots\!24}{62\!\cdots\!99}a^{14}+\frac{30\!\cdots\!22}{62\!\cdots\!99}a^{13}+\frac{10\!\cdots\!81}{62\!\cdots\!99}a^{12}+\frac{26\!\cdots\!83}{62\!\cdots\!99}a^{11}+\frac{11\!\cdots\!41}{62\!\cdots\!99}a^{10}+\frac{20\!\cdots\!11}{48\!\cdots\!29}a^{9}+\frac{39\!\cdots\!80}{62\!\cdots\!99}a^{8}+\frac{77\!\cdots\!58}{62\!\cdots\!99}a^{7}+\frac{13\!\cdots\!06}{62\!\cdots\!99}a^{6}-\frac{28\!\cdots\!07}{62\!\cdots\!99}a^{5}+\frac{31\!\cdots\!91}{62\!\cdots\!99}a^{4}-\frac{29\!\cdots\!17}{62\!\cdots\!99}a^{3}+\frac{72\!\cdots\!78}{62\!\cdots\!99}a^{2}+\frac{52\!\cdots\!18}{62\!\cdots\!99}a+\frac{65\!\cdots\!00}{62\!\cdots\!99}$, $\frac{99\!\cdots\!91}{62\!\cdots\!99}a^{39}-\frac{99\!\cdots\!51}{62\!\cdots\!99}a^{38}+\frac{10\!\cdots\!86}{62\!\cdots\!99}a^{37}-\frac{10\!\cdots\!36}{62\!\cdots\!99}a^{36}+\frac{76\!\cdots\!02}{62\!\cdots\!99}a^{35}-\frac{75\!\cdots\!60}{62\!\cdots\!99}a^{34}+\frac{43\!\cdots\!14}{62\!\cdots\!99}a^{33}-\frac{42\!\cdots\!04}{62\!\cdots\!99}a^{32}+\frac{22\!\cdots\!83}{62\!\cdots\!99}a^{31}-\frac{21\!\cdots\!67}{62\!\cdots\!99}a^{30}+\frac{91\!\cdots\!03}{62\!\cdots\!99}a^{29}-\frac{87\!\cdots\!83}{62\!\cdots\!99}a^{28}+\frac{32\!\cdots\!03}{62\!\cdots\!99}a^{27}-\frac{30\!\cdots\!03}{62\!\cdots\!99}a^{26}+\frac{10\!\cdots\!72}{62\!\cdots\!99}a^{25}-\frac{94\!\cdots\!10}{62\!\cdots\!99}a^{24}+\frac{28\!\cdots\!87}{62\!\cdots\!99}a^{23}-\frac{26\!\cdots\!18}{62\!\cdots\!99}a^{22}+\frac{65\!\cdots\!19}{62\!\cdots\!99}a^{21}-\frac{59\!\cdots\!62}{62\!\cdots\!99}a^{20}+\frac{13\!\cdots\!12}{62\!\cdots\!99}a^{19}-\frac{11\!\cdots\!13}{62\!\cdots\!99}a^{18}+\frac{23\!\cdots\!52}{62\!\cdots\!99}a^{17}-\frac{19\!\cdots\!36}{62\!\cdots\!99}a^{16}+\frac{33\!\cdots\!43}{62\!\cdots\!99}a^{15}-\frac{24\!\cdots\!63}{62\!\cdots\!99}a^{14}+\frac{32\!\cdots\!72}{62\!\cdots\!99}a^{13}-\frac{18\!\cdots\!09}{62\!\cdots\!99}a^{12}+\frac{23\!\cdots\!75}{62\!\cdots\!99}a^{11}-\frac{55\!\cdots\!55}{62\!\cdots\!99}a^{10}+\frac{85\!\cdots\!19}{62\!\cdots\!99}a^{9}-\frac{26\!\cdots\!97}{62\!\cdots\!99}a^{8}+\frac{35\!\cdots\!00}{62\!\cdots\!99}a^{7}-\frac{15\!\cdots\!29}{62\!\cdots\!99}a^{6}+\frac{65\!\cdots\!84}{48\!\cdots\!29}a^{5}-\frac{25\!\cdots\!91}{62\!\cdots\!99}a^{4}+\frac{11\!\cdots\!97}{48\!\cdots\!29}a^{3}-\frac{13\!\cdots\!22}{62\!\cdots\!99}a^{2}+\frac{13\!\cdots\!82}{62\!\cdots\!99}a-\frac{11\!\cdots\!62}{62\!\cdots\!99}$, $\frac{12\!\cdots\!86}{62\!\cdots\!99}a^{39}-\frac{25\!\cdots\!68}{62\!\cdots\!99}a^{38}+\frac{12\!\cdots\!65}{62\!\cdots\!99}a^{37}-\frac{33\!\cdots\!72}{62\!\cdots\!99}a^{36}+\frac{82\!\cdots\!69}{62\!\cdots\!99}a^{35}-\frac{14\!\cdots\!49}{62\!\cdots\!99}a^{34}+\frac{45\!\cdots\!01}{62\!\cdots\!99}a^{33}-\frac{36\!\cdots\!19}{62\!\cdots\!99}a^{32}+\frac{22\!\cdots\!89}{62\!\cdots\!99}a^{31}-\frac{25\!\cdots\!41}{62\!\cdots\!99}a^{30}+\frac{89\!\cdots\!84}{62\!\cdots\!99}a^{29}+\frac{77\!\cdots\!26}{62\!\cdots\!99}a^{28}+\frac{30\!\cdots\!04}{62\!\cdots\!99}a^{27}+\frac{85\!\cdots\!02}{62\!\cdots\!99}a^{26}+\frac{92\!\cdots\!56}{62\!\cdots\!99}a^{25}+\frac{38\!\cdots\!71}{62\!\cdots\!99}a^{24}+\frac{24\!\cdots\!43}{62\!\cdots\!99}a^{23}+\frac{13\!\cdots\!43}{62\!\cdots\!99}a^{22}+\frac{51\!\cdots\!09}{62\!\cdots\!99}a^{21}+\frac{44\!\cdots\!41}{62\!\cdots\!99}a^{20}+\frac{97\!\cdots\!31}{62\!\cdots\!99}a^{19}+\frac{13\!\cdots\!34}{62\!\cdots\!99}a^{18}+\frac{15\!\cdots\!57}{62\!\cdots\!99}a^{17}+\frac{35\!\cdots\!39}{62\!\cdots\!99}a^{16}+\frac{20\!\cdots\!54}{62\!\cdots\!99}a^{15}+\frac{82\!\cdots\!31}{62\!\cdots\!99}a^{14}+\frac{13\!\cdots\!50}{62\!\cdots\!99}a^{13}+\frac{15\!\cdots\!02}{62\!\cdots\!99}a^{12}+\frac{98\!\cdots\!42}{62\!\cdots\!99}a^{11}+\frac{20\!\cdots\!09}{62\!\cdots\!99}a^{10}+\frac{62\!\cdots\!69}{62\!\cdots\!99}a^{9}+\frac{68\!\cdots\!30}{62\!\cdots\!99}a^{8}+\frac{22\!\cdots\!84}{62\!\cdots\!99}a^{7}+\frac{21\!\cdots\!85}{62\!\cdots\!99}a^{6}-\frac{47\!\cdots\!65}{62\!\cdots\!99}a^{5}+\frac{58\!\cdots\!85}{62\!\cdots\!99}a^{4}-\frac{55\!\cdots\!23}{62\!\cdots\!99}a^{3}+\frac{11\!\cdots\!26}{62\!\cdots\!99}a^{2}-\frac{48\!\cdots\!52}{62\!\cdots\!99}a+\frac{43\!\cdots\!29}{62\!\cdots\!99}$, $\frac{68\!\cdots\!05}{62\!\cdots\!99}a^{39}-\frac{18\!\cdots\!95}{62\!\cdots\!99}a^{38}+\frac{66\!\cdots\!55}{62\!\cdots\!99}a^{37}-\frac{13\!\cdots\!65}{62\!\cdots\!99}a^{36}+\frac{43\!\cdots\!85}{62\!\cdots\!99}a^{35}+\frac{62\!\cdots\!25}{62\!\cdots\!99}a^{34}+\frac{23\!\cdots\!25}{62\!\cdots\!99}a^{33}+\frac{64\!\cdots\!95}{62\!\cdots\!99}a^{32}+\frac{11\!\cdots\!95}{62\!\cdots\!99}a^{31}+\frac{42\!\cdots\!26}{62\!\cdots\!99}a^{30}+\frac{44\!\cdots\!45}{62\!\cdots\!99}a^{29}+\frac{23\!\cdots\!45}{62\!\cdots\!99}a^{28}+\frac{14\!\cdots\!50}{62\!\cdots\!99}a^{27}+\frac{12\!\cdots\!15}{62\!\cdots\!99}a^{26}+\frac{44\!\cdots\!49}{62\!\cdots\!99}a^{25}+\frac{44\!\cdots\!00}{62\!\cdots\!99}a^{24}+\frac{11\!\cdots\!45}{62\!\cdots\!99}a^{23}+\frac{14\!\cdots\!50}{62\!\cdots\!99}a^{22}+\frac{23\!\cdots\!90}{62\!\cdots\!99}a^{21}+\frac{41\!\cdots\!09}{62\!\cdots\!99}a^{20}+\frac{42\!\cdots\!50}{62\!\cdots\!99}a^{19}+\frac{10\!\cdots\!50}{62\!\cdots\!99}a^{18}+\frac{65\!\cdots\!25}{62\!\cdots\!99}a^{17}+\frac{25\!\cdots\!75}{62\!\cdots\!99}a^{16}+\frac{73\!\cdots\!70}{62\!\cdots\!99}a^{15}+\frac{52\!\cdots\!50}{62\!\cdots\!99}a^{14}+\frac{23\!\cdots\!20}{62\!\cdots\!99}a^{13}+\frac{88\!\cdots\!20}{62\!\cdots\!99}a^{12}+\frac{15\!\cdots\!35}{62\!\cdots\!99}a^{11}+\frac{98\!\cdots\!84}{62\!\cdots\!99}a^{10}+\frac{59\!\cdots\!80}{62\!\cdots\!99}a^{9}+\frac{99\!\cdots\!55}{62\!\cdots\!99}a^{8}-\frac{35\!\cdots\!20}{62\!\cdots\!99}a^{7}+\frac{30\!\cdots\!05}{62\!\cdots\!99}a^{6}-\frac{85\!\cdots\!95}{62\!\cdots\!99}a^{5}+\frac{72\!\cdots\!05}{62\!\cdots\!99}a^{4}-\frac{68\!\cdots\!35}{62\!\cdots\!99}a^{3}+\frac{64\!\cdots\!60}{62\!\cdots\!99}a^{2}-\frac{59\!\cdots\!40}{62\!\cdots\!99}a-\frac{16\!\cdots\!61}{62\!\cdots\!99}$, $\frac{69\!\cdots\!00}{62\!\cdots\!99}a^{39}+\frac{22\!\cdots\!65}{62\!\cdots\!99}a^{38}+\frac{67\!\cdots\!25}{62\!\cdots\!99}a^{37}+\frac{15\!\cdots\!55}{62\!\cdots\!99}a^{36}+\frac{44\!\cdots\!10}{62\!\cdots\!99}a^{35}+\frac{72\!\cdots\!05}{62\!\cdots\!99}a^{34}+\frac{24\!\cdots\!95}{62\!\cdots\!99}a^{33}+\frac{70\!\cdots\!40}{62\!\cdots\!99}a^{32}+\frac{11\!\cdots\!90}{62\!\cdots\!99}a^{31}+\frac{45\!\cdots\!15}{62\!\cdots\!99}a^{30}+\frac{45\!\cdots\!20}{62\!\cdots\!99}a^{29}+\frac{24\!\cdots\!90}{62\!\cdots\!99}a^{28}+\frac{15\!\cdots\!15}{62\!\cdots\!99}a^{27}+\frac{12\!\cdots\!25}{62\!\cdots\!99}a^{26}+\frac{45\!\cdots\!75}{62\!\cdots\!99}a^{25}+\frac{46\!\cdots\!15}{62\!\cdots\!99}a^{24}+\frac{11\!\cdots\!45}{62\!\cdots\!99}a^{23}+\frac{14\!\cdots\!05}{62\!\cdots\!99}a^{22}+\frac{23\!\cdots\!20}{62\!\cdots\!99}a^{21}+\frac{41\!\cdots\!05}{62\!\cdots\!99}a^{20}+\frac{43\!\cdots\!65}{62\!\cdots\!99}a^{19}+\frac{10\!\cdots\!90}{62\!\cdots\!99}a^{18}+\frac{67\!\cdots\!55}{62\!\cdots\!99}a^{17}+\frac{25\!\cdots\!60}{62\!\cdots\!99}a^{16}+\frac{74\!\cdots\!35}{62\!\cdots\!99}a^{15}+\frac{53\!\cdots\!90}{62\!\cdots\!99}a^{14}+\frac{24\!\cdots\!75}{62\!\cdots\!99}a^{13}+\frac{90\!\cdots\!55}{62\!\cdots\!99}a^{12}+\frac{15\!\cdots\!90}{62\!\cdots\!99}a^{11}+\frac{10\!\cdots\!99}{62\!\cdots\!99}a^{10}+\frac{60\!\cdots\!55}{62\!\cdots\!99}a^{9}+\frac{10\!\cdots\!45}{62\!\cdots\!99}a^{8}-\frac{35\!\cdots\!85}{62\!\cdots\!99}a^{7}+\frac{31\!\cdots\!75}{62\!\cdots\!99}a^{6}-\frac{77\!\cdots\!40}{62\!\cdots\!99}a^{5}+\frac{73\!\cdots\!10}{62\!\cdots\!99}a^{4}-\frac{69\!\cdots\!55}{62\!\cdots\!99}a^{3}+\frac{66\!\cdots\!30}{62\!\cdots\!99}a^{2}-\frac{60\!\cdots\!50}{62\!\cdots\!99}a-\frac{17\!\cdots\!92}{62\!\cdots\!99}$, $\frac{33\!\cdots\!15}{62\!\cdots\!99}a^{39}+\frac{23\!\cdots\!35}{62\!\cdots\!99}a^{38}+\frac{32\!\cdots\!65}{62\!\cdots\!99}a^{37}+\frac{16\!\cdots\!45}{62\!\cdots\!99}a^{36}+\frac{21\!\cdots\!56}{62\!\cdots\!99}a^{35}+\frac{43\!\cdots\!15}{62\!\cdots\!99}a^{34}+\frac{11\!\cdots\!35}{62\!\cdots\!99}a^{33}+\frac{38\!\cdots\!05}{62\!\cdots\!99}a^{32}+\frac{57\!\cdots\!05}{62\!\cdots\!99}a^{31}+\frac{24\!\cdots\!26}{62\!\cdots\!99}a^{30}+\frac{22\!\cdots\!95}{62\!\cdots\!99}a^{29}+\frac{12\!\cdots\!55}{62\!\cdots\!99}a^{28}+\frac{73\!\cdots\!70}{62\!\cdots\!99}a^{27}+\frac{62\!\cdots\!45}{62\!\cdots\!99}a^{26}+\frac{21\!\cdots\!98}{62\!\cdots\!99}a^{25}+\frac{22\!\cdots\!20}{62\!\cdots\!99}a^{24}+\frac{57\!\cdots\!95}{62\!\cdots\!99}a^{23}+\frac{72\!\cdots\!90}{62\!\cdots\!99}a^{22}+\frac{11\!\cdots\!30}{62\!\cdots\!99}a^{21}+\frac{20\!\cdots\!43}{62\!\cdots\!99}a^{20}+\frac{20\!\cdots\!70}{62\!\cdots\!99}a^{19}+\frac{53\!\cdots\!70}{62\!\cdots\!99}a^{18}+\frac{32\!\cdots\!15}{62\!\cdots\!99}a^{17}+\frac{12\!\cdots\!05}{62\!\cdots\!99}a^{16}+\frac{36\!\cdots\!38}{62\!\cdots\!99}a^{15}+\frac{25\!\cdots\!70}{62\!\cdots\!99}a^{14}+\frac{11\!\cdots\!60}{62\!\cdots\!99}a^{13}+\frac{43\!\cdots\!00}{62\!\cdots\!99}a^{12}+\frac{75\!\cdots\!25}{62\!\cdots\!99}a^{11}+\frac{49\!\cdots\!72}{62\!\cdots\!99}a^{10}+\frac{29\!\cdots\!80}{62\!\cdots\!99}a^{9}+\frac{49\!\cdots\!25}{62\!\cdots\!99}a^{8}-\frac{17\!\cdots\!40}{62\!\cdots\!99}a^{7}+\frac{15\!\cdots\!15}{62\!\cdots\!99}a^{6}-\frac{29\!\cdots\!59}{62\!\cdots\!99}a^{5}+\frac{35\!\cdots\!95}{62\!\cdots\!99}a^{4}-\frac{33\!\cdots\!45}{62\!\cdots\!99}a^{3}+\frac{32\!\cdots\!20}{62\!\cdots\!99}a^{2}-\frac{29\!\cdots\!20}{62\!\cdots\!99}a-\frac{12\!\cdots\!41}{62\!\cdots\!99}$, $\frac{22\!\cdots\!64}{62\!\cdots\!99}a^{39}-\frac{19\!\cdots\!89}{62\!\cdots\!99}a^{38}+\frac{24\!\cdots\!88}{62\!\cdots\!99}a^{37}-\frac{21\!\cdots\!73}{62\!\cdots\!99}a^{36}+\frac{16\!\cdots\!01}{62\!\cdots\!99}a^{35}-\frac{15\!\cdots\!69}{62\!\cdots\!99}a^{34}+\frac{96\!\cdots\!01}{62\!\cdots\!99}a^{33}-\frac{84\!\cdots\!19}{62\!\cdots\!99}a^{32}+\frac{48\!\cdots\!38}{62\!\cdots\!99}a^{31}-\frac{42\!\cdots\!38}{62\!\cdots\!99}a^{30}+\frac{20\!\cdots\!04}{62\!\cdots\!99}a^{29}-\frac{17\!\cdots\!63}{62\!\cdots\!99}a^{28}+\frac{71\!\cdots\!70}{62\!\cdots\!99}a^{27}-\frac{60\!\cdots\!64}{62\!\cdots\!99}a^{26}+\frac{22\!\cdots\!07}{62\!\cdots\!99}a^{25}-\frac{18\!\cdots\!83}{62\!\cdots\!99}a^{24}+\frac{61\!\cdots\!67}{62\!\cdots\!99}a^{23}-\frac{52\!\cdots\!37}{62\!\cdots\!99}a^{22}+\frac{14\!\cdots\!65}{62\!\cdots\!99}a^{21}-\frac{11\!\cdots\!05}{62\!\cdots\!99}a^{20}+\frac{28\!\cdots\!70}{62\!\cdots\!99}a^{19}-\frac{22\!\cdots\!26}{62\!\cdots\!99}a^{18}+\frac{50\!\cdots\!15}{62\!\cdots\!99}a^{17}-\frac{37\!\cdots\!60}{62\!\cdots\!99}a^{16}+\frac{71\!\cdots\!65}{62\!\cdots\!99}a^{15}-\frac{48\!\cdots\!60}{62\!\cdots\!99}a^{14}+\frac{69\!\cdots\!95}{62\!\cdots\!99}a^{13}-\frac{33\!\cdots\!64}{62\!\cdots\!99}a^{12}+\frac{49\!\cdots\!11}{62\!\cdots\!99}a^{11}-\frac{72\!\cdots\!71}{62\!\cdots\!99}a^{10}+\frac{19\!\cdots\!34}{62\!\cdots\!99}a^{9}-\frac{38\!\cdots\!90}{62\!\cdots\!99}a^{8}+\frac{76\!\cdots\!43}{62\!\cdots\!99}a^{7}-\frac{26\!\cdots\!07}{62\!\cdots\!99}a^{6}+\frac{17\!\cdots\!57}{62\!\cdots\!99}a^{5}-\frac{43\!\cdots\!49}{62\!\cdots\!99}a^{4}+\frac{29\!\cdots\!45}{62\!\cdots\!99}a^{3}-\frac{38\!\cdots\!12}{62\!\cdots\!99}a^{2}+\frac{69\!\cdots\!19}{62\!\cdots\!99}a-\frac{65\!\cdots\!83}{62\!\cdots\!99}$, $\frac{19\!\cdots\!20}{62\!\cdots\!99}a^{39}-\frac{17\!\cdots\!83}{62\!\cdots\!99}a^{38}+\frac{21\!\cdots\!96}{62\!\cdots\!99}a^{37}-\frac{19\!\cdots\!89}{62\!\cdots\!99}a^{36}+\frac{14\!\cdots\!05}{62\!\cdots\!99}a^{35}-\frac{13\!\cdots\!25}{62\!\cdots\!99}a^{34}+\frac{83\!\cdots\!07}{62\!\cdots\!99}a^{33}-\frac{75\!\cdots\!07}{62\!\cdots\!99}a^{32}+\frac{42\!\cdots\!50}{62\!\cdots\!99}a^{31}-\frac{38\!\cdots\!14}{62\!\cdots\!99}a^{30}+\frac{17\!\cdots\!52}{62\!\cdots\!99}a^{29}-\frac{15\!\cdots\!80}{62\!\cdots\!99}a^{28}+\frac{61\!\cdots\!70}{62\!\cdots\!99}a^{27}-\frac{54\!\cdots\!16}{62\!\cdots\!99}a^{26}+\frac{19\!\cdots\!87}{62\!\cdots\!99}a^{25}-\frac{16\!\cdots\!71}{62\!\cdots\!99}a^{24}+\frac{53\!\cdots\!64}{62\!\cdots\!99}a^{23}-\frac{46\!\cdots\!77}{62\!\cdots\!99}a^{22}+\frac{12\!\cdots\!61}{62\!\cdots\!99}a^{21}-\frac{10\!\cdots\!41}{62\!\cdots\!99}a^{20}+\frac{24\!\cdots\!34}{62\!\cdots\!99}a^{19}-\frac{20\!\cdots\!17}{62\!\cdots\!99}a^{18}+\frac{43\!\cdots\!95}{62\!\cdots\!99}a^{17}-\frac{33\!\cdots\!48}{62\!\cdots\!99}a^{16}+\frac{62\!\cdots\!09}{62\!\cdots\!99}a^{15}-\frac{43\!\cdots\!84}{62\!\cdots\!99}a^{14}+\frac{60\!\cdots\!99}{62\!\cdots\!99}a^{13}-\frac{30\!\cdots\!40}{62\!\cdots\!99}a^{12}+\frac{42\!\cdots\!99}{62\!\cdots\!99}a^{11}-\frac{69\!\cdots\!07}{62\!\cdots\!99}a^{10}+\frac{15\!\cdots\!18}{62\!\cdots\!99}a^{9}-\frac{36\!\cdots\!23}{62\!\cdots\!99}a^{8}+\frac{64\!\cdots\!31}{62\!\cdots\!99}a^{7}-\frac{24\!\cdots\!55}{62\!\cdots\!99}a^{6}+\frac{14\!\cdots\!21}{62\!\cdots\!99}a^{5}-\frac{37\!\cdots\!49}{62\!\cdots\!99}a^{4}+\frac{24\!\cdots\!24}{62\!\cdots\!99}a^{3}-\frac{33\!\cdots\!40}{62\!\cdots\!99}a^{2}+\frac{96\!\cdots\!11}{62\!\cdots\!99}a-\frac{82\!\cdots\!75}{62\!\cdots\!99}$, $\frac{97\!\cdots\!48}{62\!\cdots\!99}a^{39}-\frac{88\!\cdots\!22}{62\!\cdots\!99}a^{38}+\frac{10\!\cdots\!30}{62\!\cdots\!99}a^{37}-\frac{97\!\cdots\!05}{62\!\cdots\!99}a^{36}+\frac{74\!\cdots\!17}{62\!\cdots\!99}a^{35}-\frac{67\!\cdots\!09}{62\!\cdots\!99}a^{34}+\frac{42\!\cdots\!89}{62\!\cdots\!99}a^{33}-\frac{37\!\cdots\!27}{62\!\cdots\!99}a^{32}+\frac{21\!\cdots\!02}{62\!\cdots\!99}a^{31}-\frac{19\!\cdots\!20}{62\!\cdots\!99}a^{30}+\frac{87\!\cdots\!98}{62\!\cdots\!99}a^{29}-\frac{78\!\cdots\!67}{62\!\cdots\!99}a^{28}+\frac{31\!\cdots\!36}{62\!\cdots\!99}a^{27}-\frac{27\!\cdots\!30}{62\!\cdots\!99}a^{26}+\frac{97\!\cdots\!49}{62\!\cdots\!99}a^{25}-\frac{84\!\cdots\!95}{62\!\cdots\!99}a^{24}+\frac{27\!\cdots\!49}{62\!\cdots\!99}a^{23}-\frac{23\!\cdots\!75}{62\!\cdots\!99}a^{22}+\frac{62\!\cdots\!05}{62\!\cdots\!99}a^{21}-\frac{52\!\cdots\!75}{62\!\cdots\!99}a^{20}+\frac{12\!\cdots\!62}{62\!\cdots\!99}a^{19}-\frac{10\!\cdots\!30}{62\!\cdots\!99}a^{18}+\frac{21\!\cdots\!57}{62\!\cdots\!99}a^{17}-\frac{16\!\cdots\!80}{62\!\cdots\!99}a^{16}+\frac{31\!\cdots\!05}{62\!\cdots\!99}a^{15}-\frac{21\!\cdots\!78}{62\!\cdots\!99}a^{14}+\frac{30\!\cdots\!67}{62\!\cdots\!99}a^{13}-\frac{15\!\cdots\!76}{62\!\cdots\!99}a^{12}+\frac{21\!\cdots\!73}{62\!\cdots\!99}a^{11}-\frac{34\!\cdots\!61}{62\!\cdots\!99}a^{10}+\frac{80\!\cdots\!92}{62\!\cdots\!99}a^{9}-\frac{18\!\cdots\!48}{62\!\cdots\!99}a^{8}+\frac{32\!\cdots\!69}{62\!\cdots\!99}a^{7}-\frac{12\!\cdots\!23}{62\!\cdots\!99}a^{6}+\frac{72\!\cdots\!55}{62\!\cdots\!99}a^{5}-\frac{18\!\cdots\!09}{62\!\cdots\!99}a^{4}+\frac{12\!\cdots\!60}{62\!\cdots\!99}a^{3}-\frac{16\!\cdots\!26}{62\!\cdots\!99}a^{2}+\frac{81\!\cdots\!23}{62\!\cdots\!99}a-\frac{72\!\cdots\!87}{62\!\cdots\!99}$, $\frac{66\!\cdots\!40}{62\!\cdots\!99}a^{39}-\frac{59\!\cdots\!73}{62\!\cdots\!99}a^{38}+\frac{72\!\cdots\!60}{62\!\cdots\!99}a^{37}-\frac{65\!\cdots\!61}{62\!\cdots\!99}a^{36}+\frac{50\!\cdots\!65}{62\!\cdots\!99}a^{35}-\frac{45\!\cdots\!13}{62\!\cdots\!99}a^{34}+\frac{28\!\cdots\!07}{62\!\cdots\!99}a^{33}-\frac{25\!\cdots\!72}{62\!\cdots\!99}a^{32}+\frac{14\!\cdots\!74}{62\!\cdots\!99}a^{31}-\frac{12\!\cdots\!12}{62\!\cdots\!99}a^{30}+\frac{59\!\cdots\!66}{62\!\cdots\!99}a^{29}-\frac{52\!\cdots\!34}{62\!\cdots\!99}a^{28}+\frac{21\!\cdots\!76}{62\!\cdots\!99}a^{27}-\frac{18\!\cdots\!98}{62\!\cdots\!99}a^{26}+\frac{66\!\cdots\!95}{62\!\cdots\!99}a^{25}-\frac{57\!\cdots\!77}{62\!\cdots\!99}a^{24}+\frac{18\!\cdots\!02}{62\!\cdots\!99}a^{23}-\frac{15\!\cdots\!53}{62\!\cdots\!99}a^{22}+\frac{42\!\cdots\!59}{62\!\cdots\!99}a^{21}-\frac{35\!\cdots\!89}{62\!\cdots\!99}a^{20}+\frac{85\!\cdots\!30}{62\!\cdots\!99}a^{19}-\frac{68\!\cdots\!19}{62\!\cdots\!99}a^{18}+\frac{14\!\cdots\!19}{62\!\cdots\!99}a^{17}-\frac{11\!\cdots\!76}{62\!\cdots\!99}a^{16}+\frac{21\!\cdots\!71}{62\!\cdots\!99}a^{15}-\frac{14\!\cdots\!70}{62\!\cdots\!99}a^{14}+\frac{20\!\cdots\!47}{62\!\cdots\!99}a^{13}-\frac{10\!\cdots\!06}{62\!\cdots\!99}a^{12}+\frac{14\!\cdots\!17}{62\!\cdots\!99}a^{11}-\frac{22\!\cdots\!25}{62\!\cdots\!99}a^{10}+\frac{55\!\cdots\!26}{62\!\cdots\!99}a^{9}-\frac{12\!\cdots\!66}{62\!\cdots\!99}a^{8}+\frac{22\!\cdots\!43}{62\!\cdots\!99}a^{7}-\frac{80\!\cdots\!53}{62\!\cdots\!99}a^{6}+\frac{49\!\cdots\!85}{62\!\cdots\!99}a^{5}-\frac{12\!\cdots\!99}{62\!\cdots\!99}a^{4}+\frac{84\!\cdots\!06}{62\!\cdots\!99}a^{3}-\frac{45\!\cdots\!68}{62\!\cdots\!99}a^{2}+\frac{76\!\cdots\!15}{62\!\cdots\!99}a-\frac{64\!\cdots\!70}{62\!\cdots\!99}$, $\frac{10\!\cdots\!81}{62\!\cdots\!99}a^{39}-\frac{48\!\cdots\!93}{62\!\cdots\!99}a^{38}+\frac{10\!\cdots\!71}{62\!\cdots\!99}a^{37}-\frac{43\!\cdots\!62}{51\!\cdots\!09}a^{36}+\frac{71\!\cdots\!11}{62\!\cdots\!99}a^{35}-\frac{36\!\cdots\!64}{62\!\cdots\!99}a^{34}+\frac{39\!\cdots\!29}{62\!\cdots\!99}a^{33}-\frac{20\!\cdots\!79}{62\!\cdots\!99}a^{32}+\frac{20\!\cdots\!13}{62\!\cdots\!99}a^{31}-\frac{10\!\cdots\!85}{62\!\cdots\!99}a^{30}+\frac{80\!\cdots\!97}{62\!\cdots\!99}a^{29}-\frac{40\!\cdots\!47}{62\!\cdots\!99}a^{28}+\frac{27\!\cdots\!29}{62\!\cdots\!99}a^{27}-\frac{13\!\cdots\!47}{62\!\cdots\!99}a^{26}+\frac{85\!\cdots\!03}{62\!\cdots\!99}a^{25}-\frac{41\!\cdots\!18}{62\!\cdots\!99}a^{24}+\frac{23\!\cdots\!15}{62\!\cdots\!99}a^{23}-\frac{11\!\cdots\!79}{62\!\cdots\!99}a^{22}+\frac{51\!\cdots\!86}{62\!\cdots\!99}a^{21}-\frac{24\!\cdots\!27}{62\!\cdots\!99}a^{20}+\frac{99\!\cdots\!84}{62\!\cdots\!99}a^{19}-\frac{45\!\cdots\!98}{62\!\cdots\!99}a^{18}+\frac{16\!\cdots\!41}{62\!\cdots\!99}a^{17}-\frac{68\!\cdots\!64}{62\!\cdots\!99}a^{16}+\frac{22\!\cdots\!41}{62\!\cdots\!99}a^{15}-\frac{70\!\cdots\!10}{62\!\cdots\!99}a^{14}+\frac{18\!\cdots\!38}{62\!\cdots\!99}a^{13}-\frac{20\!\cdots\!42}{62\!\cdots\!99}a^{12}+\frac{12\!\cdots\!35}{62\!\cdots\!99}a^{11}+\frac{73\!\cdots\!75}{62\!\cdots\!99}a^{10}+\frac{47\!\cdots\!98}{62\!\cdots\!99}a^{9}+\frac{19\!\cdots\!69}{62\!\cdots\!99}a^{8}+\frac{17\!\cdots\!93}{62\!\cdots\!99}a^{7}+\frac{37\!\cdots\!44}{62\!\cdots\!99}a^{6}-\frac{99\!\cdots\!24}{62\!\cdots\!99}a^{5}+\frac{23\!\cdots\!57}{62\!\cdots\!99}a^{4}-\frac{22\!\cdots\!18}{62\!\cdots\!99}a^{3}+\frac{71\!\cdots\!58}{62\!\cdots\!99}a^{2}-\frac{13\!\cdots\!81}{62\!\cdots\!99}a+\frac{64\!\cdots\!37}{62\!\cdots\!99}$ (assuming GRH)
Regulator:		\( 276594299286030.2 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 276594299286030.2 \cdot 8420}{10\cdot\sqrt{100383397447978918530459891214693626269465146363712847232818603515625}}\cr\approx \mathstrut & 0.213757685524020 \end{aligned}\] (assuming GRH)

Galois group

$C_2\times C_{20}$ (as 40T2):

An abelian group of order 40

The 40 conjugacy class representatives for $C_2\times C_{20}$

Character table for $C_2\times C_{20}$

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{5}, \sqrt{33})\), 4.0.136125.2, \(\Q(\zeta_{5})\), \(\Q(\zeta_{11})^+\), 8.0.18530015625.3, \(\Q(\zeta_{33})^+\), 10.10.669871503125.1, 10.10.1790566527853125.1, 20.20.3206128490667995866421572265625.1, 20.0.10019151533337487082567413330078125.1, 20.0.1402274470934209014892578125.1

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	$20^{2}$	R	R	$20^{2}$	R	$20^{2}$	$20^{2}$	${\href{/padicField/19.10.0.1}{10} }^{4}$	${\href{/padicField/23.4.0.1}{4} }^{10}$	${\href{/padicField/29.10.0.1}{10} }^{4}$	${\href{/padicField/31.5.0.1}{5} }^{8}$	$20^{2}$	${\href{/padicField/41.5.0.1}{5} }^{8}$	${\href{/padicField/43.4.0.1}{4} }^{10}$	$20^{2}$	$20^{2}$	${\href{/padicField/59.10.0.1}{10} }^{4}$

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
\(3\)		Deg $40$	$2$	$20$	$20$
\(5\)		Deg $40$	$4$	$10$	$30$
\(11\)	11.10.9.1	$x^{10} + 110$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$
	11.10.9.1	$x^{10} + 110$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$
	11.10.9.1	$x^{10} + 110$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$
	11.10.9.1	$x^{10} + 110$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$