Number field 40.0.67330470569637410331240297486633042732017118041235378120152501254022721.1

• Label: 40.0.673...721.1
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $6.733\times 10^{70}$
• Root discriminant: \(58.98\)
• Ramified primes: $3,41$
• Class number: $69632$ (GRH)
• Class group: [8, 8, 8, 136] (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

x^40 - x^39 + 20*x^38 - 17*x^37 + 226*x^36 - 172*x^35 + 1735*x^34 - 1174*x^33 + 9997*x^32 - 6058*x^31 + 44934*x^30 - 24189*x^29 + 161911*x^28 - 77302*x^27 + 472811*x^26 - 197858*x^25 + 1127150*x^24 - 410723*x^23 + 2190299*x^22 - 682496*x^21 + 3457565*x^20 - 911552*x^19 + 4383590*x^18 - 948272*x^17 + 4405640*x^16 - 773063*x^15 + 3426005*x^14 - 457001*x^13 + 2008708*x^12 - 207142*x^11 + 846197*x^10 - 53834*x^9 + 244002*x^8 - 13431*x^7 + 43098*x^6 - 33*x^5 + 4180*x^4 - 220*x^3 + 155*x^2 + 10*x + 1

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(67330470569637410331240297486633042732017118041235378120152501254022721\) \(\medspace = 3^{20}\cdot 41^{38}\)
Root discriminant:		\(58.98\)
Galois root discriminant:		$3^{1/2}41^{19/20}\approx 58.98007103060146$
Ramified primes:		\(3\), \(41\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$40$
This field is Galois and abelian over $\Q$.
Conductor:		\(123=3\cdot 41\)
Dirichlet character group:		$\lbrace$$\chi_{123}(1,·)$, $\chi_{123}(2,·)$, $\chi_{123}(4,·)$, $\chi_{123}(5,·)$, $\chi_{123}(8,·)$, $\chi_{123}(10,·)$, $\chi_{123}(16,·)$, $\chi_{123}(20,·)$, $\chi_{123}(23,·)$, $\chi_{123}(25,·)$, $\chi_{123}(31,·)$, $\chi_{123}(32,·)$, $\chi_{123}(37,·)$, $\chi_{123}(40,·)$, $\chi_{123}(43,·)$, $\chi_{123}(46,·)$, $\chi_{123}(49,·)$, $\chi_{123}(50,·)$, $\chi_{123}(59,·)$, $\chi_{123}(61,·)$, $\chi_{123}(62,·)$, $\chi_{123}(64,·)$, $\chi_{123}(73,·)$, $\chi_{123}(74,·)$, $\chi_{123}(77,·)$, $\chi_{123}(80,·)$, $\chi_{123}(83,·)$, $\chi_{123}(86,·)$, $\chi_{123}(91,·)$, $\chi_{123}(92,·)$, $\chi_{123}(98,·)$, $\chi_{123}(100,·)$, $\chi_{123}(103,·)$, $\chi_{123}(107,·)$, $\chi_{123}(113,·)$, $\chi_{123}(115,·)$, $\chi_{123}(118,·)$, $\chi_{123}(119,·)$, $\chi_{123}(121,·)$, $\chi_{123}(122,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{163}a^{37}-\frac{1}{163}a^{36}+\frac{20}{163}a^{35}-\frac{30}{163}a^{34}+\frac{76}{163}a^{33}+\frac{57}{163}a^{32}+\frac{6}{163}a^{31}-\frac{43}{163}a^{30}-\frac{35}{163}a^{29}+\frac{58}{163}a^{28}+\frac{16}{163}a^{27}+\frac{64}{163}a^{26}-\frac{50}{163}a^{25}+\frac{78}{163}a^{24}-\frac{69}{163}a^{23}+\frac{22}{163}a^{22}-\frac{31}{163}a^{21}-\frac{44}{163}a^{20}-\frac{55}{163}a^{19}+\frac{62}{163}a^{18}-\frac{71}{163}a^{17}+\frac{7}{163}a^{16}+\frac{40}{163}a^{15}+\frac{7}{163}a^{14}-\frac{15}{163}a^{13}+\frac{15}{163}a^{12}-\frac{20}{163}a^{11}-\frac{80}{163}a^{10}+\frac{27}{163}a^{9}-\frac{35}{163}a^{8}-\frac{37}{163}a^{7}-\frac{69}{163}a^{6}-\frac{43}{163}a^{5}-\frac{73}{163}a^{4}-\frac{15}{163}a^{3}+\frac{37}{163}a^{2}+\frac{76}{163}a-\frac{25}{163}$, $\frac{1}{163}a^{38}+\frac{19}{163}a^{36}-\frac{10}{163}a^{35}+\frac{46}{163}a^{34}-\frac{30}{163}a^{33}+\frac{63}{163}a^{32}-\frac{37}{163}a^{31}-\frac{78}{163}a^{30}+\frac{23}{163}a^{29}+\frac{74}{163}a^{28}+\frac{80}{163}a^{27}+\frac{14}{163}a^{26}+\frac{28}{163}a^{25}+\frac{9}{163}a^{24}-\frac{47}{163}a^{23}-\frac{9}{163}a^{22}-\frac{75}{163}a^{21}+\frac{64}{163}a^{20}+\frac{7}{163}a^{19}-\frac{9}{163}a^{18}-\frac{64}{163}a^{17}+\frac{47}{163}a^{16}+\frac{47}{163}a^{15}-\frac{8}{163}a^{14}-\frac{5}{163}a^{12}+\frac{63}{163}a^{11}-\frac{53}{163}a^{10}-\frac{8}{163}a^{9}-\frac{72}{163}a^{8}+\frac{57}{163}a^{7}+\frac{51}{163}a^{6}+\frac{47}{163}a^{5}+\frac{75}{163}a^{4}+\frac{22}{163}a^{3}-\frac{50}{163}a^{2}+\frac{51}{163}a-\frac{25}{163}$, $\frac{1}{80\!\cdots\!33}a^{39}+\frac{15\!\cdots\!08}{80\!\cdots\!33}a^{38}-\frac{26\!\cdots\!85}{80\!\cdots\!33}a^{37}-\frac{37\!\cdots\!95}{80\!\cdots\!33}a^{36}-\frac{30\!\cdots\!11}{80\!\cdots\!33}a^{35}+\frac{33\!\cdots\!90}{80\!\cdots\!33}a^{34}-\frac{31\!\cdots\!25}{80\!\cdots\!33}a^{33}+\frac{35\!\cdots\!87}{80\!\cdots\!33}a^{32}+\frac{77\!\cdots\!75}{80\!\cdots\!33}a^{31}+\frac{15\!\cdots\!73}{80\!\cdots\!33}a^{30}-\frac{12\!\cdots\!89}{80\!\cdots\!33}a^{29}+\frac{19\!\cdots\!71}{80\!\cdots\!33}a^{28}+\frac{32\!\cdots\!60}{80\!\cdots\!33}a^{27}+\frac{91\!\cdots\!77}{80\!\cdots\!33}a^{26}-\frac{11\!\cdots\!34}{80\!\cdots\!33}a^{25}+\frac{35\!\cdots\!42}{80\!\cdots\!33}a^{24}-\frac{32\!\cdots\!37}{80\!\cdots\!33}a^{23}+\frac{31\!\cdots\!60}{80\!\cdots\!33}a^{22}-\frac{36\!\cdots\!15}{80\!\cdots\!33}a^{21}+\frac{20\!\cdots\!33}{80\!\cdots\!33}a^{20}-\frac{77\!\cdots\!01}{80\!\cdots\!33}a^{19}-\frac{74\!\cdots\!27}{80\!\cdots\!33}a^{18}-\frac{24\!\cdots\!35}{80\!\cdots\!33}a^{17}+\frac{29\!\cdots\!92}{80\!\cdots\!33}a^{16}+\frac{23\!\cdots\!81}{80\!\cdots\!33}a^{15}-\frac{50\!\cdots\!84}{80\!\cdots\!33}a^{14}+\frac{30\!\cdots\!40}{80\!\cdots\!33}a^{13}-\frac{30\!\cdots\!43}{80\!\cdots\!33}a^{12}+\frac{28\!\cdots\!95}{80\!\cdots\!33}a^{11}-\frac{14\!\cdots\!92}{80\!\cdots\!33}a^{10}+\frac{37\!\cdots\!90}{80\!\cdots\!33}a^{9}+\frac{16\!\cdots\!19}{80\!\cdots\!33}a^{8}-\frac{14\!\cdots\!29}{80\!\cdots\!33}a^{7}-\frac{34\!\cdots\!64}{80\!\cdots\!33}a^{6}+\frac{19\!\cdots\!39}{80\!\cdots\!33}a^{5}+\frac{18\!\cdots\!09}{80\!\cdots\!33}a^{4}-\frac{29\!\cdots\!98}{80\!\cdots\!33}a^{3}+\frac{15\!\cdots\!92}{80\!\cdots\!33}a^{2}-\frac{40\!\cdots\!03}{80\!\cdots\!33}a+\frac{23\!\cdots\!14}{80\!\cdots\!33}$

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

Class group and class number

$C_{8}\times C_{8}\times C_{8}\times C_{136}$, which has order $69632$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( \frac{679863214873483707149961740986160510907426966780545356889332423567766640984489580}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{39} - \frac{711690862085424325045424231689095359976306904969528246508164634700779905267781651}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{38} + \frac{13614071289145396980582151788211085578505424027483242705001603452868382031040548381}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{37} - \frac{12179518842869215154767441054204775316989685528362414372556732735927393466246712560}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{36} + \frac{153891639535650325397313969052773082859114453187966761897803818997919455274916091812}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{35} - \frac{123882125480299755948136120263316935085895587209571202941153182430225183631637901073}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{34} + \frac{1181679109800932737070880937015306447490806021159025865962946969006045258272447293751}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{33} - \frac{850896502182410843390842623467566597393003872506199826769974760705530689627016545674}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{32} + \frac{6808303941330816749451213053892869514542875856179682512690972925969708739520676299860}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{31} - \frac{4419994886810899198590815774072408473040125266378726495370979025828119740678519226813}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{30} + \frac{30594679418220150548565699599961374393388828028664385382551033487445951596510516917919}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{29} - \frac{17789498770457984568205926785274634990994787329835420364980237255133509772287449844606}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{28} + \frac{110189533328119881231083775396429468296606763895419553972741688296509755864478824363743}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{27} - \frac{57369166618895443426868929122477144964848200007068175728551671291815613769039844725787}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{26} + \frac{321551131636069124869268030823427256140987010470952562039775637308623371349108000466477}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{25} - \frac{148496076564507834719177989433633377891191164176734402568086831652800440676177434404783}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{24} + \frac{765771410512761268033658226023972962674019953657656348336728531140116544034419353066755}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{23} - \frac{312419838744094940137245866484136431621711478298286299528855322048068104459183315993393}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{22} + \frac{1486011123941435853365755540301796313476877348161881219765851053035286307695708245362423}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{21} - \frac{528243804141761206432492531646758349511400328848798135501563925114542866639621967891781}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{20} + \frac{2341275311338940673618519479937244367643969719961291837192289120397626192209437204349841}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{19} - \frac{720930784616853567512872864489520255080325184665413558888673370949130614088164583852776}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{18} + \frac{2960687889586077311779312494972419501720260863303675295946501175274382236685120498536771}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{17} - \frac{772811963122211017530263280396988209747248486031323741776895879818746904775591285625202}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{16} + \frac{2964726359792869999538730763788125502457326710226202718005901216649145093068222182764216}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{15} - \frac{654559779281559812966471960341220598309475972880658248860934644865411027017029393201874}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{14} + \frac{2293946065511732888664049407042549467162118530894816067922267746740107753810750335060164}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{13} - \frac{411235406025526840739804230861173894774362337626783770993594362200508040206244844736337}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{12} + \frac{1334794063772327239225677705200397427003033710054515302072132544742138877755615869378697}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{11} - \frac{200415663608049036319555137692628789689639015317952759584853470934031879337706513437973}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{10} + \frac{556173604417562199381320710668724383873923401076720288263393285046200707057767416570410}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{9} - \frac{61939039908514558574121083926684029612006781532383482435073289736761666543725908994681}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{8} + \frac{157321005959595250707656825042629359507808492024885735915966776606239291934975252084003}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{7} - \frac{16787046375399506648840531303094290830737448903511360941054363586420833696004887321029}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{6} + \frac{26983608895716260869309676155128763592124493570538004308307818819182953603034378297110}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{5} - \frac{1396460504237557926732179453524095335974696182587678270069651048643793643409079443570}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{4} + \frac{2424939504184881007107554773901087495682539546225026307049072533514692686380663342217}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{3} - \frac{341874504094605942297510222587143828685681356785809362608000200965839015230011665881}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a^{2} + \frac{80443827675904038640705178628044306254394211640994497637408281039196926080689628098}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} a + \frac{5029987067585514761332664763082464586173941869075899219967787188480463229717308428}{8051921367310432482588950816162189901255578764284871302581700957317939085956805533} \)  (order $6$)
Fundamental units:		$\frac{22\!\cdots\!96}{80\!\cdots\!33}a^{39}-\frac{11\!\cdots\!72}{80\!\cdots\!33}a^{38}+\frac{43\!\cdots\!97}{80\!\cdots\!33}a^{37}-\frac{17\!\cdots\!82}{80\!\cdots\!33}a^{36}+\frac{48\!\cdots\!61}{80\!\cdots\!33}a^{35}-\frac{15\!\cdots\!59}{80\!\cdots\!33}a^{34}+\frac{36\!\cdots\!20}{80\!\cdots\!33}a^{33}-\frac{85\!\cdots\!87}{80\!\cdots\!33}a^{32}+\frac{21\!\cdots\!86}{80\!\cdots\!33}a^{31}-\frac{33\!\cdots\!32}{80\!\cdots\!33}a^{30}+\frac{93\!\cdots\!65}{80\!\cdots\!33}a^{29}-\frac{86\!\cdots\!63}{80\!\cdots\!33}a^{28}+\frac{33\!\cdots\!92}{80\!\cdots\!33}a^{27}-\frac{10\!\cdots\!22}{80\!\cdots\!33}a^{26}+\frac{97\!\cdots\!17}{80\!\cdots\!33}a^{25}+\frac{28\!\cdots\!88}{80\!\cdots\!33}a^{24}+\frac{23\!\cdots\!86}{80\!\cdots\!33}a^{23}+\frac{19\!\cdots\!31}{80\!\cdots\!33}a^{22}+\frac{44\!\cdots\!21}{80\!\cdots\!33}a^{21}+\frac{61\!\cdots\!98}{80\!\cdots\!33}a^{20}+\frac{70\!\cdots\!26}{80\!\cdots\!33}a^{19}+\frac{13\!\cdots\!82}{80\!\cdots\!33}a^{18}+\frac{89\!\cdots\!54}{80\!\cdots\!33}a^{17}+\frac{20\!\cdots\!87}{80\!\cdots\!33}a^{16}+\frac{90\!\cdots\!30}{80\!\cdots\!33}a^{15}+\frac{14\!\cdots\!85}{49\!\cdots\!91}a^{14}+\frac{70\!\cdots\!42}{80\!\cdots\!33}a^{13}+\frac{21\!\cdots\!48}{80\!\cdots\!33}a^{12}+\frac{41\!\cdots\!18}{80\!\cdots\!33}a^{11}+\frac{13\!\cdots\!15}{80\!\cdots\!33}a^{10}+\frac{17\!\cdots\!39}{80\!\cdots\!33}a^{9}+\frac{58\!\cdots\!79}{80\!\cdots\!33}a^{8}+\frac{53\!\cdots\!22}{80\!\cdots\!33}a^{7}+\frac{15\!\cdots\!44}{80\!\cdots\!33}a^{6}+\frac{95\!\cdots\!47}{80\!\cdots\!33}a^{5}+\frac{28\!\cdots\!06}{80\!\cdots\!33}a^{4}+\frac{12\!\cdots\!22}{80\!\cdots\!33}a^{3}+\frac{17\!\cdots\!49}{80\!\cdots\!33}a^{2}+\frac{12\!\cdots\!16}{80\!\cdots\!33}a+\frac{47\!\cdots\!28}{80\!\cdots\!33}$, $\frac{48\!\cdots\!11}{80\!\cdots\!33}a^{39}-\frac{34\!\cdots\!71}{80\!\cdots\!33}a^{38}+\frac{95\!\cdots\!63}{80\!\cdots\!33}a^{37}-\frac{55\!\cdots\!04}{80\!\cdots\!33}a^{36}+\frac{10\!\cdots\!16}{80\!\cdots\!33}a^{35}-\frac{52\!\cdots\!59}{80\!\cdots\!33}a^{34}+\frac{81\!\cdots\!34}{80\!\cdots\!33}a^{33}-\frac{33\!\cdots\!04}{80\!\cdots\!33}a^{32}+\frac{46\!\cdots\!28}{80\!\cdots\!33}a^{31}-\frac{15\!\cdots\!86}{80\!\cdots\!33}a^{30}+\frac{20\!\cdots\!07}{80\!\cdots\!33}a^{29}-\frac{57\!\cdots\!49}{80\!\cdots\!33}a^{28}+\frac{75\!\cdots\!98}{80\!\cdots\!33}a^{27}-\frac{15\!\cdots\!73}{80\!\cdots\!33}a^{26}+\frac{21\!\cdots\!38}{80\!\cdots\!33}a^{25}-\frac{32\!\cdots\!34}{80\!\cdots\!33}a^{24}+\frac{52\!\cdots\!01}{80\!\cdots\!33}a^{23}-\frac{48\!\cdots\!49}{80\!\cdots\!33}a^{22}+\frac{10\!\cdots\!68}{80\!\cdots\!33}a^{21}-\frac{39\!\cdots\!23}{80\!\cdots\!33}a^{20}+\frac{15\!\cdots\!80}{80\!\cdots\!33}a^{19}+\frac{17\!\cdots\!18}{80\!\cdots\!33}a^{18}+\frac{20\!\cdots\!03}{80\!\cdots\!33}a^{17}+\frac{12\!\cdots\!32}{80\!\cdots\!33}a^{16}+\frac{20\!\cdots\!53}{80\!\cdots\!33}a^{15}+\frac{20\!\cdots\!41}{80\!\cdots\!33}a^{14}+\frac{15\!\cdots\!45}{80\!\cdots\!33}a^{13}+\frac{23\!\cdots\!40}{80\!\cdots\!33}a^{12}+\frac{92\!\cdots\!55}{80\!\cdots\!33}a^{11}+\frac{16\!\cdots\!05}{80\!\cdots\!33}a^{10}+\frac{39\!\cdots\!28}{80\!\cdots\!33}a^{9}+\frac{83\!\cdots\!26}{80\!\cdots\!33}a^{8}+\frac{11\!\cdots\!21}{80\!\cdots\!33}a^{7}+\frac{24\!\cdots\!96}{80\!\cdots\!33}a^{6}+\frac{19\!\cdots\!50}{80\!\cdots\!33}a^{5}+\frac{50\!\cdots\!78}{80\!\cdots\!33}a^{4}+\frac{19\!\cdots\!90}{80\!\cdots\!33}a^{3}+\frac{32\!\cdots\!67}{80\!\cdots\!33}a^{2}+\frac{23\!\cdots\!85}{80\!\cdots\!33}a+\frac{10\!\cdots\!55}{80\!\cdots\!33}$, $\frac{10\!\cdots\!25}{80\!\cdots\!33}a^{39}-\frac{10\!\cdots\!97}{80\!\cdots\!33}a^{38}+\frac{21\!\cdots\!03}{80\!\cdots\!33}a^{37}-\frac{18\!\cdots\!20}{80\!\cdots\!33}a^{36}+\frac{23\!\cdots\!92}{80\!\cdots\!33}a^{35}-\frac{19\!\cdots\!19}{80\!\cdots\!33}a^{34}+\frac{18\!\cdots\!27}{80\!\cdots\!33}a^{33}-\frac{13\!\cdots\!60}{80\!\cdots\!33}a^{32}+\frac{10\!\cdots\!46}{80\!\cdots\!33}a^{31}-\frac{67\!\cdots\!81}{80\!\cdots\!33}a^{30}+\frac{47\!\cdots\!81}{80\!\cdots\!33}a^{29}-\frac{27\!\cdots\!43}{80\!\cdots\!33}a^{28}+\frac{17\!\cdots\!85}{80\!\cdots\!33}a^{27}-\frac{88\!\cdots\!27}{80\!\cdots\!33}a^{26}+\frac{50\!\cdots\!08}{80\!\cdots\!33}a^{25}-\frac{22\!\cdots\!49}{80\!\cdots\!33}a^{24}+\frac{12\!\cdots\!49}{80\!\cdots\!33}a^{23}-\frac{47\!\cdots\!59}{80\!\cdots\!33}a^{22}+\frac{23\!\cdots\!61}{80\!\cdots\!33}a^{21}-\frac{81\!\cdots\!99}{80\!\cdots\!33}a^{20}+\frac{36\!\cdots\!90}{80\!\cdots\!33}a^{19}-\frac{11\!\cdots\!22}{80\!\cdots\!33}a^{18}+\frac{46\!\cdots\!25}{80\!\cdots\!33}a^{17}-\frac{11\!\cdots\!89}{80\!\cdots\!33}a^{16}+\frac{47\!\cdots\!56}{80\!\cdots\!33}a^{15}-\frac{99\!\cdots\!91}{80\!\cdots\!33}a^{14}+\frac{36\!\cdots\!43}{80\!\cdots\!33}a^{13}-\frac{62\!\cdots\!82}{80\!\cdots\!33}a^{12}+\frac{21\!\cdots\!57}{80\!\cdots\!33}a^{11}-\frac{29\!\cdots\!11}{80\!\cdots\!33}a^{10}+\frac{91\!\cdots\!42}{80\!\cdots\!33}a^{9}-\frac{88\!\cdots\!84}{80\!\cdots\!33}a^{8}+\frac{26\!\cdots\!71}{80\!\cdots\!33}a^{7}-\frac{22\!\cdots\!49}{80\!\cdots\!33}a^{6}+\frac{46\!\cdots\!78}{80\!\cdots\!33}a^{5}-\frac{12\!\cdots\!50}{80\!\cdots\!33}a^{4}+\frac{43\!\cdots\!37}{80\!\cdots\!33}a^{3}-\frac{24\!\cdots\!87}{80\!\cdots\!33}a^{2}+\frac{15\!\cdots\!23}{80\!\cdots\!33}a+\frac{98\!\cdots\!55}{80\!\cdots\!33}$, $\frac{69\!\cdots\!74}{80\!\cdots\!33}a^{39}-\frac{76\!\cdots\!63}{80\!\cdots\!33}a^{38}+\frac{13\!\cdots\!38}{80\!\cdots\!33}a^{37}-\frac{13\!\cdots\!70}{80\!\cdots\!33}a^{36}+\frac{15\!\cdots\!45}{80\!\cdots\!33}a^{35}-\frac{13\!\cdots\!58}{80\!\cdots\!33}a^{34}+\frac{12\!\cdots\!61}{80\!\cdots\!33}a^{33}-\frac{93\!\cdots\!12}{80\!\cdots\!33}a^{32}+\frac{69\!\cdots\!52}{80\!\cdots\!33}a^{31}-\frac{48\!\cdots\!77}{80\!\cdots\!33}a^{30}+\frac{31\!\cdots\!44}{80\!\cdots\!33}a^{29}-\frac{19\!\cdots\!77}{80\!\cdots\!33}a^{28}+\frac{11\!\cdots\!51}{80\!\cdots\!33}a^{27}-\frac{64\!\cdots\!05}{80\!\cdots\!33}a^{26}+\frac{33\!\cdots\!28}{80\!\cdots\!33}a^{25}-\frac{16\!\cdots\!15}{80\!\cdots\!33}a^{24}+\frac{78\!\cdots\!73}{80\!\cdots\!33}a^{23}-\frac{35\!\cdots\!16}{80\!\cdots\!33}a^{22}+\frac{15\!\cdots\!14}{80\!\cdots\!33}a^{21}-\frac{61\!\cdots\!71}{80\!\cdots\!33}a^{20}+\frac{24\!\cdots\!33}{80\!\cdots\!33}a^{19}-\frac{85\!\cdots\!60}{80\!\cdots\!33}a^{18}+\frac{30\!\cdots\!07}{80\!\cdots\!33}a^{17}-\frac{93\!\cdots\!73}{80\!\cdots\!33}a^{16}+\frac{30\!\cdots\!62}{80\!\cdots\!33}a^{15}-\frac{80\!\cdots\!53}{80\!\cdots\!33}a^{14}+\frac{23\!\cdots\!84}{80\!\cdots\!33}a^{13}-\frac{52\!\cdots\!97}{80\!\cdots\!33}a^{12}+\frac{13\!\cdots\!11}{80\!\cdots\!33}a^{11}-\frac{26\!\cdots\!22}{80\!\cdots\!33}a^{10}+\frac{57\!\cdots\!36}{80\!\cdots\!33}a^{9}-\frac{85\!\cdots\!88}{80\!\cdots\!33}a^{8}+\frac{16\!\cdots\!61}{80\!\cdots\!33}a^{7}-\frac{22\!\cdots\!91}{80\!\cdots\!33}a^{6}+\frac{28\!\cdots\!19}{80\!\cdots\!33}a^{5}-\frac{21\!\cdots\!30}{80\!\cdots\!33}a^{4}+\frac{16\!\cdots\!89}{49\!\cdots\!91}a^{3}-\frac{39\!\cdots\!88}{80\!\cdots\!33}a^{2}+\frac{14\!\cdots\!82}{80\!\cdots\!33}a-\frac{34\!\cdots\!85}{80\!\cdots\!33}$, $\frac{13\!\cdots\!84}{80\!\cdots\!33}a^{39}-\frac{13\!\cdots\!30}{80\!\cdots\!33}a^{38}+\frac{26\!\cdots\!86}{80\!\cdots\!33}a^{37}-\frac{23\!\cdots\!88}{80\!\cdots\!33}a^{36}+\frac{30\!\cdots\!48}{80\!\cdots\!33}a^{35}-\frac{24\!\cdots\!30}{80\!\cdots\!33}a^{34}+\frac{23\!\cdots\!74}{80\!\cdots\!33}a^{33}-\frac{16\!\cdots\!12}{80\!\cdots\!33}a^{32}+\frac{13\!\cdots\!68}{80\!\cdots\!33}a^{31}-\frac{86\!\cdots\!14}{80\!\cdots\!33}a^{30}+\frac{59\!\cdots\!86}{80\!\cdots\!33}a^{29}-\frac{34\!\cdots\!24}{80\!\cdots\!33}a^{28}+\frac{21\!\cdots\!78}{80\!\cdots\!33}a^{27}-\frac{11\!\cdots\!94}{80\!\cdots\!33}a^{26}+\frac{63\!\cdots\!42}{80\!\cdots\!33}a^{25}-\frac{29\!\cdots\!58}{80\!\cdots\!33}a^{24}+\frac{15\!\cdots\!26}{80\!\cdots\!33}a^{23}-\frac{61\!\cdots\!14}{80\!\cdots\!33}a^{22}+\frac{29\!\cdots\!10}{80\!\cdots\!33}a^{21}-\frac{10\!\cdots\!50}{80\!\cdots\!33}a^{20}+\frac{45\!\cdots\!66}{80\!\cdots\!33}a^{19}-\frac{14\!\cdots\!16}{80\!\cdots\!33}a^{18}+\frac{58\!\cdots\!06}{80\!\cdots\!33}a^{17}-\frac{15\!\cdots\!44}{80\!\cdots\!33}a^{16}+\frac{58\!\cdots\!36}{80\!\cdots\!33}a^{15}-\frac{12\!\cdots\!16}{80\!\cdots\!33}a^{14}+\frac{45\!\cdots\!72}{80\!\cdots\!33}a^{13}-\frac{80\!\cdots\!46}{80\!\cdots\!33}a^{12}+\frac{26\!\cdots\!54}{80\!\cdots\!33}a^{11}-\frac{39\!\cdots\!82}{80\!\cdots\!33}a^{10}+\frac{10\!\cdots\!52}{80\!\cdots\!33}a^{9}-\frac{12\!\cdots\!79}{80\!\cdots\!33}a^{8}+\frac{31\!\cdots\!18}{80\!\cdots\!33}a^{7}-\frac{32\!\cdots\!50}{80\!\cdots\!33}a^{6}+\frac{53\!\cdots\!08}{80\!\cdots\!33}a^{5}-\frac{26\!\cdots\!88}{80\!\cdots\!33}a^{4}+\frac{48\!\cdots\!34}{80\!\cdots\!33}a^{3}-\frac{55\!\cdots\!94}{80\!\cdots\!33}a^{2}+\frac{16\!\cdots\!76}{80\!\cdots\!33}a+\frac{10\!\cdots\!20}{80\!\cdots\!33}$, $\frac{27\!\cdots\!51}{80\!\cdots\!33}a^{39}-\frac{21\!\cdots\!37}{80\!\cdots\!33}a^{38}+\frac{53\!\cdots\!36}{80\!\cdots\!33}a^{37}-\frac{34\!\cdots\!32}{80\!\cdots\!33}a^{36}+\frac{59\!\cdots\!09}{80\!\cdots\!33}a^{35}-\frac{32\!\cdots\!79}{80\!\cdots\!33}a^{34}+\frac{44\!\cdots\!37}{80\!\cdots\!33}a^{33}-\frac{21\!\cdots\!04}{80\!\cdots\!33}a^{32}+\frac{25\!\cdots\!27}{80\!\cdots\!33}a^{31}-\frac{10\!\cdots\!66}{80\!\cdots\!33}a^{30}+\frac{11\!\cdots\!87}{80\!\cdots\!33}a^{29}-\frac{38\!\cdots\!86}{80\!\cdots\!33}a^{28}+\frac{40\!\cdots\!71}{80\!\cdots\!33}a^{27}-\frac{11\!\cdots\!63}{80\!\cdots\!33}a^{26}+\frac{11\!\cdots\!97}{80\!\cdots\!33}a^{25}-\frac{25\!\cdots\!89}{80\!\cdots\!33}a^{24}+\frac{26\!\cdots\!63}{80\!\cdots\!33}a^{23}-\frac{45\!\cdots\!31}{80\!\cdots\!33}a^{22}+\frac{50\!\cdots\!36}{80\!\cdots\!33}a^{21}-\frac{60\!\cdots\!97}{80\!\cdots\!33}a^{20}+\frac{77\!\cdots\!97}{80\!\cdots\!33}a^{19}-\frac{58\!\cdots\!42}{80\!\cdots\!33}a^{18}+\frac{94\!\cdots\!79}{80\!\cdots\!33}a^{17}-\frac{32\!\cdots\!98}{80\!\cdots\!33}a^{16}+\frac{90\!\cdots\!91}{80\!\cdots\!33}a^{15}-\frac{41\!\cdots\!03}{80\!\cdots\!33}a^{14}+\frac{65\!\cdots\!56}{80\!\cdots\!33}a^{13}+\frac{15\!\cdots\!77}{80\!\cdots\!33}a^{12}+\frac{35\!\cdots\!03}{80\!\cdots\!33}a^{11}+\frac{10\!\cdots\!45}{80\!\cdots\!33}a^{10}+\frac{13\!\cdots\!52}{80\!\cdots\!33}a^{9}+\frac{59\!\cdots\!29}{80\!\cdots\!33}a^{8}+\frac{33\!\cdots\!67}{80\!\cdots\!33}a^{7}+\frac{20\!\cdots\!97}{80\!\cdots\!33}a^{6}+\frac{48\!\cdots\!62}{80\!\cdots\!33}a^{5}+\frac{70\!\cdots\!54}{80\!\cdots\!33}a^{4}+\frac{35\!\cdots\!06}{80\!\cdots\!33}a^{3}+\frac{56\!\cdots\!45}{80\!\cdots\!33}a^{2}+\frac{41\!\cdots\!08}{80\!\cdots\!33}a-\frac{40\!\cdots\!14}{80\!\cdots\!33}$, $\frac{13\!\cdots\!70}{80\!\cdots\!33}a^{39}-\frac{14\!\cdots\!82}{80\!\cdots\!33}a^{38}+\frac{26\!\cdots\!87}{80\!\cdots\!33}a^{37}-\frac{24\!\cdots\!87}{80\!\cdots\!33}a^{36}+\frac{30\!\cdots\!63}{80\!\cdots\!33}a^{35}-\frac{24\!\cdots\!15}{80\!\cdots\!33}a^{34}+\frac{23\!\cdots\!48}{80\!\cdots\!33}a^{33}-\frac{16\!\cdots\!34}{80\!\cdots\!33}a^{32}+\frac{13\!\cdots\!01}{80\!\cdots\!33}a^{31}-\frac{87\!\cdots\!98}{80\!\cdots\!33}a^{30}+\frac{60\!\cdots\!82}{80\!\cdots\!33}a^{29}-\frac{35\!\cdots\!58}{80\!\cdots\!33}a^{28}+\frac{21\!\cdots\!41}{80\!\cdots\!33}a^{27}-\frac{11\!\cdots\!36}{80\!\cdots\!33}a^{26}+\frac{63\!\cdots\!91}{80\!\cdots\!33}a^{25}-\frac{29\!\cdots\!69}{80\!\cdots\!33}a^{24}+\frac{15\!\cdots\!80}{80\!\cdots\!33}a^{23}-\frac{61\!\cdots\!58}{80\!\cdots\!33}a^{22}+\frac{29\!\cdots\!71}{80\!\cdots\!33}a^{21}-\frac{10\!\cdots\!63}{80\!\cdots\!33}a^{20}+\frac{28\!\cdots\!47}{49\!\cdots\!91}a^{19}-\frac{14\!\cdots\!04}{80\!\cdots\!33}a^{18}+\frac{58\!\cdots\!46}{80\!\cdots\!33}a^{17}-\frac{15\!\cdots\!98}{80\!\cdots\!33}a^{16}+\frac{58\!\cdots\!23}{80\!\cdots\!33}a^{15}-\frac{13\!\cdots\!50}{80\!\cdots\!33}a^{14}+\frac{45\!\cdots\!14}{80\!\cdots\!33}a^{13}-\frac{81\!\cdots\!19}{80\!\cdots\!33}a^{12}+\frac{26\!\cdots\!62}{80\!\cdots\!33}a^{11}-\frac{39\!\cdots\!20}{80\!\cdots\!33}a^{10}+\frac{11\!\cdots\!49}{80\!\cdots\!33}a^{9}-\frac{12\!\cdots\!69}{80\!\cdots\!33}a^{8}+\frac{31\!\cdots\!75}{80\!\cdots\!33}a^{7}-\frac{33\!\cdots\!30}{80\!\cdots\!33}a^{6}+\frac{53\!\cdots\!94}{80\!\cdots\!33}a^{5}-\frac{28\!\cdots\!41}{80\!\cdots\!33}a^{4}+\frac{48\!\cdots\!12}{80\!\cdots\!33}a^{3}-\frac{68\!\cdots\!29}{80\!\cdots\!33}a^{2}+\frac{16\!\cdots\!26}{80\!\cdots\!33}a-\frac{60\!\cdots\!39}{80\!\cdots\!33}$, $\frac{39\!\cdots\!37}{80\!\cdots\!33}a^{39}-\frac{24\!\cdots\!10}{80\!\cdots\!33}a^{38}+\frac{77\!\cdots\!46}{80\!\cdots\!33}a^{37}-\frac{37\!\cdots\!75}{80\!\cdots\!33}a^{36}+\frac{86\!\cdots\!57}{80\!\cdots\!33}a^{35}-\frac{34\!\cdots\!29}{80\!\cdots\!33}a^{34}+\frac{66\!\cdots\!00}{80\!\cdots\!33}a^{33}-\frac{20\!\cdots\!59}{80\!\cdots\!33}a^{32}+\frac{37\!\cdots\!65}{80\!\cdots\!33}a^{31}-\frac{91\!\cdots\!86}{80\!\cdots\!33}a^{30}+\frac{16\!\cdots\!11}{80\!\cdots\!33}a^{29}-\frac{29\!\cdots\!29}{80\!\cdots\!33}a^{28}+\frac{60\!\cdots\!54}{80\!\cdots\!33}a^{27}-\frac{67\!\cdots\!98}{80\!\cdots\!33}a^{26}+\frac{17\!\cdots\!47}{80\!\cdots\!33}a^{25}-\frac{90\!\cdots\!56}{80\!\cdots\!33}a^{24}+\frac{41\!\cdots\!03}{80\!\cdots\!33}a^{23}+\frac{18\!\cdots\!11}{80\!\cdots\!33}a^{22}+\frac{81\!\cdots\!61}{80\!\cdots\!33}a^{21}+\frac{47\!\cdots\!72}{80\!\cdots\!33}a^{20}+\frac{12\!\cdots\!98}{80\!\cdots\!33}a^{19}+\frac{13\!\cdots\!66}{80\!\cdots\!33}a^{18}+\frac{16\!\cdots\!28}{80\!\cdots\!33}a^{17}+\frac{25\!\cdots\!32}{80\!\cdots\!33}a^{16}+\frac{16\!\cdots\!62}{80\!\cdots\!33}a^{15}+\frac{31\!\cdots\!21}{80\!\cdots\!33}a^{14}+\frac{12\!\cdots\!02}{80\!\cdots\!33}a^{13}+\frac{29\!\cdots\!10}{80\!\cdots\!33}a^{12}+\frac{75\!\cdots\!08}{80\!\cdots\!33}a^{11}+\frac{19\!\cdots\!80}{80\!\cdots\!33}a^{10}+\frac{32\!\cdots\!65}{80\!\cdots\!33}a^{9}+\frac{89\!\cdots\!07}{80\!\cdots\!33}a^{8}+\frac{95\!\cdots\!41}{80\!\cdots\!33}a^{7}+\frac{25\!\cdots\!86}{80\!\cdots\!33}a^{6}+\frac{16\!\cdots\!89}{80\!\cdots\!33}a^{5}+\frac{47\!\cdots\!75}{80\!\cdots\!33}a^{4}+\frac{18\!\cdots\!71}{80\!\cdots\!33}a^{3}+\frac{29\!\cdots\!57}{80\!\cdots\!33}a^{2}+\frac{20\!\cdots\!48}{80\!\cdots\!33}a+\frac{16\!\cdots\!16}{80\!\cdots\!33}$, $\frac{47\!\cdots\!67}{80\!\cdots\!33}a^{39}-\frac{48\!\cdots\!98}{80\!\cdots\!33}a^{38}+\frac{95\!\cdots\!17}{80\!\cdots\!33}a^{37}-\frac{83\!\cdots\!61}{80\!\cdots\!33}a^{36}+\frac{10\!\cdots\!97}{80\!\cdots\!33}a^{35}-\frac{85\!\cdots\!51}{80\!\cdots\!33}a^{34}+\frac{82\!\cdots\!72}{80\!\cdots\!33}a^{33}-\frac{58\!\cdots\!28}{80\!\cdots\!33}a^{32}+\frac{47\!\cdots\!23}{80\!\cdots\!33}a^{31}-\frac{30\!\cdots\!45}{80\!\cdots\!33}a^{30}+\frac{21\!\cdots\!48}{80\!\cdots\!33}a^{29}-\frac{12\!\cdots\!55}{80\!\cdots\!33}a^{28}+\frac{77\!\cdots\!84}{80\!\cdots\!33}a^{27}-\frac{39\!\cdots\!74}{80\!\cdots\!33}a^{26}+\frac{22\!\cdots\!82}{80\!\cdots\!33}a^{25}-\frac{10\!\cdots\!35}{80\!\cdots\!33}a^{24}+\frac{54\!\cdots\!74}{80\!\cdots\!33}a^{23}-\frac{21\!\cdots\!02}{80\!\cdots\!33}a^{22}+\frac{10\!\cdots\!04}{80\!\cdots\!33}a^{21}-\frac{36\!\cdots\!39}{80\!\cdots\!33}a^{20}+\frac{16\!\cdots\!44}{80\!\cdots\!33}a^{19}-\frac{49\!\cdots\!28}{80\!\cdots\!33}a^{18}+\frac{21\!\cdots\!32}{80\!\cdots\!33}a^{17}-\frac{53\!\cdots\!68}{80\!\cdots\!33}a^{16}+\frac{21\!\cdots\!08}{80\!\cdots\!33}a^{15}-\frac{45\!\cdots\!62}{80\!\cdots\!33}a^{14}+\frac{16\!\cdots\!33}{80\!\cdots\!33}a^{13}-\frac{28\!\cdots\!96}{80\!\cdots\!33}a^{12}+\frac{10\!\cdots\!92}{80\!\cdots\!33}a^{11}-\frac{13\!\cdots\!87}{80\!\cdots\!33}a^{10}+\frac{42\!\cdots\!07}{80\!\cdots\!33}a^{9}-\frac{40\!\cdots\!46}{80\!\cdots\!33}a^{8}+\frac{12\!\cdots\!51}{80\!\cdots\!33}a^{7}-\frac{96\!\cdots\!34}{80\!\cdots\!33}a^{6}+\frac{22\!\cdots\!58}{80\!\cdots\!33}a^{5}-\frac{37\!\cdots\!17}{80\!\cdots\!33}a^{4}+\frac{21\!\cdots\!03}{80\!\cdots\!33}a^{3}+\frac{61\!\cdots\!09}{80\!\cdots\!33}a^{2}+\frac{76\!\cdots\!29}{80\!\cdots\!33}a+\frac{49\!\cdots\!00}{80\!\cdots\!33}$, $\frac{45\!\cdots\!56}{80\!\cdots\!33}a^{39}-\frac{41\!\cdots\!27}{80\!\cdots\!33}a^{38}+\frac{89\!\cdots\!27}{80\!\cdots\!33}a^{37}-\frac{68\!\cdots\!84}{80\!\cdots\!33}a^{36}+\frac{10\!\cdots\!48}{80\!\cdots\!33}a^{35}-\frac{68\!\cdots\!53}{80\!\cdots\!33}a^{34}+\frac{77\!\cdots\!90}{80\!\cdots\!33}a^{33}-\frac{46\!\cdots\!16}{80\!\cdots\!33}a^{32}+\frac{44\!\cdots\!96}{80\!\cdots\!33}a^{31}-\frac{23\!\cdots\!91}{80\!\cdots\!33}a^{30}+\frac{19\!\cdots\!85}{80\!\cdots\!33}a^{29}-\frac{90\!\cdots\!33}{80\!\cdots\!33}a^{28}+\frac{71\!\cdots\!39}{80\!\cdots\!33}a^{27}-\frac{28\!\cdots\!81}{80\!\cdots\!33}a^{26}+\frac{20\!\cdots\!34}{80\!\cdots\!33}a^{25}-\frac{69\!\cdots\!35}{80\!\cdots\!33}a^{24}+\frac{49\!\cdots\!85}{80\!\cdots\!33}a^{23}-\frac{13\!\cdots\!73}{80\!\cdots\!33}a^{22}+\frac{95\!\cdots\!71}{80\!\cdots\!33}a^{21}-\frac{21\!\cdots\!13}{80\!\cdots\!33}a^{20}+\frac{14\!\cdots\!74}{80\!\cdots\!33}a^{19}-\frac{26\!\cdots\!88}{80\!\cdots\!33}a^{18}+\frac{18\!\cdots\!17}{80\!\cdots\!33}a^{17}-\frac{23\!\cdots\!12}{80\!\cdots\!33}a^{16}+\frac{18\!\cdots\!28}{80\!\cdots\!33}a^{15}-\frac{15\!\cdots\!49}{80\!\cdots\!33}a^{14}+\frac{14\!\cdots\!01}{80\!\cdots\!33}a^{13}-\frac{52\!\cdots\!45}{80\!\cdots\!33}a^{12}+\frac{82\!\cdots\!99}{80\!\cdots\!33}a^{11}-\frac{45\!\cdots\!21}{80\!\cdots\!33}a^{10}+\frac{33\!\cdots\!64}{80\!\cdots\!33}a^{9}+\frac{17\!\cdots\!74}{80\!\cdots\!33}a^{8}+\frac{92\!\cdots\!11}{80\!\cdots\!33}a^{7}+\frac{67\!\cdots\!84}{80\!\cdots\!33}a^{6}+\frac{14\!\cdots\!46}{80\!\cdots\!33}a^{5}+\frac{26\!\cdots\!50}{80\!\cdots\!33}a^{4}+\frac{12\!\cdots\!55}{80\!\cdots\!33}a^{3}+\frac{19\!\cdots\!73}{80\!\cdots\!33}a^{2}+\frac{14\!\cdots\!87}{80\!\cdots\!33}a+\frac{14\!\cdots\!37}{80\!\cdots\!33}$, $\frac{28\!\cdots\!42}{80\!\cdots\!33}a^{39}-\frac{15\!\cdots\!78}{80\!\cdots\!33}a^{38}+\frac{55\!\cdots\!57}{80\!\cdots\!33}a^{37}-\frac{23\!\cdots\!46}{80\!\cdots\!33}a^{36}+\frac{61\!\cdots\!43}{80\!\cdots\!33}a^{35}-\frac{20\!\cdots\!93}{80\!\cdots\!33}a^{34}+\frac{47\!\cdots\!16}{80\!\cdots\!33}a^{33}-\frac{11\!\cdots\!07}{80\!\cdots\!33}a^{32}+\frac{26\!\cdots\!28}{80\!\cdots\!33}a^{31}-\frac{47\!\cdots\!62}{80\!\cdots\!33}a^{30}+\frac{12\!\cdots\!13}{80\!\cdots\!33}a^{29}-\frac{12\!\cdots\!17}{80\!\cdots\!33}a^{28}+\frac{43\!\cdots\!18}{80\!\cdots\!33}a^{27}-\frac{19\!\cdots\!40}{80\!\cdots\!33}a^{26}+\frac{12\!\cdots\!03}{80\!\cdots\!33}a^{25}+\frac{17\!\cdots\!62}{80\!\cdots\!33}a^{24}+\frac{29\!\cdots\!68}{80\!\cdots\!33}a^{23}+\frac{20\!\cdots\!37}{80\!\cdots\!33}a^{22}+\frac{57\!\cdots\!07}{80\!\cdots\!33}a^{21}+\frac{70\!\cdots\!92}{80\!\cdots\!33}a^{20}+\frac{90\!\cdots\!74}{80\!\cdots\!33}a^{19}+\frac{15\!\cdots\!36}{80\!\cdots\!33}a^{18}+\frac{11\!\cdots\!50}{80\!\cdots\!33}a^{17}+\frac{24\!\cdots\!11}{80\!\cdots\!33}a^{16}+\frac{11\!\cdots\!54}{80\!\cdots\!33}a^{15}+\frac{29\!\cdots\!03}{80\!\cdots\!33}a^{14}+\frac{90\!\cdots\!04}{80\!\cdots\!33}a^{13}+\frac{25\!\cdots\!66}{80\!\cdots\!33}a^{12}+\frac{53\!\cdots\!72}{80\!\cdots\!33}a^{11}+\frac{16\!\cdots\!11}{80\!\cdots\!33}a^{10}+\frac{22\!\cdots\!62}{80\!\cdots\!33}a^{9}+\frac{73\!\cdots\!29}{80\!\cdots\!33}a^{8}+\frac{68\!\cdots\!68}{80\!\cdots\!33}a^{7}+\frac{19\!\cdots\!68}{80\!\cdots\!33}a^{6}+\frac{12\!\cdots\!63}{80\!\cdots\!33}a^{5}+\frac{36\!\cdots\!92}{80\!\cdots\!33}a^{4}+\frac{15\!\cdots\!36}{80\!\cdots\!33}a^{3}+\frac{21\!\cdots\!93}{80\!\cdots\!33}a^{2}+\frac{15\!\cdots\!36}{80\!\cdots\!33}a+\frac{60\!\cdots\!12}{80\!\cdots\!33}$, $\frac{22\!\cdots\!96}{80\!\cdots\!33}a^{39}-\frac{11\!\cdots\!72}{80\!\cdots\!33}a^{38}+\frac{43\!\cdots\!97}{80\!\cdots\!33}a^{37}-\frac{17\!\cdots\!82}{80\!\cdots\!33}a^{36}+\frac{48\!\cdots\!61}{80\!\cdots\!33}a^{35}-\frac{15\!\cdots\!59}{80\!\cdots\!33}a^{34}+\frac{36\!\cdots\!20}{80\!\cdots\!33}a^{33}-\frac{85\!\cdots\!87}{80\!\cdots\!33}a^{32}+\frac{21\!\cdots\!86}{80\!\cdots\!33}a^{31}-\frac{33\!\cdots\!32}{80\!\cdots\!33}a^{30}+\frac{93\!\cdots\!65}{80\!\cdots\!33}a^{29}-\frac{86\!\cdots\!63}{80\!\cdots\!33}a^{28}+\frac{33\!\cdots\!92}{80\!\cdots\!33}a^{27}-\frac{10\!\cdots\!22}{80\!\cdots\!33}a^{26}+\frac{97\!\cdots\!17}{80\!\cdots\!33}a^{25}+\frac{28\!\cdots\!88}{80\!\cdots\!33}a^{24}+\frac{23\!\cdots\!86}{80\!\cdots\!33}a^{23}+\frac{19\!\cdots\!31}{80\!\cdots\!33}a^{22}+\frac{44\!\cdots\!21}{80\!\cdots\!33}a^{21}+\frac{61\!\cdots\!98}{80\!\cdots\!33}a^{20}+\frac{70\!\cdots\!26}{80\!\cdots\!33}a^{19}+\frac{13\!\cdots\!82}{80\!\cdots\!33}a^{18}+\frac{89\!\cdots\!54}{80\!\cdots\!33}a^{17}+\frac{20\!\cdots\!87}{80\!\cdots\!33}a^{16}+\frac{90\!\cdots\!30}{80\!\cdots\!33}a^{15}+\frac{14\!\cdots\!85}{49\!\cdots\!91}a^{14}+\frac{70\!\cdots\!42}{80\!\cdots\!33}a^{13}+\frac{21\!\cdots\!48}{80\!\cdots\!33}a^{12}+\frac{41\!\cdots\!18}{80\!\cdots\!33}a^{11}+\frac{13\!\cdots\!15}{80\!\cdots\!33}a^{10}+\frac{17\!\cdots\!39}{80\!\cdots\!33}a^{9}+\frac{58\!\cdots\!79}{80\!\cdots\!33}a^{8}+\frac{53\!\cdots\!22}{80\!\cdots\!33}a^{7}+\frac{15\!\cdots\!44}{80\!\cdots\!33}a^{6}+\frac{95\!\cdots\!47}{80\!\cdots\!33}a^{5}+\frac{28\!\cdots\!06}{80\!\cdots\!33}a^{4}+\frac{12\!\cdots\!22}{80\!\cdots\!33}a^{3}+\frac{17\!\cdots\!49}{80\!\cdots\!33}a^{2}+\frac{12\!\cdots\!16}{80\!\cdots\!33}a-\frac{33\!\cdots\!05}{80\!\cdots\!33}$, $\frac{85\!\cdots\!11}{80\!\cdots\!33}a^{39}-\frac{99\!\cdots\!59}{80\!\cdots\!33}a^{38}+\frac{17\!\cdots\!84}{80\!\cdots\!33}a^{37}-\frac{17\!\cdots\!65}{80\!\cdots\!33}a^{36}+\frac{19\!\cdots\!33}{80\!\cdots\!33}a^{35}-\frac{17\!\cdots\!46}{80\!\cdots\!33}a^{34}+\frac{15\!\cdots\!97}{80\!\cdots\!33}a^{33}-\frac{12\!\cdots\!57}{80\!\cdots\!33}a^{32}+\frac{87\!\cdots\!63}{80\!\cdots\!33}a^{31}-\frac{65\!\cdots\!32}{80\!\cdots\!33}a^{30}+\frac{39\!\cdots\!29}{80\!\cdots\!33}a^{29}-\frac{26\!\cdots\!73}{80\!\cdots\!33}a^{28}+\frac{14\!\cdots\!01}{80\!\cdots\!33}a^{27}-\frac{88\!\cdots\!03}{80\!\cdots\!33}a^{26}+\frac{41\!\cdots\!05}{80\!\cdots\!33}a^{25}-\frac{23\!\cdots\!48}{80\!\cdots\!33}a^{24}+\frac{99\!\cdots\!43}{80\!\cdots\!33}a^{23}-\frac{50\!\cdots\!09}{80\!\cdots\!33}a^{22}+\frac{19\!\cdots\!29}{80\!\cdots\!33}a^{21}-\frac{87\!\cdots\!22}{80\!\cdots\!33}a^{20}+\frac{30\!\cdots\!19}{80\!\cdots\!33}a^{19}-\frac{12\!\cdots\!56}{80\!\cdots\!33}a^{18}+\frac{38\!\cdots\!17}{80\!\cdots\!33}a^{17}-\frac{13\!\cdots\!10}{80\!\cdots\!33}a^{16}+\frac{38\!\cdots\!24}{80\!\cdots\!33}a^{15}-\frac{12\!\cdots\!60}{80\!\cdots\!33}a^{14}+\frac{30\!\cdots\!73}{80\!\cdots\!33}a^{13}-\frac{83\!\cdots\!39}{80\!\cdots\!33}a^{12}+\frac{10\!\cdots\!81}{49\!\cdots\!91}a^{11}-\frac{43\!\cdots\!22}{80\!\cdots\!33}a^{10}+\frac{73\!\cdots\!29}{80\!\cdots\!33}a^{9}-\frac{15\!\cdots\!49}{80\!\cdots\!33}a^{8}+\frac{20\!\cdots\!28}{80\!\cdots\!33}a^{7}-\frac{40\!\cdots\!69}{80\!\cdots\!33}a^{6}+\frac{35\!\cdots\!96}{80\!\cdots\!33}a^{5}-\frac{46\!\cdots\!19}{80\!\cdots\!33}a^{4}+\frac{28\!\cdots\!94}{80\!\cdots\!33}a^{3}-\frac{50\!\cdots\!58}{80\!\cdots\!33}a^{2}+\frac{74\!\cdots\!07}{80\!\cdots\!33}a+\frac{42\!\cdots\!96}{80\!\cdots\!33}$, $\frac{41\!\cdots\!85}{80\!\cdots\!33}a^{39}-\frac{39\!\cdots\!46}{80\!\cdots\!33}a^{38}+\frac{83\!\cdots\!27}{80\!\cdots\!33}a^{37}-\frac{66\!\cdots\!52}{80\!\cdots\!33}a^{36}+\frac{93\!\cdots\!35}{80\!\cdots\!33}a^{35}-\frac{66\!\cdots\!02}{80\!\cdots\!33}a^{34}+\frac{71\!\cdots\!36}{80\!\cdots\!33}a^{33}-\frac{44\!\cdots\!16}{80\!\cdots\!33}a^{32}+\frac{41\!\cdots\!23}{80\!\cdots\!33}a^{31}-\frac{22\!\cdots\!92}{80\!\cdots\!33}a^{30}+\frac{18\!\cdots\!99}{80\!\cdots\!33}a^{29}-\frac{89\!\cdots\!70}{80\!\cdots\!33}a^{28}+\frac{66\!\cdots\!12}{80\!\cdots\!33}a^{27}-\frac{28\!\cdots\!84}{80\!\cdots\!33}a^{26}+\frac{19\!\cdots\!91}{80\!\cdots\!33}a^{25}-\frac{70\!\cdots\!80}{80\!\cdots\!33}a^{24}+\frac{45\!\cdots\!32}{80\!\cdots\!33}a^{23}-\frac{14\!\cdots\!70}{80\!\cdots\!33}a^{22}+\frac{88\!\cdots\!70}{80\!\cdots\!33}a^{21}-\frac{22\!\cdots\!72}{80\!\cdots\!33}a^{20}+\frac{13\!\cdots\!58}{80\!\cdots\!33}a^{19}-\frac{28\!\cdots\!17}{80\!\cdots\!33}a^{18}+\frac{17\!\cdots\!75}{80\!\cdots\!33}a^{17}-\frac{26\!\cdots\!96}{80\!\cdots\!33}a^{16}+\frac{17\!\cdots\!94}{80\!\cdots\!33}a^{15}-\frac{19\!\cdots\!85}{80\!\cdots\!33}a^{14}+\frac{13\!\cdots\!44}{80\!\cdots\!33}a^{13}-\frac{83\!\cdots\!88}{80\!\cdots\!33}a^{12}+\frac{46\!\cdots\!22}{49\!\cdots\!91}a^{11}-\frac{19\!\cdots\!71}{80\!\cdots\!33}a^{10}+\frac{31\!\cdots\!03}{80\!\cdots\!33}a^{9}+\frac{85\!\cdots\!17}{80\!\cdots\!33}a^{8}+\frac{84\!\cdots\!62}{80\!\cdots\!33}a^{7}+\frac{44\!\cdots\!54}{80\!\cdots\!33}a^{6}+\frac{13\!\cdots\!16}{80\!\cdots\!33}a^{5}+\frac{22\!\cdots\!67}{80\!\cdots\!33}a^{4}+\frac{10\!\cdots\!74}{80\!\cdots\!33}a^{3}+\frac{16\!\cdots\!71}{80\!\cdots\!33}a^{2}+\frac{12\!\cdots\!15}{80\!\cdots\!33}a+\frac{13\!\cdots\!57}{80\!\cdots\!33}$, $\frac{70\!\cdots\!47}{80\!\cdots\!33}a^{39}-\frac{79\!\cdots\!04}{80\!\cdots\!33}a^{38}+\frac{14\!\cdots\!24}{80\!\cdots\!33}a^{37}-\frac{13\!\cdots\!25}{80\!\cdots\!33}a^{36}+\frac{16\!\cdots\!95}{80\!\cdots\!33}a^{35}-\frac{14\!\cdots\!25}{80\!\cdots\!33}a^{34}+\frac{12\!\cdots\!09}{80\!\cdots\!33}a^{33}-\frac{98\!\cdots\!24}{80\!\cdots\!33}a^{32}+\frac{71\!\cdots\!40}{80\!\cdots\!33}a^{31}-\frac{51\!\cdots\!77}{80\!\cdots\!33}a^{30}+\frac{32\!\cdots\!14}{80\!\cdots\!33}a^{29}-\frac{21\!\cdots\!73}{80\!\cdots\!33}a^{28}+\frac{11\!\cdots\!26}{80\!\cdots\!33}a^{27}-\frac{69\!\cdots\!50}{80\!\cdots\!33}a^{26}+\frac{34\!\cdots\!88}{80\!\cdots\!33}a^{25}-\frac{18\!\cdots\!97}{80\!\cdots\!33}a^{24}+\frac{82\!\cdots\!72}{80\!\cdots\!33}a^{23}-\frac{39\!\cdots\!32}{80\!\cdots\!33}a^{22}+\frac{16\!\cdots\!94}{80\!\cdots\!33}a^{21}-\frac{68\!\cdots\!37}{80\!\cdots\!33}a^{20}+\frac{25\!\cdots\!80}{80\!\cdots\!33}a^{19}-\frac{95\!\cdots\!22}{80\!\cdots\!33}a^{18}+\frac{32\!\cdots\!86}{80\!\cdots\!33}a^{17}-\frac{10\!\cdots\!53}{80\!\cdots\!33}a^{16}+\frac{32\!\cdots\!32}{80\!\cdots\!33}a^{15}-\frac{95\!\cdots\!63}{80\!\cdots\!33}a^{14}+\frac{25\!\cdots\!07}{80\!\cdots\!33}a^{13}-\frac{64\!\cdots\!58}{80\!\cdots\!33}a^{12}+\frac{15\!\cdots\!64}{80\!\cdots\!33}a^{11}-\frac{33\!\cdots\!74}{80\!\cdots\!33}a^{10}+\frac{64\!\cdots\!45}{80\!\cdots\!33}a^{9}-\frac{11\!\cdots\!03}{80\!\cdots\!33}a^{8}+\frac{18\!\cdots\!43}{80\!\cdots\!33}a^{7}-\frac{32\!\cdots\!87}{80\!\cdots\!33}a^{6}+\frac{34\!\cdots\!16}{80\!\cdots\!33}a^{5}-\frac{42\!\cdots\!09}{80\!\cdots\!33}a^{4}+\frac{33\!\cdots\!67}{80\!\cdots\!33}a^{3}-\frac{52\!\cdots\!77}{80\!\cdots\!33}a^{2}+\frac{15\!\cdots\!46}{80\!\cdots\!33}a-\frac{44\!\cdots\!07}{80\!\cdots\!33}$, $\frac{58\!\cdots\!97}{80\!\cdots\!33}a^{39}-\frac{65\!\cdots\!83}{80\!\cdots\!33}a^{38}+\frac{11\!\cdots\!84}{80\!\cdots\!33}a^{37}-\frac{11\!\cdots\!85}{80\!\cdots\!33}a^{36}+\frac{13\!\cdots\!24}{80\!\cdots\!33}a^{35}-\frac{11\!\cdots\!24}{80\!\cdots\!33}a^{34}+\frac{10\!\cdots\!43}{80\!\cdots\!33}a^{33}-\frac{81\!\cdots\!63}{80\!\cdots\!33}a^{32}+\frac{58\!\cdots\!49}{80\!\cdots\!33}a^{31}-\frac{42\!\cdots\!12}{80\!\cdots\!33}a^{30}+\frac{26\!\cdots\!05}{80\!\cdots\!33}a^{29}-\frac{17\!\cdots\!72}{80\!\cdots\!33}a^{28}+\frac{95\!\cdots\!20}{80\!\cdots\!33}a^{27}-\frac{56\!\cdots\!69}{80\!\cdots\!33}a^{26}+\frac{27\!\cdots\!25}{80\!\cdots\!33}a^{25}-\frac{14\!\cdots\!48}{80\!\cdots\!33}a^{24}+\frac{66\!\cdots\!54}{80\!\cdots\!33}a^{23}-\frac{31\!\cdots\!49}{80\!\cdots\!33}a^{22}+\frac{12\!\cdots\!79}{80\!\cdots\!33}a^{21}-\frac{55\!\cdots\!29}{80\!\cdots\!33}a^{20}+\frac{20\!\cdots\!61}{80\!\cdots\!33}a^{19}-\frac{77\!\cdots\!59}{80\!\cdots\!33}a^{18}+\frac{25\!\cdots\!71}{80\!\cdots\!33}a^{17}-\frac{85\!\cdots\!63}{80\!\cdots\!33}a^{16}+\frac{25\!\cdots\!42}{80\!\cdots\!33}a^{15}-\frac{75\!\cdots\!46}{80\!\cdots\!33}a^{14}+\frac{19\!\cdots\!58}{80\!\cdots\!33}a^{13}-\frac{49\!\cdots\!78}{80\!\cdots\!33}a^{12}+\frac{11\!\cdots\!04}{80\!\cdots\!33}a^{11}-\frac{25\!\cdots\!64}{80\!\cdots\!33}a^{10}+\frac{47\!\cdots\!60}{80\!\cdots\!33}a^{9}-\frac{86\!\cdots\!03}{80\!\cdots\!33}a^{8}+\frac{13\!\cdots\!94}{80\!\cdots\!33}a^{7}-\frac{23\!\cdots\!25}{80\!\cdots\!33}a^{6}+\frac{22\!\cdots\!49}{80\!\cdots\!33}a^{5}-\frac{26\!\cdots\!41}{80\!\cdots\!33}a^{4}+\frac{18\!\cdots\!07}{80\!\cdots\!33}a^{3}-\frac{41\!\cdots\!07}{80\!\cdots\!33}a^{2}+\frac{51\!\cdots\!54}{80\!\cdots\!33}a+\frac{30\!\cdots\!78}{80\!\cdots\!33}$, $\frac{67\!\cdots\!03}{80\!\cdots\!33}a^{39}-\frac{70\!\cdots\!15}{80\!\cdots\!33}a^{38}+\frac{13\!\cdots\!35}{80\!\cdots\!33}a^{37}-\frac{12\!\cdots\!42}{80\!\cdots\!33}a^{36}+\frac{15\!\cdots\!94}{80\!\cdots\!33}a^{35}-\frac{12\!\cdots\!35}{80\!\cdots\!33}a^{34}+\frac{11\!\cdots\!64}{80\!\cdots\!33}a^{33}-\frac{83\!\cdots\!50}{80\!\cdots\!33}a^{32}+\frac{67\!\cdots\!87}{80\!\cdots\!33}a^{31}-\frac{43\!\cdots\!08}{80\!\cdots\!33}a^{30}+\frac{30\!\cdots\!85}{80\!\cdots\!33}a^{29}-\frac{17\!\cdots\!93}{80\!\cdots\!33}a^{28}+\frac{10\!\cdots\!00}{80\!\cdots\!33}a^{27}-\frac{56\!\cdots\!79}{80\!\cdots\!33}a^{26}+\frac{31\!\cdots\!78}{80\!\cdots\!33}a^{25}-\frac{14\!\cdots\!85}{80\!\cdots\!33}a^{24}+\frac{75\!\cdots\!07}{80\!\cdots\!33}a^{23}-\frac{30\!\cdots\!83}{80\!\cdots\!33}a^{22}+\frac{14\!\cdots\!12}{80\!\cdots\!33}a^{21}-\frac{52\!\cdots\!05}{80\!\cdots\!33}a^{20}+\frac{23\!\cdots\!40}{80\!\cdots\!33}a^{19}-\frac{71\!\cdots\!00}{80\!\cdots\!33}a^{18}+\frac{29\!\cdots\!63}{80\!\cdots\!33}a^{17}-\frac{76\!\cdots\!30}{80\!\cdots\!33}a^{16}+\frac{29\!\cdots\!63}{80\!\cdots\!33}a^{15}-\frac{64\!\cdots\!35}{80\!\cdots\!33}a^{14}+\frac{22\!\cdots\!19}{80\!\cdots\!33}a^{13}-\frac{40\!\cdots\!42}{80\!\cdots\!33}a^{12}+\frac{13\!\cdots\!13}{80\!\cdots\!33}a^{11}-\frac{19\!\cdots\!33}{80\!\cdots\!33}a^{10}+\frac{55\!\cdots\!10}{80\!\cdots\!33}a^{9}-\frac{61\!\cdots\!27}{80\!\cdots\!33}a^{8}+\frac{15\!\cdots\!83}{80\!\cdots\!33}a^{7}-\frac{16\!\cdots\!27}{80\!\cdots\!33}a^{6}+\frac{26\!\cdots\!08}{80\!\cdots\!33}a^{5}-\frac{13\!\cdots\!51}{80\!\cdots\!33}a^{4}+\frac{24\!\cdots\!12}{80\!\cdots\!33}a^{3}-\frac{27\!\cdots\!40}{80\!\cdots\!33}a^{2}+\frac{80\!\cdots\!93}{80\!\cdots\!33}a+\frac{50\!\cdots\!66}{80\!\cdots\!33}$, $\frac{55\!\cdots\!19}{80\!\cdots\!33}a^{39}-\frac{16\!\cdots\!17}{80\!\cdots\!33}a^{38}+\frac{11\!\cdots\!81}{80\!\cdots\!33}a^{37}-\frac{31\!\cdots\!09}{80\!\cdots\!33}a^{36}+\frac{13\!\cdots\!91}{80\!\cdots\!33}a^{35}-\frac{34\!\cdots\!24}{80\!\cdots\!33}a^{34}+\frac{10\!\cdots\!90}{80\!\cdots\!33}a^{33}-\frac{25\!\cdots\!57}{80\!\cdots\!33}a^{32}+\frac{63\!\cdots\!31}{80\!\cdots\!33}a^{31}-\frac{14\!\cdots\!05}{80\!\cdots\!33}a^{30}+\frac{28\!\cdots\!74}{80\!\cdots\!33}a^{29}-\frac{61\!\cdots\!04}{80\!\cdots\!33}a^{28}+\frac{10\!\cdots\!85}{80\!\cdots\!33}a^{27}-\frac{21\!\cdots\!83}{80\!\cdots\!33}a^{26}+\frac{30\!\cdots\!87}{80\!\cdots\!33}a^{25}-\frac{60\!\cdots\!79}{80\!\cdots\!33}a^{24}+\frac{71\!\cdots\!82}{80\!\cdots\!33}a^{23}-\frac{14\!\cdots\!84}{80\!\cdots\!33}a^{22}+\frac{13\!\cdots\!10}{80\!\cdots\!33}a^{21}-\frac{26\!\cdots\!85}{80\!\cdots\!33}a^{20}+\frac{20\!\cdots\!76}{80\!\cdots\!33}a^{19}-\frac{40\!\cdots\!54}{80\!\cdots\!33}a^{18}+\frac{25\!\cdots\!89}{80\!\cdots\!33}a^{17}-\frac{48\!\cdots\!44}{80\!\cdots\!33}a^{16}+\frac{23\!\cdots\!66}{80\!\cdots\!33}a^{15}-\frac{47\!\cdots\!70}{80\!\cdots\!33}a^{14}+\frac{15\!\cdots\!99}{80\!\cdots\!33}a^{13}-\frac{34\!\cdots\!55}{80\!\cdots\!33}a^{12}+\frac{69\!\cdots\!91}{80\!\cdots\!33}a^{11}-\frac{19\!\cdots\!01}{80\!\cdots\!33}a^{10}+\frac{15\!\cdots\!99}{80\!\cdots\!33}a^{9}-\frac{72\!\cdots\!33}{80\!\cdots\!33}a^{8}-\frac{28\!\cdots\!30}{80\!\cdots\!33}a^{7}-\frac{18\!\cdots\!02}{80\!\cdots\!33}a^{6}-\frac{21\!\cdots\!43}{80\!\cdots\!33}a^{5}-\frac{21\!\cdots\!25}{80\!\cdots\!33}a^{4}-\frac{65\!\cdots\!16}{80\!\cdots\!33}a^{3}-\frac{97\!\cdots\!84}{80\!\cdots\!33}a^{2}-\frac{65\!\cdots\!61}{80\!\cdots\!33}a-\frac{21\!\cdots\!79}{80\!\cdots\!33}$, $\frac{38\!\cdots\!87}{80\!\cdots\!33}a^{39}-\frac{35\!\cdots\!97}{80\!\cdots\!33}a^{38}+\frac{75\!\cdots\!53}{80\!\cdots\!33}a^{37}-\frac{59\!\cdots\!55}{80\!\cdots\!33}a^{36}+\frac{85\!\cdots\!48}{80\!\cdots\!33}a^{35}-\frac{59\!\cdots\!50}{80\!\cdots\!33}a^{34}+\frac{65\!\cdots\!17}{80\!\cdots\!33}a^{33}-\frac{40\!\cdots\!90}{80\!\cdots\!33}a^{32}+\frac{37\!\cdots\!36}{80\!\cdots\!33}a^{31}-\frac{20\!\cdots\!56}{80\!\cdots\!33}a^{30}+\frac{16\!\cdots\!11}{80\!\cdots\!33}a^{29}-\frac{79\!\cdots\!86}{80\!\cdots\!33}a^{28}+\frac{60\!\cdots\!81}{80\!\cdots\!33}a^{27}-\frac{24\!\cdots\!61}{80\!\cdots\!33}a^{26}+\frac{17\!\cdots\!62}{80\!\cdots\!33}a^{25}-\frac{62\!\cdots\!92}{80\!\cdots\!33}a^{24}+\frac{41\!\cdots\!97}{80\!\cdots\!33}a^{23}-\frac{12\!\cdots\!15}{80\!\cdots\!33}a^{22}+\frac{80\!\cdots\!31}{80\!\cdots\!33}a^{21}-\frac{19\!\cdots\!88}{80\!\cdots\!33}a^{20}+\frac{12\!\cdots\!64}{80\!\cdots\!33}a^{19}-\frac{24\!\cdots\!58}{80\!\cdots\!33}a^{18}+\frac{15\!\cdots\!75}{80\!\cdots\!33}a^{17}-\frac{23\!\cdots\!56}{80\!\cdots\!33}a^{16}+\frac{15\!\cdots\!92}{80\!\cdots\!33}a^{15}-\frac{16\!\cdots\!45}{80\!\cdots\!33}a^{14}+\frac{11\!\cdots\!88}{80\!\cdots\!33}a^{13}-\frac{66\!\cdots\!09}{80\!\cdots\!33}a^{12}+\frac{68\!\cdots\!47}{80\!\cdots\!33}a^{11}-\frac{13\!\cdots\!81}{80\!\cdots\!33}a^{10}+\frac{27\!\cdots\!59}{80\!\cdots\!33}a^{9}+\frac{87\!\cdots\!26}{80\!\cdots\!33}a^{8}+\frac{75\!\cdots\!18}{80\!\cdots\!33}a^{7}+\frac{40\!\cdots\!63}{80\!\cdots\!33}a^{6}+\frac{11\!\cdots\!72}{80\!\cdots\!33}a^{5}+\frac{20\!\cdots\!75}{80\!\cdots\!33}a^{4}+\frac{95\!\cdots\!28}{80\!\cdots\!33}a^{3}+\frac{15\!\cdots\!57}{80\!\cdots\!33}a^{2}+\frac{11\!\cdots\!32}{80\!\cdots\!33}a+\frac{14\!\cdots\!18}{80\!\cdots\!33}$ (assuming GRH)
Regulator:		\( 310417721980536.1 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 310417721980536.1 \cdot 69632}{6\cdot\sqrt{67330470569637410331240297486633042732017118041235378120152501254022721}}\cr\approx \mathstrut & 0.127672002064966 \end{aligned}\] (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

\(\Q(\sqrt{-123}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{41})\), 4.0.620289.1, 4.4.68921.1, 5.5.2825761.1, 8.0.384758443521.1, 10.0.79553810057732523.1, 10.10.327381934393961.1, 10.0.1940336830676403.1, 20.0.6328808694701784334458304211945529.1, 20.0.259481156482773157712790472689766689.1, \(\Q(\zeta_{41})^+\)

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	${\href{/padicField/2.10.0.1}{10} }^{4}$	R	${\href{/padicField/5.10.0.1}{10} }^{4}$	$20^{2}$	$20^{2}$	$20^{2}$	$20^{2}$	$20^{2}$	${\href{/padicField/23.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/padicField/31.5.0.1}{5} }^{8}$	${\href{/padicField/37.5.0.1}{5} }^{8}$	R	${\href{/padicField/43.10.0.1}{10} }^{4}$	$20^{2}$	$20^{2}$	${\href{/padicField/59.10.0.1}{10} }^{4}$

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
\(3\)	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
\(41\)		Deg $40$	$20$	$2$	$38$