Number field 40.0.100383397447978918530459891214693626269465146363712847232818603515625.3

• Label: 40.0.100...625.3
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $1.004\times 10^{68}$
• Root discriminant: \(50.12\)
• Ramified primes: $3,5,11$
• Class number: $18524$ (GRH)
• Class group: [2, 9262] (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

x^40 - x^39 + 21*x^38 - 18*x^37 + 249*x^36 - 191*x^35 + 2008*x^34 - 1359*x^33 + 12160*x^32 - 7243*x^31 + 57502*x^30 - 29496*x^29 + 218159*x^28 - 94389*x^27 + 671689*x^26 - 234893*x^25 + 1690845*x^24 - 452550*x^23 + 3479454*x^22 - 637891*x^21 + 5838877*x^20 - 598784*x^19 + 7925819*x^18 - 188442*x^17 + 8602640*x^16 + 356201*x^15 + 7327393*x^14 + 688488*x^13 + 4739830*x^12 + 533615*x^11 + 2212864*x^10 + 232193*x^9 + 670719*x^8 + 9081*x^7 + 109203*x^6 - 5639*x^5 + 12525*x^4 - 994*x^3 + 252*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$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(4,·)$, $\chi_{165}(7,·)$, $\chi_{165}(8,·)$, $\chi_{165}(13,·)$, $\chi_{165}(14,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(146,·)$, $\chi_{165}(26,·)$, $\chi_{165}(28,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(34,·)$, $\chi_{165}(43,·)$, $\chi_{165}(49,·)$, $\chi_{165}(52,·)$, $\chi_{165}(56,·)$, $\chi_{165}(59,·)$, $\chi_{165}(62,·)$, $\chi_{165}(64,·)$, $\chi_{165}(68,·)$, $\chi_{165}(71,·)$, $\chi_{165}(73,·)$, $\chi_{165}(83,·)$, $\chi_{165}(142,·)$, $\chi_{165}(86,·)$, $\chi_{165}(89,·)$, $\chi_{165}(91,·)$, $\chi_{165}(98,·)$, $\chi_{165}(104,·)$, $\chi_{165}(107,·)$, $\chi_{165}(112,·)$, $\chi_{165}(118,·)$, $\chi_{165}(119,·)$, $\chi_{165}(136,·)$, $\chi_{165}(124,·)$, $\chi_{165}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{109}a^{34}+\frac{31}{109}a^{33}+\frac{19}{109}a^{32}+\frac{50}{109}a^{31}-\frac{53}{109}a^{30}-\frac{9}{109}a^{29}+\frac{39}{109}a^{28}-\frac{50}{109}a^{27}+\frac{1}{109}a^{26}-\frac{26}{109}a^{25}+\frac{8}{109}a^{24}-\frac{6}{109}a^{23}+\frac{38}{109}a^{22}-\frac{17}{109}a^{21}-\frac{46}{109}a^{20}-\frac{48}{109}a^{19}-\frac{39}{109}a^{18}-\frac{34}{109}a^{17}+\frac{14}{109}a^{16}+\frac{18}{109}a^{15}+\frac{51}{109}a^{14}+\frac{44}{109}a^{13}+\frac{43}{109}a^{12}+\frac{4}{109}a^{11}-\frac{24}{109}a^{10}-\frac{8}{109}a^{9}+\frac{46}{109}a^{8}-\frac{3}{109}a^{7}-\frac{8}{109}a^{5}-\frac{33}{109}a^{4}+\frac{21}{109}a^{3}-\frac{18}{109}a^{2}-\frac{52}{109}a+\frac{46}{109}$, $\frac{1}{109}a^{35}+\frac{39}{109}a^{33}+\frac{6}{109}a^{32}+\frac{32}{109}a^{31}-\frac{1}{109}a^{30}-\frac{9}{109}a^{29}+\frac{49}{109}a^{28}+\frac{25}{109}a^{27}+\frac{52}{109}a^{26}+\frac{51}{109}a^{25}-\frac{36}{109}a^{24}+\frac{6}{109}a^{23}+\frac{4}{109}a^{22}+\frac{45}{109}a^{21}-\frac{39}{109}a^{20}+\frac{32}{109}a^{19}-\frac{24}{109}a^{18}-\frac{22}{109}a^{17}+\frac{20}{109}a^{16}+\frac{38}{109}a^{15}-\frac{11}{109}a^{14}-\frac{13}{109}a^{13}-\frac{21}{109}a^{12}-\frac{39}{109}a^{11}-\frac{27}{109}a^{10}-\frac{33}{109}a^{9}-\frac{12}{109}a^{8}-\frac{16}{109}a^{7}-\frac{8}{109}a^{6}-\frac{3}{109}a^{5}-\frac{46}{109}a^{4}-\frac{15}{109}a^{3}-\frac{39}{109}a^{2}+\frac{23}{109}a-\frac{9}{109}$, $\frac{1}{109}a^{36}-\frac{4}{109}a^{33}+\frac{54}{109}a^{32}+\frac{11}{109}a^{31}-\frac{13}{109}a^{30}-\frac{36}{109}a^{29}+\frac{30}{109}a^{28}+\frac{40}{109}a^{27}+\frac{12}{109}a^{26}-\frac{3}{109}a^{25}+\frac{21}{109}a^{24}+\frac{20}{109}a^{23}-\frac{20}{109}a^{22}-\frac{30}{109}a^{21}-\frac{27}{109}a^{20}-\frac{5}{109}a^{19}-\frac{27}{109}a^{18}+\frac{38}{109}a^{17}+\frac{37}{109}a^{16}+\frac{50}{109}a^{15}-\frac{40}{109}a^{14}+\frac{7}{109}a^{13}+\frac{28}{109}a^{12}+\frac{35}{109}a^{11}+\frac{31}{109}a^{10}-\frac{27}{109}a^{9}+\frac{43}{109}a^{8}-\frac{3}{109}a^{6}+\frac{48}{109}a^{5}-\frac{36}{109}a^{4}+\frac{14}{109}a^{3}-\frac{38}{109}a^{2}-\frac{52}{109}a-\frac{50}{109}$, $\frac{1}{109}a^{37}-\frac{40}{109}a^{33}-\frac{22}{109}a^{32}-\frac{31}{109}a^{31}-\frac{30}{109}a^{30}-\frac{6}{109}a^{29}-\frac{22}{109}a^{28}+\frac{30}{109}a^{27}+\frac{1}{109}a^{26}+\frac{26}{109}a^{25}+\frac{52}{109}a^{24}-\frac{44}{109}a^{23}+\frac{13}{109}a^{22}+\frac{14}{109}a^{21}+\frac{29}{109}a^{20}-\frac{1}{109}a^{19}-\frac{9}{109}a^{18}+\frac{10}{109}a^{17}-\frac{3}{109}a^{16}+\frac{32}{109}a^{15}-\frac{7}{109}a^{14}-\frac{14}{109}a^{13}-\frac{11}{109}a^{12}+\frac{47}{109}a^{11}-\frac{14}{109}a^{10}+\frac{11}{109}a^{9}-\frac{34}{109}a^{8}-\frac{15}{109}a^{7}+\frac{48}{109}a^{6}+\frac{41}{109}a^{5}-\frac{9}{109}a^{4}+\frac{46}{109}a^{3}-\frac{15}{109}a^{2}-\frac{40}{109}a-\frac{34}{109}$, $\frac{1}{109}a^{38}+\frac{19}{109}a^{33}-\frac{34}{109}a^{32}+\frac{8}{109}a^{31}+\frac{54}{109}a^{30}+\frac{54}{109}a^{29}-\frac{45}{109}a^{28}-\frac{37}{109}a^{27}-\frac{43}{109}a^{26}-\frac{7}{109}a^{25}-\frac{51}{109}a^{24}-\frac{9}{109}a^{23}+\frac{8}{109}a^{22}+\frac{3}{109}a^{21}+\frac{12}{109}a^{20}+\frac{33}{109}a^{19}-\frac{24}{109}a^{18}+\frac{54}{109}a^{17}+\frac{47}{109}a^{16}-\frac{50}{109}a^{15}-\frac{45}{109}a^{14}+\frac{5}{109}a^{13}+\frac{23}{109}a^{12}+\frac{37}{109}a^{11}+\frac{32}{109}a^{10}-\frac{27}{109}a^{9}-\frac{28}{109}a^{8}+\frac{37}{109}a^{7}+\frac{41}{109}a^{6}-\frac{2}{109}a^{5}+\frac{34}{109}a^{4}-\frac{47}{109}a^{3}+\frac{3}{109}a^{2}-\frac{43}{109}a-\frac{13}{109}$, $\frac{1}{48\!\cdots\!71}a^{39}-\frac{17\!\cdots\!75}{48\!\cdots\!71}a^{38}-\frac{82\!\cdots\!42}{48\!\cdots\!71}a^{37}-\frac{21\!\cdots\!05}{48\!\cdots\!71}a^{36}-\frac{77\!\cdots\!49}{48\!\cdots\!71}a^{35}+\frac{31\!\cdots\!17}{48\!\cdots\!71}a^{34}+\frac{41\!\cdots\!45}{48\!\cdots\!71}a^{33}+\frac{15\!\cdots\!86}{48\!\cdots\!71}a^{32}-\frac{16\!\cdots\!17}{48\!\cdots\!71}a^{31}-\frac{26\!\cdots\!67}{48\!\cdots\!71}a^{30}+\frac{46\!\cdots\!19}{48\!\cdots\!71}a^{29}+\frac{24\!\cdots\!78}{48\!\cdots\!71}a^{28}-\frac{22\!\cdots\!81}{48\!\cdots\!71}a^{27}+\frac{17\!\cdots\!76}{48\!\cdots\!71}a^{26}+\frac{23\!\cdots\!45}{48\!\cdots\!71}a^{25}+\frac{37\!\cdots\!01}{48\!\cdots\!71}a^{24}-\frac{10\!\cdots\!21}{48\!\cdots\!71}a^{23}-\frac{22\!\cdots\!72}{48\!\cdots\!71}a^{22}+\frac{17\!\cdots\!49}{48\!\cdots\!71}a^{21}-\frac{13\!\cdots\!54}{48\!\cdots\!71}a^{20}+\frac{11\!\cdots\!05}{48\!\cdots\!71}a^{19}+\frac{71\!\cdots\!45}{48\!\cdots\!71}a^{18}+\frac{11\!\cdots\!39}{48\!\cdots\!71}a^{17}-\frac{12\!\cdots\!67}{48\!\cdots\!71}a^{16}-\frac{48\!\cdots\!04}{48\!\cdots\!71}a^{15}-\frac{14\!\cdots\!88}{48\!\cdots\!71}a^{14}+\frac{21\!\cdots\!27}{48\!\cdots\!71}a^{13}+\frac{21\!\cdots\!17}{48\!\cdots\!71}a^{12}-\frac{10\!\cdots\!53}{48\!\cdots\!71}a^{11}+\frac{34\!\cdots\!84}{48\!\cdots\!71}a^{10}-\frac{22\!\cdots\!63}{48\!\cdots\!71}a^{9}+\frac{19\!\cdots\!52}{48\!\cdots\!71}a^{8}+\frac{10\!\cdots\!87}{48\!\cdots\!71}a^{7}-\frac{18\!\cdots\!22}{48\!\cdots\!71}a^{6}-\frac{23\!\cdots\!60}{48\!\cdots\!71}a^{5}-\frac{22\!\cdots\!88}{48\!\cdots\!71}a^{4}-\frac{30\!\cdots\!65}{14\!\cdots\!41}a^{3}+\frac{18\!\cdots\!53}{48\!\cdots\!71}a^{2}+\frac{23\!\cdots\!18}{48\!\cdots\!71}a+\frac{16\!\cdots\!57}{48\!\cdots\!71}$

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

Class group and class number

$C_{2}\times C_{9262}$, which has order $18524$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( \frac{26281091788646389710510413001060931990337284849179492587352630174565543118306577}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{39} - \frac{28191838206702436115071206864593211495557662405737192671535310756070322999185815}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{38} + \frac{553530774918741203155061821921596781505547770228316063714225382953157618710700775}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{37} - \frac{512861206840069341910611165784833834327423771140132820934010040191208610990794843}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{36} + \frac{6572421852155275578212342572163776511341464200998616305535239765795149413440716032}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{35} - \frac{5489535195768518730344695930199393237149228436896344222566318715773554631192077720}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{34} + \frac{53066597676019725724158738201429444259260499487311294293648097185297595389577262041}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{33} - \frac{39489022677069587940889518690225633053458730948177620792944152083405446286653657633}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{32} + \frac{321603563030319538124771062045217638515104532718139439540475602650991080789709850489}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{31} - \frac{213127303686564493414032469132153690592633271946958754460694144157394818391061468132}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{30} + \frac{1521596896838623255256631508943894470630850947652871095866375427150098203006646960929}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{29} - \frac{882552448617677337055791221632616433929656995454099000295355473984845330514656043504}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{28} + \frac{5773480474136659925336623607329973792286699616748933462542868787610945062136931381811}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{27} - \frac{2887023480164321999980050330572695765919200524293031216214020394010340710005594810112}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{26} + \frac{17771224849974749780939168644149657517041564139345819859411851817685017420684535022317}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{25} - \frac{7422041199873113562758260478990607859689109309319782784293143104426282248204660376523}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{24} + \frac{44696242420947962152154445958929818378212450656271455916368893106004686162413472303319}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{23} - \frac{15033830806129978162714936328453172765311781071788316459671561724016701711024702534627}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{22} + \frac{91832951141033064885127613110989081608809450385283652826401694158553446781927274439522}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{21} - \frac{23225331321836529701668660051575260088699106055172094207769143606847084552355062630406}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{20} + \frac{153698626916630279250667913426645520737811179379855596836037229551646450196560927244500}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{19} - \frac{26592427619805872820697217135062371481347537304419344151381669248007065713386823732299}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{18} + \frac{207826854550735515722097607609679974782355749751268675128073408256722629735074659490676}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{17} - \frac{19728185952563327519086714540077342439345036601866864397769827999967205143503923206231}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{16} + \frac{224279882195865677915577540168702749826879172065526263599681502065804604147708099510960}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{15} - \frac{6756747791223244311676690777428213080311047366619149653609033433255122399099081668344}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{14} + \frac{189579205010169420509303594648260512507849152678196770413831561655981516799156662158910}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{13} + \frac{4276914723075017763490085547747757009045958079557089107548005065939680562401453975801}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{12} + \frac{121325496855133725395412209222278961890287794884495049742831568553563655113124155558058}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{11} + \frac{5008843691213336135558957882548651626210291138051190268674264428974783397780021800029}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{10} + \frac{55926775036531069719434688627395053672945215142030116054832518668951668338210393680875}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{9} + \frac{1870010615639541315517151775822738624320897699057617453644002464291237670070274570249}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{8} + \frac{16637871378949282474502532225298829036666017743671783140962468181252697426065325950698}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{7} - \frac{1046746838637205453752493394911220988046929279711493035498847579846836362829710889699}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{6} + \frac{2695739928797475796640664959613889931372382943502219801443536500225109639688093572503}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{5} - \frac{344916366740290421726946267800867614340496972873203725962369813260732216435305591018}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{4} + \frac{951263246209682559036212236186901557697434956120345523991315088693175415354859854}{1454099433870901730639760652006540566616914944088210541317156482260900090629941} a^{3} - \frac{48940211131362596503019300047876770314739494837452870611858312199096147967851869533}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{2} + \frac{5724689276670063775400749057471533927627025753492628873171742541336710223594264361}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a + \frac{2428368981139489205271122647392097059898538377170735756246700232513697907776590}{4415659748727233695795970420313439702295402261405483386935585280994109449527619} \)  (order $6$)
Fundamental units:		$\frac{15\!\cdots\!74}{48\!\cdots\!71}a^{39}-\frac{86\!\cdots\!30}{48\!\cdots\!71}a^{38}+\frac{30\!\cdots\!68}{48\!\cdots\!71}a^{37}-\frac{13\!\cdots\!31}{48\!\cdots\!71}a^{36}+\frac{35\!\cdots\!47}{48\!\cdots\!71}a^{35}-\frac{12\!\cdots\!54}{48\!\cdots\!71}a^{34}+\frac{28\!\cdots\!15}{48\!\cdots\!71}a^{33}-\frac{70\!\cdots\!71}{48\!\cdots\!71}a^{32}+\frac{16\!\cdots\!05}{48\!\cdots\!71}a^{31}-\frac{27\!\cdots\!65}{48\!\cdots\!71}a^{30}+\frac{77\!\cdots\!32}{48\!\cdots\!71}a^{29}-\frac{58\!\cdots\!60}{48\!\cdots\!71}a^{28}+\frac{28\!\cdots\!16}{48\!\cdots\!71}a^{27}+\frac{39\!\cdots\!12}{48\!\cdots\!71}a^{26}+\frac{87\!\cdots\!69}{48\!\cdots\!71}a^{25}+\frac{93\!\cdots\!36}{48\!\cdots\!71}a^{24}+\frac{21\!\cdots\!95}{48\!\cdots\!71}a^{23}+\frac{43\!\cdots\!48}{48\!\cdots\!71}a^{22}+\frac{43\!\cdots\!50}{48\!\cdots\!71}a^{21}+\frac{12\!\cdots\!70}{48\!\cdots\!71}a^{20}+\frac{71\!\cdots\!03}{48\!\cdots\!71}a^{19}+\frac{27\!\cdots\!44}{48\!\cdots\!71}a^{18}+\frac{95\!\cdots\!02}{48\!\cdots\!71}a^{17}+\frac{45\!\cdots\!97}{48\!\cdots\!71}a^{16}+\frac{10\!\cdots\!74}{48\!\cdots\!71}a^{15}+\frac{54\!\cdots\!85}{48\!\cdots\!71}a^{14}+\frac{83\!\cdots\!99}{48\!\cdots\!71}a^{13}+\frac{49\!\cdots\!66}{48\!\cdots\!71}a^{12}+\frac{50\!\cdots\!35}{48\!\cdots\!71}a^{11}+\frac{30\!\cdots\!90}{48\!\cdots\!71}a^{10}+\frac{20\!\cdots\!62}{48\!\cdots\!71}a^{9}+\frac{12\!\cdots\!71}{48\!\cdots\!71}a^{8}+\frac{42\!\cdots\!61}{48\!\cdots\!71}a^{7}+\frac{24\!\cdots\!76}{48\!\cdots\!71}a^{6}-\frac{46\!\cdots\!32}{48\!\cdots\!71}a^{5}+\frac{31\!\cdots\!48}{48\!\cdots\!71}a^{4}-\frac{52\!\cdots\!95}{14\!\cdots\!41}a^{3}+\frac{33\!\cdots\!91}{48\!\cdots\!71}a^{2}-\frac{50\!\cdots\!56}{48\!\cdots\!71}a+\frac{14\!\cdots\!71}{48\!\cdots\!71}$, $\frac{46\!\cdots\!31}{48\!\cdots\!71}a^{39}-\frac{47\!\cdots\!58}{48\!\cdots\!71}a^{38}+\frac{98\!\cdots\!39}{48\!\cdots\!71}a^{37}-\frac{84\!\cdots\!49}{48\!\cdots\!71}a^{36}+\frac{11\!\cdots\!70}{48\!\cdots\!71}a^{35}-\frac{90\!\cdots\!54}{48\!\cdots\!71}a^{34}+\frac{93\!\cdots\!91}{48\!\cdots\!71}a^{33}-\frac{64\!\cdots\!67}{48\!\cdots\!71}a^{32}+\frac{56\!\cdots\!95}{48\!\cdots\!71}a^{31}-\frac{34\!\cdots\!32}{48\!\cdots\!71}a^{30}+\frac{26\!\cdots\!04}{48\!\cdots\!71}a^{29}-\frac{13\!\cdots\!83}{48\!\cdots\!71}a^{28}+\frac{10\!\cdots\!11}{48\!\cdots\!71}a^{27}-\frac{44\!\cdots\!40}{48\!\cdots\!71}a^{26}+\frac{31\!\cdots\!52}{48\!\cdots\!71}a^{25}-\frac{11\!\cdots\!57}{48\!\cdots\!71}a^{24}+\frac{78\!\cdots\!56}{48\!\cdots\!71}a^{23}-\frac{21\!\cdots\!00}{48\!\cdots\!71}a^{22}+\frac{16\!\cdots\!61}{48\!\cdots\!71}a^{21}-\frac{31\!\cdots\!15}{48\!\cdots\!71}a^{20}+\frac{27\!\cdots\!17}{48\!\cdots\!71}a^{19}-\frac{30\!\cdots\!88}{48\!\cdots\!71}a^{18}+\frac{36\!\cdots\!45}{48\!\cdots\!71}a^{17}-\frac{12\!\cdots\!24}{48\!\cdots\!71}a^{16}+\frac{40\!\cdots\!06}{48\!\cdots\!71}a^{15}+\frac{12\!\cdots\!20}{48\!\cdots\!71}a^{14}+\frac{34\!\cdots\!46}{48\!\cdots\!71}a^{13}+\frac{28\!\cdots\!69}{48\!\cdots\!71}a^{12}+\frac{21\!\cdots\!72}{48\!\cdots\!71}a^{11}+\frac{22\!\cdots\!18}{48\!\cdots\!71}a^{10}+\frac{10\!\cdots\!46}{48\!\cdots\!71}a^{9}+\frac{96\!\cdots\!89}{48\!\cdots\!71}a^{8}+\frac{30\!\cdots\!44}{48\!\cdots\!71}a^{7}+\frac{44\!\cdots\!25}{48\!\cdots\!71}a^{6}+\frac{49\!\cdots\!06}{48\!\cdots\!71}a^{5}-\frac{31\!\cdots\!97}{48\!\cdots\!71}a^{4}+\frac{17\!\cdots\!58}{14\!\cdots\!41}a^{3}-\frac{50\!\cdots\!48}{48\!\cdots\!71}a^{2}+\frac{11\!\cdots\!68}{48\!\cdots\!71}a+\frac{53\!\cdots\!78}{48\!\cdots\!71}$, $\frac{21\!\cdots\!53}{48\!\cdots\!71}a^{39}-\frac{11\!\cdots\!37}{48\!\cdots\!71}a^{38}+\frac{44\!\cdots\!36}{48\!\cdots\!71}a^{37}-\frac{17\!\cdots\!32}{48\!\cdots\!71}a^{36}+\frac{51\!\cdots\!85}{48\!\cdots\!71}a^{35}-\frac{15\!\cdots\!61}{48\!\cdots\!71}a^{34}+\frac{40\!\cdots\!41}{48\!\cdots\!71}a^{33}-\frac{79\!\cdots\!84}{48\!\cdots\!71}a^{32}+\frac{23\!\cdots\!23}{48\!\cdots\!71}a^{31}-\frac{25\!\cdots\!81}{48\!\cdots\!71}a^{30}+\frac{11\!\cdots\!79}{48\!\cdots\!71}a^{29}-\frac{11\!\cdots\!00}{48\!\cdots\!71}a^{28}+\frac{41\!\cdots\!26}{48\!\cdots\!71}a^{27}+\frac{34\!\cdots\!41}{48\!\cdots\!71}a^{26}+\frac{12\!\cdots\!57}{48\!\cdots\!71}a^{25}+\frac{23\!\cdots\!86}{48\!\cdots\!71}a^{24}+\frac{30\!\cdots\!72}{48\!\cdots\!71}a^{23}+\frac{87\!\cdots\!81}{48\!\cdots\!71}a^{22}+\frac{60\!\cdots\!43}{48\!\cdots\!71}a^{21}+\frac{24\!\cdots\!26}{48\!\cdots\!71}a^{20}+\frac{99\!\cdots\!13}{48\!\cdots\!71}a^{19}+\frac{49\!\cdots\!65}{48\!\cdots\!71}a^{18}+\frac{13\!\cdots\!87}{48\!\cdots\!71}a^{17}+\frac{78\!\cdots\!32}{48\!\cdots\!71}a^{16}+\frac{13\!\cdots\!92}{48\!\cdots\!71}a^{15}+\frac{94\!\cdots\!22}{48\!\cdots\!71}a^{14}+\frac{11\!\cdots\!69}{48\!\cdots\!71}a^{13}+\frac{84\!\cdots\!68}{48\!\cdots\!71}a^{12}+\frac{65\!\cdots\!34}{48\!\cdots\!71}a^{11}+\frac{52\!\cdots\!69}{48\!\cdots\!71}a^{10}+\frac{24\!\cdots\!19}{48\!\cdots\!71}a^{9}+\frac{22\!\cdots\!19}{48\!\cdots\!71}a^{8}+\frac{33\!\cdots\!47}{48\!\cdots\!71}a^{7}+\frac{46\!\cdots\!97}{48\!\cdots\!71}a^{6}-\frac{16\!\cdots\!23}{48\!\cdots\!71}a^{5}+\frac{63\!\cdots\!91}{48\!\cdots\!71}a^{4}-\frac{12\!\cdots\!35}{14\!\cdots\!41}a^{3}+\frac{78\!\cdots\!22}{48\!\cdots\!71}a^{2}-\frac{65\!\cdots\!56}{48\!\cdots\!71}a+\frac{33\!\cdots\!71}{48\!\cdots\!71}$, $\frac{81\!\cdots\!21}{20\!\cdots\!49}a^{39}-\frac{65\!\cdots\!79}{20\!\cdots\!49}a^{38}+\frac{17\!\cdots\!32}{20\!\cdots\!49}a^{37}-\frac{11\!\cdots\!84}{20\!\cdots\!49}a^{36}+\frac{20\!\cdots\!68}{20\!\cdots\!49}a^{35}-\frac{11\!\cdots\!27}{20\!\cdots\!49}a^{34}+\frac{16\!\cdots\!82}{20\!\cdots\!49}a^{33}-\frac{78\!\cdots\!33}{20\!\cdots\!49}a^{32}+\frac{97\!\cdots\!26}{20\!\cdots\!49}a^{31}-\frac{39\!\cdots\!70}{20\!\cdots\!49}a^{30}+\frac{45\!\cdots\!53}{20\!\cdots\!49}a^{29}-\frac{14\!\cdots\!45}{20\!\cdots\!49}a^{28}+\frac{17\!\cdots\!02}{20\!\cdots\!49}a^{27}-\frac{42\!\cdots\!53}{20\!\cdots\!49}a^{26}+\frac{53\!\cdots\!59}{20\!\cdots\!49}a^{25}-\frac{84\!\cdots\!38}{20\!\cdots\!49}a^{24}+\frac{13\!\cdots\!24}{20\!\cdots\!49}a^{23}-\frac{99\!\cdots\!18}{20\!\cdots\!49}a^{22}+\frac{27\!\cdots\!66}{20\!\cdots\!49}a^{21}+\frac{33\!\cdots\!29}{20\!\cdots\!49}a^{20}+\frac{46\!\cdots\!26}{20\!\cdots\!49}a^{19}+\frac{43\!\cdots\!30}{20\!\cdots\!49}a^{18}+\frac{64\!\cdots\!19}{20\!\cdots\!49}a^{17}+\frac{10\!\cdots\!89}{20\!\cdots\!49}a^{16}+\frac{70\!\cdots\!25}{20\!\cdots\!49}a^{15}+\frac{16\!\cdots\!19}{20\!\cdots\!49}a^{14}+\frac{60\!\cdots\!18}{20\!\cdots\!49}a^{13}+\frac{17\!\cdots\!96}{20\!\cdots\!49}a^{12}+\frac{40\!\cdots\!53}{20\!\cdots\!49}a^{11}+\frac{11\!\cdots\!97}{20\!\cdots\!49}a^{10}+\frac{19\!\cdots\!53}{20\!\cdots\!49}a^{9}+\frac{51\!\cdots\!08}{20\!\cdots\!49}a^{8}+\frac{58\!\cdots\!74}{20\!\cdots\!49}a^{7}+\frac{10\!\cdots\!49}{20\!\cdots\!49}a^{6}+\frac{89\!\cdots\!55}{20\!\cdots\!49}a^{5}+\frac{82\!\cdots\!02}{20\!\cdots\!49}a^{4}+\frac{26\!\cdots\!75}{60\!\cdots\!79}a^{3}+\frac{44\!\cdots\!94}{20\!\cdots\!49}a^{2}+\frac{24\!\cdots\!03}{20\!\cdots\!49}a-\frac{21\!\cdots\!38}{20\!\cdots\!49}$, $\frac{16\!\cdots\!74}{48\!\cdots\!71}a^{39}-\frac{17\!\cdots\!53}{48\!\cdots\!71}a^{38}+\frac{34\!\cdots\!89}{48\!\cdots\!71}a^{37}-\frac{31\!\cdots\!86}{48\!\cdots\!71}a^{36}+\frac{41\!\cdots\!27}{48\!\cdots\!71}a^{35}-\frac{30\!\cdots\!26}{44\!\cdots\!19}a^{34}+\frac{33\!\cdots\!50}{48\!\cdots\!71}a^{33}-\frac{23\!\cdots\!49}{48\!\cdots\!71}a^{32}+\frac{20\!\cdots\!91}{48\!\cdots\!71}a^{31}-\frac{12\!\cdots\!41}{48\!\cdots\!71}a^{30}+\frac{95\!\cdots\!25}{48\!\cdots\!71}a^{29}-\frac{52\!\cdots\!29}{48\!\cdots\!71}a^{28}+\frac{36\!\cdots\!35}{48\!\cdots\!71}a^{27}-\frac{17\!\cdots\!53}{48\!\cdots\!71}a^{26}+\frac{11\!\cdots\!64}{48\!\cdots\!71}a^{25}-\frac{43\!\cdots\!84}{48\!\cdots\!71}a^{24}+\frac{28\!\cdots\!22}{48\!\cdots\!71}a^{23}-\frac{86\!\cdots\!08}{48\!\cdots\!71}a^{22}+\frac{57\!\cdots\!79}{48\!\cdots\!71}a^{21}-\frac{12\!\cdots\!37}{48\!\cdots\!71}a^{20}+\frac{96\!\cdots\!37}{48\!\cdots\!71}a^{19}-\frac{13\!\cdots\!14}{48\!\cdots\!71}a^{18}+\frac{13\!\cdots\!39}{48\!\cdots\!71}a^{17}-\frac{85\!\cdots\!13}{48\!\cdots\!71}a^{16}+\frac{14\!\cdots\!03}{48\!\cdots\!71}a^{15}-\frac{22\!\cdots\!41}{48\!\cdots\!71}a^{14}+\frac{11\!\cdots\!16}{48\!\cdots\!71}a^{13}+\frac{59\!\cdots\!83}{48\!\cdots\!71}a^{12}+\frac{76\!\cdots\!99}{48\!\cdots\!71}a^{11}+\frac{51\!\cdots\!75}{48\!\cdots\!71}a^{10}+\frac{35\!\cdots\!06}{48\!\cdots\!71}a^{9}+\frac{19\!\cdots\!30}{48\!\cdots\!71}a^{8}+\frac{10\!\cdots\!51}{48\!\cdots\!71}a^{7}-\frac{49\!\cdots\!21}{48\!\cdots\!71}a^{6}+\frac{16\!\cdots\!51}{48\!\cdots\!71}a^{5}-\frac{21\!\cdots\!09}{48\!\cdots\!71}a^{4}+\frac{56\!\cdots\!89}{14\!\cdots\!41}a^{3}-\frac{28\!\cdots\!30}{48\!\cdots\!71}a^{2}+\frac{28\!\cdots\!67}{48\!\cdots\!71}a-\frac{12\!\cdots\!42}{48\!\cdots\!71}$, $\frac{10\!\cdots\!50}{10\!\cdots\!11}a^{39}-\frac{86\!\cdots\!04}{10\!\cdots\!11}a^{38}+\frac{20\!\cdots\!80}{10\!\cdots\!11}a^{37}-\frac{15\!\cdots\!65}{10\!\cdots\!11}a^{36}+\frac{24\!\cdots\!12}{10\!\cdots\!11}a^{35}-\frac{15\!\cdots\!42}{10\!\cdots\!11}a^{34}+\frac{19\!\cdots\!21}{10\!\cdots\!11}a^{33}-\frac{10\!\cdots\!15}{10\!\cdots\!11}a^{32}+\frac{11\!\cdots\!66}{10\!\cdots\!11}a^{31}-\frac{56\!\cdots\!31}{10\!\cdots\!11}a^{30}+\frac{56\!\cdots\!94}{10\!\cdots\!11}a^{29}-\frac{21\!\cdots\!14}{10\!\cdots\!11}a^{28}+\frac{21\!\cdots\!64}{10\!\cdots\!11}a^{27}-\frac{64\!\cdots\!12}{10\!\cdots\!11}a^{26}+\frac{65\!\cdots\!27}{10\!\cdots\!11}a^{25}-\frac{14\!\cdots\!69}{10\!\cdots\!11}a^{24}+\frac{16\!\cdots\!08}{10\!\cdots\!11}a^{23}-\frac{22\!\cdots\!28}{10\!\cdots\!11}a^{22}+\frac{34\!\cdots\!60}{10\!\cdots\!11}a^{21}-\frac{16\!\cdots\!58}{10\!\cdots\!11}a^{20}+\frac{57\!\cdots\!77}{10\!\cdots\!11}a^{19}+\frac{18\!\cdots\!65}{10\!\cdots\!11}a^{18}+\frac{78\!\cdots\!33}{10\!\cdots\!11}a^{17}+\frac{87\!\cdots\!14}{10\!\cdots\!11}a^{16}+\frac{85\!\cdots\!49}{10\!\cdots\!11}a^{15}+\frac{14\!\cdots\!73}{10\!\cdots\!11}a^{14}+\frac{73\!\cdots\!53}{10\!\cdots\!11}a^{13}+\frac{16\!\cdots\!55}{10\!\cdots\!11}a^{12}+\frac{47\!\cdots\!85}{10\!\cdots\!11}a^{11}+\frac{11\!\cdots\!23}{10\!\cdots\!11}a^{10}+\frac{22\!\cdots\!92}{10\!\cdots\!11}a^{9}+\frac{51\!\cdots\!55}{10\!\cdots\!11}a^{8}+\frac{68\!\cdots\!66}{10\!\cdots\!11}a^{7}+\frac{91\!\cdots\!22}{10\!\cdots\!11}a^{6}+\frac{10\!\cdots\!00}{10\!\cdots\!11}a^{5}+\frac{73\!\cdots\!24}{10\!\cdots\!11}a^{4}+\frac{33\!\cdots\!45}{31\!\cdots\!81}a^{3}+\frac{47\!\cdots\!05}{10\!\cdots\!11}a^{2}+\frac{26\!\cdots\!68}{10\!\cdots\!11}a+\frac{25\!\cdots\!94}{10\!\cdots\!11}$, $\frac{48\!\cdots\!88}{48\!\cdots\!71}a^{39}-\frac{65\!\cdots\!34}{48\!\cdots\!71}a^{38}+\frac{10\!\cdots\!59}{48\!\cdots\!71}a^{37}-\frac{12\!\cdots\!31}{48\!\cdots\!71}a^{36}+\frac{12\!\cdots\!87}{48\!\cdots\!71}a^{35}-\frac{13\!\cdots\!59}{48\!\cdots\!71}a^{34}+\frac{10\!\cdots\!39}{48\!\cdots\!71}a^{33}-\frac{99\!\cdots\!71}{48\!\cdots\!71}a^{32}+\frac{61\!\cdots\!49}{48\!\cdots\!71}a^{31}-\frac{55\!\cdots\!28}{48\!\cdots\!71}a^{30}+\frac{29\!\cdots\!44}{48\!\cdots\!71}a^{29}-\frac{23\!\cdots\!62}{48\!\cdots\!71}a^{28}+\frac{11\!\cdots\!73}{48\!\cdots\!71}a^{27}-\frac{81\!\cdots\!28}{48\!\cdots\!71}a^{26}+\frac{34\!\cdots\!61}{48\!\cdots\!71}a^{25}-\frac{22\!\cdots\!63}{48\!\cdots\!71}a^{24}+\frac{86\!\cdots\!48}{48\!\cdots\!71}a^{23}-\frac{49\!\cdots\!25}{48\!\cdots\!71}a^{22}+\frac{17\!\cdots\!69}{48\!\cdots\!71}a^{21}-\frac{85\!\cdots\!21}{48\!\cdots\!71}a^{20}+\frac{29\!\cdots\!23}{48\!\cdots\!71}a^{19}-\frac{11\!\cdots\!69}{48\!\cdots\!71}a^{18}+\frac{40\!\cdots\!58}{48\!\cdots\!71}a^{17}-\frac{12\!\cdots\!87}{48\!\cdots\!71}a^{16}+\frac{42\!\cdots\!60}{48\!\cdots\!71}a^{15}-\frac{10\!\cdots\!19}{48\!\cdots\!71}a^{14}+\frac{36\!\cdots\!64}{48\!\cdots\!71}a^{13}-\frac{69\!\cdots\!05}{48\!\cdots\!71}a^{12}+\frac{22\!\cdots\!88}{48\!\cdots\!71}a^{11}-\frac{36\!\cdots\!70}{48\!\cdots\!71}a^{10}+\frac{10\!\cdots\!96}{48\!\cdots\!71}a^{9}-\frac{14\!\cdots\!31}{48\!\cdots\!71}a^{8}+\frac{33\!\cdots\!26}{48\!\cdots\!71}a^{7}-\frac{59\!\cdots\!41}{48\!\cdots\!71}a^{6}+\frac{68\!\cdots\!80}{48\!\cdots\!71}a^{5}-\frac{89\!\cdots\!83}{48\!\cdots\!71}a^{4}+\frac{28\!\cdots\!16}{14\!\cdots\!41}a^{3}-\frac{15\!\cdots\!69}{48\!\cdots\!71}a^{2}+\frac{35\!\cdots\!75}{48\!\cdots\!71}a-\frac{66\!\cdots\!45}{48\!\cdots\!71}$, $\frac{40\!\cdots\!28}{48\!\cdots\!71}a^{39}-\frac{40\!\cdots\!38}{48\!\cdots\!71}a^{38}+\frac{84\!\cdots\!43}{48\!\cdots\!71}a^{37}-\frac{73\!\cdots\!53}{48\!\cdots\!71}a^{36}+\frac{99\!\cdots\!54}{48\!\cdots\!71}a^{35}-\frac{71\!\cdots\!84}{44\!\cdots\!19}a^{34}+\frac{80\!\cdots\!69}{48\!\cdots\!71}a^{33}-\frac{55\!\cdots\!78}{48\!\cdots\!71}a^{32}+\frac{48\!\cdots\!29}{48\!\cdots\!71}a^{31}-\frac{29\!\cdots\!42}{48\!\cdots\!71}a^{30}+\frac{22\!\cdots\!87}{48\!\cdots\!71}a^{29}-\frac{11\!\cdots\!66}{48\!\cdots\!71}a^{28}+\frac{86\!\cdots\!53}{48\!\cdots\!71}a^{27}-\frac{38\!\cdots\!33}{48\!\cdots\!71}a^{26}+\frac{26\!\cdots\!96}{48\!\cdots\!71}a^{25}-\frac{94\!\cdots\!45}{48\!\cdots\!71}a^{24}+\frac{66\!\cdots\!15}{48\!\cdots\!71}a^{23}-\frac{18\!\cdots\!76}{48\!\cdots\!71}a^{22}+\frac{13\!\cdots\!74}{48\!\cdots\!71}a^{21}-\frac{25\!\cdots\!00}{48\!\cdots\!71}a^{20}+\frac{22\!\cdots\!97}{48\!\cdots\!71}a^{19}-\frac{23\!\cdots\!53}{48\!\cdots\!71}a^{18}+\frac{30\!\cdots\!04}{48\!\cdots\!71}a^{17}-\frac{58\!\cdots\!91}{48\!\cdots\!71}a^{16}+\frac{32\!\cdots\!85}{48\!\cdots\!71}a^{15}+\frac{16\!\cdots\!40}{48\!\cdots\!71}a^{14}+\frac{27\!\cdots\!52}{48\!\cdots\!71}a^{13}+\frac{29\!\cdots\!38}{48\!\cdots\!71}a^{12}+\frac{17\!\cdots\!85}{48\!\cdots\!71}a^{11}+\frac{22\!\cdots\!22}{48\!\cdots\!71}a^{10}+\frac{75\!\cdots\!34}{48\!\cdots\!71}a^{9}+\frac{97\!\cdots\!55}{48\!\cdots\!71}a^{8}+\frac{20\!\cdots\!43}{48\!\cdots\!71}a^{7}+\frac{44\!\cdots\!91}{48\!\cdots\!71}a^{6}+\frac{24\!\cdots\!14}{48\!\cdots\!71}a^{5}-\frac{18\!\cdots\!72}{48\!\cdots\!71}a^{4}+\frac{54\!\cdots\!22}{14\!\cdots\!41}a^{3}-\frac{22\!\cdots\!50}{48\!\cdots\!71}a^{2}-\frac{26\!\cdots\!59}{48\!\cdots\!71}a-\frac{10\!\cdots\!91}{48\!\cdots\!71}$, $\frac{84\!\cdots\!30}{48\!\cdots\!71}a^{39}+\frac{52\!\cdots\!39}{48\!\cdots\!71}a^{38}+\frac{15\!\cdots\!70}{48\!\cdots\!71}a^{37}+\frac{14\!\cdots\!55}{48\!\cdots\!71}a^{36}+\frac{17\!\cdots\!12}{48\!\cdots\!71}a^{35}+\frac{19\!\cdots\!84}{48\!\cdots\!71}a^{34}+\frac{12\!\cdots\!39}{48\!\cdots\!71}a^{33}+\frac{17\!\cdots\!52}{48\!\cdots\!71}a^{32}+\frac{69\!\cdots\!96}{48\!\cdots\!71}a^{31}+\frac{11\!\cdots\!34}{48\!\cdots\!71}a^{30}+\frac{29\!\cdots\!99}{48\!\cdots\!71}a^{29}+\frac{59\!\cdots\!89}{48\!\cdots\!71}a^{28}+\frac{10\!\cdots\!57}{48\!\cdots\!71}a^{27}+\frac{24\!\cdots\!81}{48\!\cdots\!71}a^{26}+\frac{27\!\cdots\!71}{48\!\cdots\!71}a^{25}+\frac{79\!\cdots\!50}{48\!\cdots\!71}a^{24}+\frac{60\!\cdots\!79}{48\!\cdots\!71}a^{23}+\frac{21\!\cdots\!42}{48\!\cdots\!71}a^{22}+\frac{10\!\cdots\!61}{48\!\cdots\!71}a^{21}+\frac{45\!\cdots\!07}{48\!\cdots\!71}a^{20}+\frac{14\!\cdots\!37}{48\!\cdots\!71}a^{19}+\frac{79\!\cdots\!50}{48\!\cdots\!71}a^{18}+\frac{14\!\cdots\!79}{48\!\cdots\!71}a^{17}+\frac{11\!\cdots\!01}{48\!\cdots\!71}a^{16}+\frac{99\!\cdots\!81}{48\!\cdots\!71}a^{15}+\frac{12\!\cdots\!36}{48\!\cdots\!71}a^{14}+\frac{96\!\cdots\!44}{48\!\cdots\!71}a^{13}+\frac{10\!\cdots\!31}{48\!\cdots\!71}a^{12}-\frac{70\!\cdots\!00}{48\!\cdots\!71}a^{11}+\frac{63\!\cdots\!24}{48\!\cdots\!71}a^{10}-\frac{10\!\cdots\!14}{48\!\cdots\!71}a^{9}+\frac{27\!\cdots\!23}{48\!\cdots\!71}a^{8}-\frac{85\!\cdots\!27}{48\!\cdots\!71}a^{7}+\frac{68\!\cdots\!03}{48\!\cdots\!71}a^{6}-\frac{42\!\cdots\!75}{48\!\cdots\!71}a^{5}+\frac{10\!\cdots\!12}{48\!\cdots\!71}a^{4}-\frac{24\!\cdots\!62}{14\!\cdots\!41}a^{3}+\frac{14\!\cdots\!21}{48\!\cdots\!71}a^{2}-\frac{79\!\cdots\!40}{48\!\cdots\!71}a+\frac{61\!\cdots\!63}{48\!\cdots\!71}$, $\frac{35\!\cdots\!81}{48\!\cdots\!71}a^{39}-\frac{36\!\cdots\!81}{48\!\cdots\!71}a^{38}+\frac{75\!\cdots\!48}{48\!\cdots\!71}a^{37}-\frac{66\!\cdots\!29}{48\!\cdots\!71}a^{36}+\frac{89\!\cdots\!21}{48\!\cdots\!71}a^{35}-\frac{70\!\cdots\!24}{48\!\cdots\!71}a^{34}+\frac{72\!\cdots\!18}{48\!\cdots\!71}a^{33}-\frac{50\!\cdots\!86}{48\!\cdots\!71}a^{32}+\frac{43\!\cdots\!78}{48\!\cdots\!71}a^{31}-\frac{27\!\cdots\!86}{48\!\cdots\!71}a^{30}+\frac{20\!\cdots\!60}{48\!\cdots\!71}a^{29}-\frac{11\!\cdots\!23}{48\!\cdots\!71}a^{28}+\frac{78\!\cdots\!30}{48\!\cdots\!71}a^{27}-\frac{35\!\cdots\!30}{48\!\cdots\!71}a^{26}+\frac{24\!\cdots\!52}{48\!\cdots\!71}a^{25}-\frac{90\!\cdots\!62}{48\!\cdots\!71}a^{24}+\frac{60\!\cdots\!54}{48\!\cdots\!71}a^{23}-\frac{17\!\cdots\!97}{48\!\cdots\!71}a^{22}+\frac{12\!\cdots\!20}{48\!\cdots\!71}a^{21}-\frac{26\!\cdots\!07}{48\!\cdots\!71}a^{20}+\frac{20\!\cdots\!07}{48\!\cdots\!71}a^{19}-\frac{27\!\cdots\!33}{48\!\cdots\!71}a^{18}+\frac{28\!\cdots\!29}{48\!\cdots\!71}a^{17}-\frac{14\!\cdots\!12}{48\!\cdots\!71}a^{16}+\frac{30\!\cdots\!91}{48\!\cdots\!71}a^{15}+\frac{44\!\cdots\!66}{48\!\cdots\!71}a^{14}+\frac{26\!\cdots\!63}{48\!\cdots\!71}a^{13}+\frac{17\!\cdots\!93}{48\!\cdots\!71}a^{12}+\frac{16\!\cdots\!07}{48\!\cdots\!71}a^{11}+\frac{14\!\cdots\!78}{48\!\cdots\!71}a^{10}+\frac{78\!\cdots\!64}{48\!\cdots\!71}a^{9}+\frac{62\!\cdots\!32}{48\!\cdots\!71}a^{8}+\frac{23\!\cdots\!20}{48\!\cdots\!71}a^{7}-\frac{27\!\cdots\!61}{48\!\cdots\!71}a^{6}+\frac{38\!\cdots\!91}{48\!\cdots\!71}a^{5}-\frac{29\!\cdots\!11}{48\!\cdots\!71}a^{4}+\frac{13\!\cdots\!14}{14\!\cdots\!41}a^{3}-\frac{46\!\cdots\!32}{48\!\cdots\!71}a^{2}+\frac{86\!\cdots\!09}{48\!\cdots\!71}a+\frac{40\!\cdots\!55}{48\!\cdots\!71}$, $\frac{50\!\cdots\!62}{48\!\cdots\!71}a^{39}-\frac{28\!\cdots\!87}{48\!\cdots\!71}a^{38}+\frac{10\!\cdots\!57}{48\!\cdots\!71}a^{37}-\frac{43\!\cdots\!44}{48\!\cdots\!71}a^{36}+\frac{11\!\cdots\!25}{48\!\cdots\!71}a^{35}-\frac{39\!\cdots\!78}{48\!\cdots\!71}a^{34}+\frac{94\!\cdots\!02}{48\!\cdots\!71}a^{33}-\frac{22\!\cdots\!25}{48\!\cdots\!71}a^{32}+\frac{51\!\cdots\!91}{44\!\cdots\!19}a^{31}-\frac{88\!\cdots\!98}{48\!\cdots\!71}a^{30}+\frac{26\!\cdots\!01}{48\!\cdots\!71}a^{29}-\frac{17\!\cdots\!97}{48\!\cdots\!71}a^{28}+\frac{97\!\cdots\!16}{48\!\cdots\!71}a^{27}+\frac{17\!\cdots\!51}{48\!\cdots\!71}a^{26}+\frac{29\!\cdots\!74}{48\!\cdots\!71}a^{25}+\frac{32\!\cdots\!37}{48\!\cdots\!71}a^{24}+\frac{73\!\cdots\!89}{48\!\cdots\!71}a^{23}+\frac{14\!\cdots\!38}{48\!\cdots\!71}a^{22}+\frac{14\!\cdots\!58}{48\!\cdots\!71}a^{21}+\frac{43\!\cdots\!06}{48\!\cdots\!71}a^{20}+\frac{24\!\cdots\!38}{48\!\cdots\!71}a^{19}+\frac{92\!\cdots\!71}{48\!\cdots\!71}a^{18}+\frac{32\!\cdots\!43}{48\!\cdots\!71}a^{17}+\frac{14\!\cdots\!35}{48\!\cdots\!71}a^{16}+\frac{34\!\cdots\!01}{48\!\cdots\!71}a^{15}+\frac{18\!\cdots\!75}{48\!\cdots\!71}a^{14}+\frac{28\!\cdots\!30}{48\!\cdots\!71}a^{13}+\frac{16\!\cdots\!96}{48\!\cdots\!71}a^{12}+\frac{17\!\cdots\!54}{48\!\cdots\!71}a^{11}+\frac{99\!\cdots\!57}{48\!\cdots\!71}a^{10}+\frac{74\!\cdots\!88}{48\!\cdots\!71}a^{9}+\frac{41\!\cdots\!14}{48\!\cdots\!71}a^{8}+\frac{16\!\cdots\!69}{48\!\cdots\!71}a^{7}+\frac{74\!\cdots\!10}{48\!\cdots\!71}a^{6}-\frac{85\!\cdots\!23}{48\!\cdots\!71}a^{5}+\frac{95\!\cdots\!01}{48\!\cdots\!71}a^{4}-\frac{13\!\cdots\!16}{14\!\cdots\!41}a^{3}+\frac{91\!\cdots\!58}{48\!\cdots\!71}a^{2}-\frac{32\!\cdots\!96}{48\!\cdots\!71}a+\frac{38\!\cdots\!08}{48\!\cdots\!71}$, $\frac{18\!\cdots\!83}{48\!\cdots\!71}a^{39}-\frac{70\!\cdots\!71}{48\!\cdots\!71}a^{38}+\frac{37\!\cdots\!59}{48\!\cdots\!71}a^{37}-\frac{83\!\cdots\!82}{48\!\cdots\!71}a^{36}+\frac{43\!\cdots\!29}{48\!\cdots\!71}a^{35}-\frac{52\!\cdots\!97}{48\!\cdots\!71}a^{34}+\frac{33\!\cdots\!65}{48\!\cdots\!71}a^{33}-\frac{47\!\cdots\!32}{48\!\cdots\!71}a^{32}+\frac{19\!\cdots\!07}{48\!\cdots\!71}a^{31}+\frac{17\!\cdots\!82}{48\!\cdots\!71}a^{30}+\frac{91\!\cdots\!85}{48\!\cdots\!71}a^{29}+\frac{17\!\cdots\!98}{48\!\cdots\!71}a^{28}+\frac{33\!\cdots\!83}{48\!\cdots\!71}a^{27}+\frac{10\!\cdots\!73}{48\!\cdots\!71}a^{26}+\frac{99\!\cdots\!59}{48\!\cdots\!71}a^{25}+\frac{42\!\cdots\!45}{48\!\cdots\!71}a^{24}+\frac{24\!\cdots\!69}{48\!\cdots\!71}a^{23}+\frac{13\!\cdots\!17}{48\!\cdots\!71}a^{22}+\frac{48\!\cdots\!74}{48\!\cdots\!71}a^{21}+\frac{32\!\cdots\!65}{48\!\cdots\!71}a^{20}+\frac{77\!\cdots\!42}{48\!\cdots\!71}a^{19}+\frac{62\!\cdots\!44}{48\!\cdots\!71}a^{18}+\frac{10\!\cdots\!89}{48\!\cdots\!71}a^{17}+\frac{94\!\cdots\!55}{48\!\cdots\!71}a^{16}+\frac{10\!\cdots\!10}{48\!\cdots\!71}a^{15}+\frac{10\!\cdots\!72}{48\!\cdots\!71}a^{14}+\frac{79\!\cdots\!45}{48\!\cdots\!71}a^{13}+\frac{96\!\cdots\!83}{48\!\cdots\!71}a^{12}+\frac{44\!\cdots\!31}{48\!\cdots\!71}a^{11}+\frac{60\!\cdots\!56}{48\!\cdots\!71}a^{10}+\frac{13\!\cdots\!07}{48\!\cdots\!71}a^{9}+\frac{25\!\cdots\!79}{48\!\cdots\!71}a^{8}-\frac{76\!\cdots\!39}{48\!\cdots\!71}a^{7}+\frac{57\!\cdots\!88}{48\!\cdots\!71}a^{6}-\frac{27\!\cdots\!21}{48\!\cdots\!71}a^{5}+\frac{82\!\cdots\!44}{48\!\cdots\!71}a^{4}-\frac{17\!\cdots\!00}{14\!\cdots\!41}a^{3}+\frac{10\!\cdots\!20}{48\!\cdots\!71}a^{2}-\frac{72\!\cdots\!23}{48\!\cdots\!71}a+\frac{46\!\cdots\!73}{48\!\cdots\!71}$, $\frac{19\!\cdots\!38}{48\!\cdots\!71}a^{39}-\frac{16\!\cdots\!58}{48\!\cdots\!71}a^{38}+\frac{39\!\cdots\!57}{48\!\cdots\!71}a^{37}-\frac{28\!\cdots\!59}{48\!\cdots\!71}a^{36}+\frac{46\!\cdots\!13}{48\!\cdots\!71}a^{35}-\frac{29\!\cdots\!25}{48\!\cdots\!71}a^{34}+\frac{37\!\cdots\!90}{48\!\cdots\!71}a^{33}-\frac{20\!\cdots\!69}{48\!\cdots\!71}a^{32}+\frac{22\!\cdots\!21}{48\!\cdots\!71}a^{31}-\frac{10\!\cdots\!75}{48\!\cdots\!71}a^{30}+\frac{10\!\cdots\!12}{48\!\cdots\!71}a^{29}-\frac{40\!\cdots\!68}{48\!\cdots\!71}a^{28}+\frac{40\!\cdots\!59}{48\!\cdots\!71}a^{27}-\frac{11\!\cdots\!64}{48\!\cdots\!71}a^{26}+\frac{12\!\cdots\!62}{48\!\cdots\!71}a^{25}-\frac{25\!\cdots\!54}{48\!\cdots\!71}a^{24}+\frac{31\!\cdots\!77}{48\!\cdots\!71}a^{23}-\frac{38\!\cdots\!64}{48\!\cdots\!71}a^{22}+\frac{64\!\cdots\!99}{48\!\cdots\!71}a^{21}-\frac{24\!\cdots\!71}{48\!\cdots\!71}a^{20}+\frac{10\!\cdots\!31}{48\!\cdots\!71}a^{19}+\frac{47\!\cdots\!87}{48\!\cdots\!71}a^{18}+\frac{14\!\cdots\!97}{48\!\cdots\!71}a^{17}+\frac{18\!\cdots\!20}{48\!\cdots\!71}a^{16}+\frac{16\!\cdots\!21}{48\!\cdots\!71}a^{15}+\frac{29\!\cdots\!51}{48\!\cdots\!71}a^{14}+\frac{13\!\cdots\!75}{48\!\cdots\!71}a^{13}+\frac{32\!\cdots\!52}{48\!\cdots\!71}a^{12}+\frac{90\!\cdots\!26}{48\!\cdots\!71}a^{11}+\frac{22\!\cdots\!53}{48\!\cdots\!71}a^{10}+\frac{42\!\cdots\!12}{48\!\cdots\!71}a^{9}+\frac{98\!\cdots\!65}{48\!\cdots\!71}a^{8}+\frac{12\!\cdots\!36}{48\!\cdots\!71}a^{7}+\frac{17\!\cdots\!28}{48\!\cdots\!71}a^{6}+\frac{19\!\cdots\!15}{48\!\cdots\!71}a^{5}+\frac{14\!\cdots\!51}{48\!\cdots\!71}a^{4}+\frac{68\!\cdots\!45}{14\!\cdots\!41}a^{3}+\frac{89\!\cdots\!43}{48\!\cdots\!71}a^{2}+\frac{50\!\cdots\!14}{48\!\cdots\!71}a+\frac{26\!\cdots\!77}{48\!\cdots\!71}$, $\frac{26\!\cdots\!33}{48\!\cdots\!71}a^{39}-\frac{22\!\cdots\!23}{48\!\cdots\!71}a^{38}+\frac{55\!\cdots\!85}{48\!\cdots\!71}a^{37}-\frac{38\!\cdots\!83}{48\!\cdots\!71}a^{36}+\frac{65\!\cdots\!76}{48\!\cdots\!71}a^{35}-\frac{39\!\cdots\!45}{48\!\cdots\!71}a^{34}+\frac{52\!\cdots\!93}{48\!\cdots\!71}a^{33}-\frac{27\!\cdots\!66}{48\!\cdots\!71}a^{32}+\frac{31\!\cdots\!24}{48\!\cdots\!71}a^{31}-\frac{13\!\cdots\!53}{48\!\cdots\!71}a^{30}+\frac{14\!\cdots\!12}{48\!\cdots\!71}a^{29}-\frac{53\!\cdots\!09}{48\!\cdots\!71}a^{28}+\frac{56\!\cdots\!84}{48\!\cdots\!71}a^{27}-\frac{15\!\cdots\!30}{48\!\cdots\!71}a^{26}+\frac{17\!\cdots\!34}{48\!\cdots\!71}a^{25}-\frac{33\!\cdots\!05}{48\!\cdots\!71}a^{24}+\frac{43\!\cdots\!06}{48\!\cdots\!71}a^{23}-\frac{46\!\cdots\!27}{48\!\cdots\!71}a^{22}+\frac{90\!\cdots\!15}{48\!\cdots\!71}a^{21}-\frac{18\!\cdots\!75}{48\!\cdots\!71}a^{20}+\frac{15\!\cdots\!23}{48\!\cdots\!71}a^{19}+\frac{94\!\cdots\!93}{48\!\cdots\!71}a^{18}+\frac{20\!\cdots\!64}{48\!\cdots\!71}a^{17}+\frac{29\!\cdots\!84}{48\!\cdots\!71}a^{16}+\frac{22\!\cdots\!90}{48\!\cdots\!71}a^{15}+\frac{46\!\cdots\!78}{48\!\cdots\!71}a^{14}+\frac{19\!\cdots\!21}{48\!\cdots\!71}a^{13}+\frac{49\!\cdots\!13}{48\!\cdots\!71}a^{12}+\frac{12\!\cdots\!42}{48\!\cdots\!71}a^{11}+\frac{33\!\cdots\!66}{48\!\cdots\!71}a^{10}+\frac{60\!\cdots\!19}{48\!\cdots\!71}a^{9}+\frac{15\!\cdots\!67}{48\!\cdots\!71}a^{8}+\frac{18\!\cdots\!52}{48\!\cdots\!71}a^{7}+\frac{28\!\cdots\!24}{48\!\cdots\!71}a^{6}+\frac{28\!\cdots\!34}{48\!\cdots\!71}a^{5}+\frac{23\!\cdots\!29}{48\!\cdots\!71}a^{4}+\frac{87\!\cdots\!04}{14\!\cdots\!41}a^{3}+\frac{13\!\cdots\!98}{48\!\cdots\!71}a^{2}+\frac{75\!\cdots\!12}{48\!\cdots\!71}a+\frac{11\!\cdots\!89}{48\!\cdots\!71}$, $\frac{53\!\cdots\!95}{48\!\cdots\!71}a^{39}-\frac{54\!\cdots\!97}{48\!\cdots\!71}a^{38}+\frac{11\!\cdots\!99}{48\!\cdots\!71}a^{37}-\frac{97\!\cdots\!40}{48\!\cdots\!71}a^{36}+\frac{13\!\cdots\!87}{48\!\cdots\!71}a^{35}-\frac{10\!\cdots\!50}{48\!\cdots\!71}a^{34}+\frac{10\!\cdots\!41}{48\!\cdots\!71}a^{33}-\frac{73\!\cdots\!78}{48\!\cdots\!71}a^{32}+\frac{65\!\cdots\!93}{48\!\cdots\!71}a^{31}-\frac{39\!\cdots\!08}{48\!\cdots\!71}a^{30}+\frac{30\!\cdots\!85}{48\!\cdots\!71}a^{29}-\frac{14\!\cdots\!13}{44\!\cdots\!19}a^{28}+\frac{11\!\cdots\!81}{48\!\cdots\!71}a^{27}-\frac{51\!\cdots\!97}{48\!\cdots\!71}a^{26}+\frac{35\!\cdots\!31}{48\!\cdots\!71}a^{25}-\frac{12\!\cdots\!59}{48\!\cdots\!71}a^{24}+\frac{90\!\cdots\!07}{48\!\cdots\!71}a^{23}-\frac{24\!\cdots\!65}{48\!\cdots\!71}a^{22}+\frac{18\!\cdots\!02}{48\!\cdots\!71}a^{21}-\frac{35\!\cdots\!29}{48\!\cdots\!71}a^{20}+\frac{31\!\cdots\!00}{48\!\cdots\!71}a^{19}-\frac{33\!\cdots\!10}{48\!\cdots\!71}a^{18}+\frac{42\!\cdots\!57}{48\!\cdots\!71}a^{17}-\frac{13\!\cdots\!09}{48\!\cdots\!71}a^{16}+\frac{45\!\cdots\!42}{48\!\cdots\!71}a^{15}+\frac{14\!\cdots\!68}{48\!\cdots\!71}a^{14}+\frac{38\!\cdots\!55}{48\!\cdots\!71}a^{13}+\frac{31\!\cdots\!59}{48\!\cdots\!71}a^{12}+\frac{24\!\cdots\!41}{48\!\cdots\!71}a^{11}+\frac{23\!\cdots\!99}{48\!\cdots\!71}a^{10}+\frac{11\!\cdots\!95}{48\!\cdots\!71}a^{9}+\frac{94\!\cdots\!63}{48\!\cdots\!71}a^{8}+\frac{32\!\cdots\!43}{48\!\cdots\!71}a^{7}-\frac{87\!\cdots\!60}{48\!\cdots\!71}a^{6}+\frac{48\!\cdots\!69}{48\!\cdots\!71}a^{5}-\frac{60\!\cdots\!63}{48\!\cdots\!71}a^{4}+\frac{15\!\cdots\!38}{14\!\cdots\!41}a^{3}-\frac{78\!\cdots\!80}{48\!\cdots\!71}a^{2}+\frac{12\!\cdots\!04}{48\!\cdots\!71}a-\frac{35\!\cdots\!75}{48\!\cdots\!71}$, $\frac{43\!\cdots\!60}{48\!\cdots\!71}a^{39}-\frac{48\!\cdots\!95}{48\!\cdots\!71}a^{38}+\frac{92\!\cdots\!20}{48\!\cdots\!71}a^{37}-\frac{88\!\cdots\!11}{48\!\cdots\!71}a^{36}+\frac{10\!\cdots\!12}{48\!\cdots\!71}a^{35}-\frac{94\!\cdots\!53}{48\!\cdots\!71}a^{34}+\frac{88\!\cdots\!76}{48\!\cdots\!71}a^{33}-\frac{68\!\cdots\!02}{48\!\cdots\!71}a^{32}+\frac{53\!\cdots\!77}{48\!\cdots\!71}a^{31}-\frac{37\!\cdots\!14}{48\!\cdots\!71}a^{30}+\frac{23\!\cdots\!47}{44\!\cdots\!19}a^{29}-\frac{15\!\cdots\!87}{48\!\cdots\!71}a^{28}+\frac{97\!\cdots\!52}{48\!\cdots\!71}a^{27}-\frac{51\!\cdots\!32}{48\!\cdots\!71}a^{26}+\frac{29\!\cdots\!06}{48\!\cdots\!71}a^{25}-\frac{13\!\cdots\!31}{48\!\cdots\!71}a^{24}+\frac{75\!\cdots\!59}{48\!\cdots\!71}a^{23}-\frac{27\!\cdots\!70}{48\!\cdots\!71}a^{22}+\frac{15\!\cdots\!44}{48\!\cdots\!71}a^{21}-\frac{43\!\cdots\!19}{48\!\cdots\!71}a^{20}+\frac{26\!\cdots\!38}{48\!\cdots\!71}a^{19}-\frac{52\!\cdots\!20}{48\!\cdots\!71}a^{18}+\frac{35\!\cdots\!38}{48\!\cdots\!71}a^{17}-\frac{43\!\cdots\!17}{48\!\cdots\!71}a^{16}+\frac{38\!\cdots\!09}{48\!\cdots\!71}a^{15}-\frac{22\!\cdots\!09}{48\!\cdots\!71}a^{14}+\frac{32\!\cdots\!05}{48\!\cdots\!71}a^{13}-\frac{24\!\cdots\!40}{48\!\cdots\!71}a^{12}+\frac{20\!\cdots\!37}{48\!\cdots\!71}a^{11}+\frac{24\!\cdots\!31}{48\!\cdots\!71}a^{10}+\frac{96\!\cdots\!48}{48\!\cdots\!71}a^{9}+\frac{49\!\cdots\!78}{48\!\cdots\!71}a^{8}+\frac{29\!\cdots\!04}{48\!\cdots\!71}a^{7}-\frac{24\!\cdots\!33}{48\!\cdots\!71}a^{6}+\frac{50\!\cdots\!29}{48\!\cdots\!71}a^{5}-\frac{72\!\cdots\!64}{48\!\cdots\!71}a^{4}+\frac{18\!\cdots\!44}{14\!\cdots\!41}a^{3}-\frac{99\!\cdots\!39}{48\!\cdots\!71}a^{2}+\frac{20\!\cdots\!92}{48\!\cdots\!71}a-\frac{43\!\cdots\!16}{48\!\cdots\!71}$, $\frac{19\!\cdots\!86}{48\!\cdots\!71}a^{39}-\frac{10\!\cdots\!09}{44\!\cdots\!19}a^{38}+\frac{40\!\cdots\!79}{48\!\cdots\!71}a^{37}-\frac{17\!\cdots\!81}{48\!\cdots\!71}a^{36}+\frac{46\!\cdots\!12}{48\!\cdots\!71}a^{35}-\frac{15\!\cdots\!70}{48\!\cdots\!71}a^{34}+\frac{36\!\cdots\!41}{48\!\cdots\!71}a^{33}-\frac{89\!\cdots\!02}{48\!\cdots\!71}a^{32}+\frac{21\!\cdots\!76}{48\!\cdots\!71}a^{31}-\frac{34\!\cdots\!07}{48\!\cdots\!71}a^{30}+\frac{10\!\cdots\!83}{48\!\cdots\!71}a^{29}-\frac{65\!\cdots\!85}{48\!\cdots\!71}a^{28}+\frac{37\!\cdots\!22}{48\!\cdots\!71}a^{27}+\frac{10\!\cdots\!81}{48\!\cdots\!71}a^{26}+\frac{11\!\cdots\!35}{48\!\cdots\!71}a^{25}+\frac{14\!\cdots\!69}{48\!\cdots\!71}a^{24}+\frac{28\!\cdots\!62}{48\!\cdots\!71}a^{23}+\frac{61\!\cdots\!76}{48\!\cdots\!71}a^{22}+\frac{56\!\cdots\!78}{48\!\cdots\!71}a^{21}+\frac{18\!\cdots\!32}{48\!\cdots\!71}a^{20}+\frac{92\!\cdots\!57}{48\!\cdots\!71}a^{19}+\frac{38\!\cdots\!89}{48\!\cdots\!71}a^{18}+\frac{12\!\cdots\!67}{48\!\cdots\!71}a^{17}+\frac{62\!\cdots\!22}{48\!\cdots\!71}a^{16}+\frac{12\!\cdots\!17}{48\!\cdots\!71}a^{15}+\frac{75\!\cdots\!94}{48\!\cdots\!71}a^{14}+\frac{10\!\cdots\!20}{48\!\cdots\!71}a^{13}+\frac{68\!\cdots\!28}{48\!\cdots\!71}a^{12}+\frac{64\!\cdots\!21}{48\!\cdots\!71}a^{11}+\frac{42\!\cdots\!69}{48\!\cdots\!71}a^{10}+\frac{25\!\cdots\!50}{48\!\cdots\!71}a^{9}+\frac{17\!\cdots\!59}{48\!\cdots\!71}a^{8}+\frac{46\!\cdots\!25}{48\!\cdots\!71}a^{7}+\frac{34\!\cdots\!56}{48\!\cdots\!71}a^{6}-\frac{90\!\cdots\!07}{48\!\cdots\!71}a^{5}+\frac{46\!\cdots\!71}{48\!\cdots\!71}a^{4}-\frac{84\!\cdots\!21}{14\!\cdots\!41}a^{3}+\frac{53\!\cdots\!17}{48\!\cdots\!71}a^{2}-\frac{58\!\cdots\!99}{48\!\cdots\!71}a+\frac{22\!\cdots\!57}{48\!\cdots\!71}$, $\frac{33\!\cdots\!57}{48\!\cdots\!71}a^{39}-\frac{15\!\cdots\!01}{48\!\cdots\!71}a^{38}+\frac{68\!\cdots\!27}{48\!\cdots\!71}a^{37}-\frac{21\!\cdots\!13}{48\!\cdots\!71}a^{36}+\frac{78\!\cdots\!76}{48\!\cdots\!71}a^{35}-\frac{17\!\cdots\!51}{48\!\cdots\!71}a^{34}+\frac{61\!\cdots\!80}{48\!\cdots\!71}a^{33}-\frac{75\!\cdots\!03}{48\!\cdots\!71}a^{32}+\frac{36\!\cdots\!12}{48\!\cdots\!71}a^{31}-\frac{10\!\cdots\!83}{48\!\cdots\!71}a^{30}+\frac{16\!\cdots\!17}{48\!\cdots\!71}a^{29}+\frac{11\!\cdots\!38}{48\!\cdots\!71}a^{28}+\frac{62\!\cdots\!99}{48\!\cdots\!71}a^{27}+\frac{10\!\cdots\!85}{48\!\cdots\!71}a^{26}+\frac{18\!\cdots\!28}{48\!\cdots\!71}a^{25}+\frac{51\!\cdots\!81}{48\!\cdots\!71}a^{24}+\frac{45\!\cdots\!64}{48\!\cdots\!71}a^{23}+\frac{17\!\cdots\!65}{48\!\cdots\!71}a^{22}+\frac{91\!\cdots\!24}{48\!\cdots\!71}a^{21}+\frac{45\!\cdots\!35}{48\!\cdots\!71}a^{20}+\frac{14\!\cdots\!45}{48\!\cdots\!71}a^{19}+\frac{90\!\cdots\!88}{48\!\cdots\!71}a^{18}+\frac{19\!\cdots\!91}{48\!\cdots\!71}a^{17}+\frac{13\!\cdots\!52}{48\!\cdots\!71}a^{16}+\frac{20\!\cdots\!84}{48\!\cdots\!71}a^{15}+\frac{16\!\cdots\!57}{48\!\cdots\!71}a^{14}+\frac{16\!\cdots\!44}{48\!\cdots\!71}a^{13}+\frac{14\!\cdots\!49}{48\!\cdots\!71}a^{12}+\frac{95\!\cdots\!66}{48\!\cdots\!71}a^{11}+\frac{90\!\cdots\!46}{48\!\cdots\!71}a^{10}+\frac{34\!\cdots\!69}{48\!\cdots\!71}a^{9}+\frac{38\!\cdots\!50}{48\!\cdots\!71}a^{8}+\frac{34\!\cdots\!22}{48\!\cdots\!71}a^{7}+\frac{81\!\cdots\!64}{48\!\cdots\!71}a^{6}-\frac{31\!\cdots\!53}{48\!\cdots\!71}a^{5}+\frac{11\!\cdots\!92}{48\!\cdots\!71}a^{4}-\frac{23\!\cdots\!95}{14\!\cdots\!41}a^{3}+\frac{14\!\cdots\!11}{48\!\cdots\!71}a^{2}-\frac{12\!\cdots\!79}{48\!\cdots\!71}a+\frac{60\!\cdots\!44}{48\!\cdots\!71}$, $\frac{22\!\cdots\!59}{48\!\cdots\!71}a^{39}-\frac{19\!\cdots\!49}{48\!\cdots\!71}a^{38}+\frac{47\!\cdots\!59}{48\!\cdots\!71}a^{37}-\frac{33\!\cdots\!09}{48\!\cdots\!71}a^{36}+\frac{55\!\cdots\!57}{48\!\cdots\!71}a^{35}-\frac{34\!\cdots\!16}{48\!\cdots\!71}a^{34}+\frac{44\!\cdots\!16}{48\!\cdots\!71}a^{33}-\frac{23\!\cdots\!48}{48\!\cdots\!71}a^{32}+\frac{27\!\cdots\!01}{48\!\cdots\!71}a^{31}-\frac{12\!\cdots\!88}{48\!\cdots\!71}a^{30}+\frac{12\!\cdots\!92}{48\!\cdots\!71}a^{29}-\frac{46\!\cdots\!05}{48\!\cdots\!71}a^{28}+\frac{48\!\cdots\!40}{48\!\cdots\!71}a^{27}-\frac{13\!\cdots\!34}{48\!\cdots\!71}a^{26}+\frac{14\!\cdots\!77}{48\!\cdots\!71}a^{25}-\frac{29\!\cdots\!29}{48\!\cdots\!71}a^{24}+\frac{37\!\cdots\!04}{48\!\cdots\!71}a^{23}-\frac{41\!\cdots\!83}{48\!\cdots\!71}a^{22}+\frac{76\!\cdots\!02}{48\!\cdots\!71}a^{21}-\frac{19\!\cdots\!81}{48\!\cdots\!71}a^{20}+\frac{12\!\cdots\!38}{48\!\cdots\!71}a^{19}+\frac{72\!\cdots\!03}{48\!\cdots\!71}a^{18}+\frac{17\!\cdots\!95}{48\!\cdots\!71}a^{17}+\frac{23\!\cdots\!53}{48\!\cdots\!71}a^{16}+\frac{19\!\cdots\!37}{48\!\cdots\!71}a^{15}+\frac{38\!\cdots\!00}{48\!\cdots\!71}a^{14}+\frac{16\!\cdots\!31}{48\!\cdots\!71}a^{13}+\frac{40\!\cdots\!53}{48\!\cdots\!71}a^{12}+\frac{10\!\cdots\!30}{48\!\cdots\!71}a^{11}+\frac{28\!\cdots\!57}{48\!\cdots\!71}a^{10}+\frac{50\!\cdots\!77}{48\!\cdots\!71}a^{9}+\frac{12\!\cdots\!55}{48\!\cdots\!71}a^{8}+\frac{15\!\cdots\!68}{48\!\cdots\!71}a^{7}+\frac{22\!\cdots\!23}{48\!\cdots\!71}a^{6}+\frac{23\!\cdots\!67}{48\!\cdots\!71}a^{5}+\frac{18\!\cdots\!99}{48\!\cdots\!71}a^{4}+\frac{75\!\cdots\!98}{14\!\cdots\!41}a^{3}+\frac{11\!\cdots\!54}{48\!\cdots\!71}a^{2}+\frac{62\!\cdots\!63}{48\!\cdots\!71}a+\frac{73\!\cdots\!30}{48\!\cdots\!71}$ (assuming GRH)
Regulator:		\( 60790762850097.93 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 60790762850097.93 \cdot 18524}{6\cdot\sqrt{100383397447978918530459891214693626269465146363712847232818603515625}}\cr\approx \mathstrut & 0.172261227385874 \end{aligned}\] (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 $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.4.15125.1, 4.0.136125.2, \(\Q(\zeta_{11})^+\), 8.0.18530015625.2, 10.0.52089208083.1, 10.10.669871503125.1, 10.0.162778775259375.1, 20.0.26496929674942114598525390625.1, \(\Q(\zeta_{55})^+\), 20.0.10019151533337487082567413330078125.1

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	$20^{2}$	$20^{2}$	${\href{/padicField/19.5.0.1}{5} }^{8}$	${\href{/padicField/23.4.0.1}{4} }^{10}$	${\href{/padicField/29.10.0.1}{10} }^{4}$	${\href{/padicField/31.5.0.1}{5} }^{8}$	$20^{2}$	${\href{/padicField/41.10.0.1}{10} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{10}$	$20^{2}$	$20^{2}$	${\href{/padicField/59.10.0.1}{10} }^{4}$

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 $40$	$4$	$10$	$30$
\(11\)	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$
	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$