Number field 36.0.7225377334561374804949923918873673793376691639423370361328125.1

• Label: 36.0.722...125.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $7.225\times 10^{60}$
• Root discriminant: \(49.04\)
• Ramified primes: $3,5$
• Class number: $2053$ (GRH)
• Class group: [2053] (GRH)
• Galois group: $C_{36}$ (as 36T1)

Normalized defining polynomial

x^36 + 9*x^34 + 54*x^32 + 273*x^30 + 1260*x^28 - x^27 + 4374*x^26 - 18*x^25 + 13050*x^24 - 189*x^23 + 34695*x^22 - 153*x^21 + 79785*x^20 + 1935*x^19 + 133435*x^18 + 11664*x^17 + 197460*x^16 + 36792*x^15 + 255879*x^14 + 40932*x^13 + 256629*x^12 + 26649*x^11 + 121878*x^10 + 26*x^9 + 55485*x^8 - 19917*x^7 + 22041*x^6 - 4752*x^5 + 6021*x^4 - 699*x^3 + 81*x^2 - 9*x + 1

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(7225377334561374804949923918873673793376691639423370361328125\) \(\medspace = 3^{88}\cdot 5^{27}\)
Root discriminant:		\(49.04\)
Galois root discriminant:		$3^{22/9}5^{3/4}\approx 49.037001213832596$
Ramified primes:		\(3\), \(5\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$36$
This field is Galois and abelian over $\Q$.
Conductor:		\(135=3^{3}\cdot 5\)
Dirichlet character group:		$\lbrace$$\chi_{135}(1,·)$, $\chi_{135}(4,·)$, $\chi_{135}(133,·)$, $\chi_{135}(7,·)$, $\chi_{135}(13,·)$, $\chi_{135}(16,·)$, $\chi_{135}(19,·)$, $\chi_{135}(22,·)$, $\chi_{135}(28,·)$, $\chi_{135}(31,·)$, $\chi_{135}(34,·)$, $\chi_{135}(37,·)$, $\chi_{135}(43,·)$, $\chi_{135}(46,·)$, $\chi_{135}(49,·)$, $\chi_{135}(52,·)$, $\chi_{135}(58,·)$, $\chi_{135}(61,·)$, $\chi_{135}(64,·)$, $\chi_{135}(67,·)$, $\chi_{135}(73,·)$, $\chi_{135}(76,·)$, $\chi_{135}(79,·)$, $\chi_{135}(82,·)$, $\chi_{135}(88,·)$, $\chi_{135}(91,·)$, $\chi_{135}(94,·)$, $\chi_{135}(97,·)$, $\chi_{135}(103,·)$, $\chi_{135}(106,·)$, $\chi_{135}(109,·)$, $\chi_{135}(112,·)$, $\chi_{135}(118,·)$, $\chi_{135}(121,·)$, $\chi_{135}(124,·)$, $\chi_{135}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{271}a^{30}-\frac{132}{271}a^{29}+\frac{118}{271}a^{28}+\frac{72}{271}a^{27}-\frac{40}{271}a^{26}-\frac{128}{271}a^{25}-\frac{7}{271}a^{24}+\frac{96}{271}a^{23}+\frac{50}{271}a^{22}+\frac{43}{271}a^{21}+\frac{73}{271}a^{20}+\frac{18}{271}a^{19}-\frac{56}{271}a^{18}+\frac{20}{271}a^{17}+\frac{13}{271}a^{16}-\frac{45}{271}a^{15}-\frac{39}{271}a^{14}-\frac{78}{271}a^{13}-\frac{21}{271}a^{12}-\frac{128}{271}a^{11}+\frac{52}{271}a^{10}+\frac{76}{271}a^{9}+\frac{16}{271}a^{8}-\frac{48}{271}a^{7}+\frac{116}{271}a^{6}+\frac{36}{271}a^{5}+\frac{19}{271}a^{4}-\frac{65}{271}a^{3}-\frac{46}{271}a^{2}-\frac{45}{271}a+\frac{10}{271}$, $\frac{1}{271}a^{31}+\frac{38}{271}a^{29}-\frac{70}{271}a^{28}-\frac{21}{271}a^{27}+\frac{12}{271}a^{26}-\frac{101}{271}a^{25}-\frac{15}{271}a^{24}-\frac{15}{271}a^{23}-\frac{132}{271}a^{22}+\frac{58}{271}a^{21}-\frac{102}{271}a^{20}-\frac{119}{271}a^{19}-\frac{55}{271}a^{18}-\frac{57}{271}a^{17}+\frac{45}{271}a^{16}-\frac{17}{271}a^{15}-\frac{77}{271}a^{14}-\frac{19}{271}a^{13}+\frac{81}{271}a^{12}-\frac{42}{271}a^{11}-\frac{106}{271}a^{10}+\frac{21}{271}a^{9}-\frac{104}{271}a^{8}+\frac{13}{271}a^{7}-\frac{99}{271}a^{6}-\frac{107}{271}a^{5}+\frac{4}{271}a^{4}+\frac{46}{271}a^{3}+\frac{116}{271}a^{2}+\frac{32}{271}a-\frac{35}{271}$, $\frac{1}{271}a^{32}+\frac{68}{271}a^{29}+\frac{102}{271}a^{28}-\frac{14}{271}a^{27}+\frac{64}{271}a^{26}-\frac{29}{271}a^{25}-\frac{20}{271}a^{24}+\frac{14}{271}a^{23}+\frac{55}{271}a^{22}-\frac{110}{271}a^{21}+\frac{88}{271}a^{20}+\frac{74}{271}a^{19}-\frac{97}{271}a^{18}+\frac{98}{271}a^{17}+\frac{31}{271}a^{16}+\frac{7}{271}a^{15}+\frac{108}{271}a^{14}+\frac{64}{271}a^{13}-\frac{57}{271}a^{12}-\frac{120}{271}a^{11}-\frac{58}{271}a^{10}-\frac{11}{271}a^{9}-\frac{53}{271}a^{8}+\frac{99}{271}a^{7}+\frac{92}{271}a^{6}-\frac{9}{271}a^{5}-\frac{134}{271}a^{4}-\frac{124}{271}a^{3}-\frac{117}{271}a^{2}+\frac{49}{271}a-\frac{109}{271}$, $\frac{1}{31\!\cdots\!91}a^{33}-\frac{95\!\cdots\!29}{31\!\cdots\!91}a^{32}+\frac{49\!\cdots\!92}{31\!\cdots\!91}a^{31}+\frac{31\!\cdots\!60}{31\!\cdots\!91}a^{30}-\frac{12\!\cdots\!72}{31\!\cdots\!91}a^{29}+\frac{24\!\cdots\!82}{31\!\cdots\!91}a^{28}-\frac{68\!\cdots\!98}{31\!\cdots\!91}a^{27}-\frac{13\!\cdots\!91}{31\!\cdots\!91}a^{26}-\frac{45\!\cdots\!90}{31\!\cdots\!91}a^{25}-\frac{91\!\cdots\!17}{31\!\cdots\!91}a^{24}-\frac{42\!\cdots\!75}{31\!\cdots\!91}a^{23}+\frac{46\!\cdots\!26}{31\!\cdots\!91}a^{22}-\frac{15\!\cdots\!27}{31\!\cdots\!91}a^{21}+\frac{11\!\cdots\!41}{31\!\cdots\!91}a^{20}+\frac{12\!\cdots\!42}{31\!\cdots\!91}a^{19}+\frac{29\!\cdots\!85}{31\!\cdots\!91}a^{18}+\frac{71\!\cdots\!46}{31\!\cdots\!91}a^{17}+\frac{47\!\cdots\!03}{31\!\cdots\!91}a^{16}+\frac{57\!\cdots\!43}{31\!\cdots\!91}a^{15}-\frac{13\!\cdots\!85}{31\!\cdots\!91}a^{14}+\frac{10\!\cdots\!46}{31\!\cdots\!91}a^{13}+\frac{65\!\cdots\!65}{31\!\cdots\!91}a^{12}-\frac{19\!\cdots\!93}{31\!\cdots\!91}a^{11}+\frac{10\!\cdots\!14}{31\!\cdots\!91}a^{10}+\frac{51\!\cdots\!18}{31\!\cdots\!91}a^{9}+\frac{94\!\cdots\!64}{31\!\cdots\!91}a^{8}-\frac{11\!\cdots\!21}{31\!\cdots\!91}a^{7}+\frac{12\!\cdots\!83}{31\!\cdots\!91}a^{6}-\frac{54\!\cdots\!14}{31\!\cdots\!91}a^{5}+\frac{53\!\cdots\!61}{31\!\cdots\!91}a^{4}-\frac{12\!\cdots\!44}{31\!\cdots\!91}a^{3}-\frac{10\!\cdots\!29}{31\!\cdots\!91}a^{2}-\frac{77\!\cdots\!83}{31\!\cdots\!91}a-\frac{14\!\cdots\!58}{31\!\cdots\!91}$, $\frac{1}{31\!\cdots\!91}a^{34}-\frac{51\!\cdots\!42}{31\!\cdots\!91}a^{32}-\frac{54\!\cdots\!95}{31\!\cdots\!91}a^{31}+\frac{60\!\cdots\!79}{31\!\cdots\!91}a^{30}-\frac{10\!\cdots\!38}{31\!\cdots\!91}a^{29}-\frac{15\!\cdots\!86}{31\!\cdots\!91}a^{28}+\frac{91\!\cdots\!45}{31\!\cdots\!91}a^{27}+\frac{14\!\cdots\!50}{31\!\cdots\!91}a^{26}+\frac{13\!\cdots\!43}{31\!\cdots\!91}a^{25}-\frac{84\!\cdots\!39}{31\!\cdots\!91}a^{24}+\frac{71\!\cdots\!65}{31\!\cdots\!91}a^{23}+\frac{13\!\cdots\!93}{31\!\cdots\!91}a^{22}-\frac{62\!\cdots\!06}{31\!\cdots\!91}a^{21}+\frac{24\!\cdots\!46}{31\!\cdots\!91}a^{20}-\frac{51\!\cdots\!71}{31\!\cdots\!91}a^{19}-\frac{12\!\cdots\!90}{31\!\cdots\!91}a^{18}+\frac{11\!\cdots\!26}{31\!\cdots\!91}a^{17}+\frac{92\!\cdots\!92}{31\!\cdots\!91}a^{16}+\frac{14\!\cdots\!50}{31\!\cdots\!91}a^{15}+\frac{10\!\cdots\!60}{31\!\cdots\!91}a^{14}+\frac{69\!\cdots\!91}{31\!\cdots\!91}a^{13}+\frac{58\!\cdots\!13}{31\!\cdots\!91}a^{12}-\frac{12\!\cdots\!88}{31\!\cdots\!91}a^{11}+\frac{92\!\cdots\!66}{31\!\cdots\!91}a^{10}+\frac{13\!\cdots\!05}{31\!\cdots\!91}a^{9}-\frac{10\!\cdots\!96}{31\!\cdots\!91}a^{8}+\frac{39\!\cdots\!82}{31\!\cdots\!91}a^{7}-\frac{77\!\cdots\!44}{31\!\cdots\!91}a^{6}+\frac{84\!\cdots\!02}{31\!\cdots\!91}a^{5}-\frac{15\!\cdots\!62}{31\!\cdots\!91}a^{4}-\frac{49\!\cdots\!98}{31\!\cdots\!91}a^{3}+\frac{37\!\cdots\!80}{31\!\cdots\!91}a^{2}+\frac{69\!\cdots\!34}{31\!\cdots\!91}a-\frac{79\!\cdots\!24}{31\!\cdots\!91}$, $\frac{1}{31\!\cdots\!91}a^{35}+\frac{37\!\cdots\!71}{31\!\cdots\!91}a^{32}+\frac{36\!\cdots\!86}{31\!\cdots\!91}a^{31}+\frac{34\!\cdots\!99}{31\!\cdots\!91}a^{30}-\frac{10\!\cdots\!65}{31\!\cdots\!91}a^{29}-\frac{33\!\cdots\!32}{31\!\cdots\!91}a^{28}-\frac{93\!\cdots\!59}{31\!\cdots\!91}a^{27}+\frac{15\!\cdots\!14}{31\!\cdots\!91}a^{26}-\frac{13\!\cdots\!35}{31\!\cdots\!91}a^{25}-\frac{10\!\cdots\!22}{31\!\cdots\!91}a^{24}-\frac{28\!\cdots\!25}{31\!\cdots\!91}a^{23}+\frac{35\!\cdots\!14}{31\!\cdots\!91}a^{22}-\frac{65\!\cdots\!99}{31\!\cdots\!91}a^{21}+\frac{16\!\cdots\!30}{31\!\cdots\!91}a^{20}+\frac{50\!\cdots\!74}{31\!\cdots\!91}a^{19}-\frac{92\!\cdots\!24}{31\!\cdots\!91}a^{18}-\frac{12\!\cdots\!69}{31\!\cdots\!91}a^{17}+\frac{37\!\cdots\!21}{31\!\cdots\!91}a^{16}+\frac{12\!\cdots\!93}{31\!\cdots\!91}a^{15}+\frac{92\!\cdots\!26}{31\!\cdots\!91}a^{14}-\frac{11\!\cdots\!87}{31\!\cdots\!91}a^{13}+\frac{12\!\cdots\!30}{31\!\cdots\!91}a^{12}-\frac{10\!\cdots\!81}{31\!\cdots\!91}a^{11}-\frac{88\!\cdots\!17}{31\!\cdots\!91}a^{10}-\frac{10\!\cdots\!80}{31\!\cdots\!91}a^{9}-\frac{14\!\cdots\!07}{31\!\cdots\!91}a^{8}+\frac{59\!\cdots\!77}{31\!\cdots\!91}a^{7}-\frac{37\!\cdots\!19}{31\!\cdots\!91}a^{6}+\frac{90\!\cdots\!66}{31\!\cdots\!91}a^{5}+\frac{13\!\cdots\!00}{31\!\cdots\!91}a^{4}-\frac{59\!\cdots\!75}{31\!\cdots\!91}a^{3}+\frac{11\!\cdots\!21}{31\!\cdots\!91}a^{2}-\frac{11\!\cdots\!06}{31\!\cdots\!91}a+\frac{64\!\cdots\!15}{31\!\cdots\!91}$

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

Class group and class number

$C_{2053}$, which has order $2053$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( -\frac{37078126722002202006861542313}{318883648678381348159186580491} a^{35} + \frac{436455875773191844058481}{318883648678381348159186580491} a^{34} - \frac{333658735026101126566580102517}{318883648678381348159186580491} a^{33} - \frac{2001832515382426292362511413692}{318883648678381348159186580491} a^{31} - \frac{1480182397289716505792610}{318883648678381348159186580491} a^{30} - \frac{10120050565304844621717574223454}{318883648678381348159186580491} a^{29} - \frac{13321641575607448552133490}{318883648678381348159186580491} a^{28} - \frac{46707036344534054552684917046940}{318883648678381348159186580491} a^{27} + \frac{36998196872548557315548741373}{318883648678381348159186580491} a^{26} - \frac{162127412195570221127548157861832}{318883648678381348159186580491} a^{25} + \frac{666462480433178568880181583177}{318883648678381348159186580491} a^{24} - \frac{483692109456341644974828709097850}{318883648678381348159186580491} a^{23} + \frac{7005114943784870297034956932647}{318883648678381348159186580491} a^{22} - \frac{1285904382591396784306293929827185}{318883648678381348159186580491} a^{21} + \frac{5658326226016319928539573401869}{318883648678381348159186580491} a^{20} - \frac{2956910950505734882506211986534510}{318883648678381348159186580491} a^{19} - \frac{71769767834304661674662912786445}{318883648678381348159186580491} a^{18} - \frac{4944438863173323636514226673501915}{318883648678381348159186580491} a^{17} - \frac{432442662214383914939411766708312}{318883648678381348159186580491} a^{16} - \frac{7316584299086287263572473621894800}{318883648678381348159186580491} a^{15} - \frac{1363704680150200757916450339527637}{318883648678381348159186580491} a^{14} - \frac{9480526120410716560175702963328627}{318883648678381348159186580491} a^{13} - \frac{1516401008627681873656285520684826}{318883648678381348159186580491} a^{12} - \frac{9506607910984621671769727368555617}{318883648678381348159186580491} a^{11} - \frac{987059800891079377668764051858217}{318883648678381348159186580491} a^{10} - \frac{4510895350525870026258225464270138}{318883648678381348159186580491} a^{9} - \frac{704698898219707051214017450308}{318883648678381348159186580491} a^{8} - \frac{2055347117445578322892423492107135}{318883648678381348159186580491} a^{7} + \frac{737751338309605317795904999397121}{318883648678381348159186580491} a^{6} - \frac{816438213882899195093317957903443}{318883648678381348159186580491} a^{5} + \frac{173912471666319478348596835458414}{318883648678381348159186580491} a^{4} - \frac{223007849754180287853732376056783}{318883648678381348159186580491} a^{3} + \frac{25889794991069670850219363349667}{318883648678381348159186580491} a^{2} - \frac{3000104427220881360006168622773}{318883648678381348159186580491} a + \frac{333343456175478416950846276587}{318883648678381348159186580491} \)  (order $10$)
Fundamental units:		$\frac{14\!\cdots\!50}{31\!\cdots\!91}a^{35}+\frac{43\!\cdots\!13}{31\!\cdots\!91}a^{34}+\frac{13\!\cdots\!50}{31\!\cdots\!91}a^{33}+\frac{39\!\cdots\!38}{31\!\cdots\!91}a^{32}+\frac{81\!\cdots\!10}{31\!\cdots\!91}a^{31}+\frac{23\!\cdots\!43}{31\!\cdots\!91}a^{30}+\frac{41\!\cdots\!20}{31\!\cdots\!91}a^{29}+\frac{11\!\cdots\!96}{31\!\cdots\!91}a^{28}+\frac{18\!\cdots\!15}{31\!\cdots\!91}a^{27}+\frac{20\!\cdots\!30}{11\!\cdots\!21}a^{26}+\frac{66\!\cdots\!48}{31\!\cdots\!91}a^{25}+\frac{43\!\cdots\!06}{73\!\cdots\!61}a^{24}+\frac{19\!\cdots\!83}{31\!\cdots\!91}a^{23}+\frac{56\!\cdots\!40}{31\!\cdots\!91}a^{22}+\frac{51\!\cdots\!77}{31\!\cdots\!91}a^{21}+\frac{14\!\cdots\!45}{31\!\cdots\!91}a^{20}+\frac{12\!\cdots\!99}{31\!\cdots\!91}a^{19}+\frac{34\!\cdots\!05}{31\!\cdots\!91}a^{18}+\frac{21\!\cdots\!35}{31\!\cdots\!91}a^{17}+\frac{59\!\cdots\!40}{31\!\cdots\!91}a^{16}+\frac{35\!\cdots\!28}{31\!\cdots\!91}a^{15}+\frac{90\!\cdots\!29}{31\!\cdots\!91}a^{14}+\frac{54\!\cdots\!74}{31\!\cdots\!91}a^{13}+\frac{11\!\cdots\!68}{31\!\cdots\!91}a^{12}+\frac{56\!\cdots\!59}{31\!\cdots\!91}a^{11}+\frac{11\!\cdots\!48}{31\!\cdots\!91}a^{10}+\frac{30\!\cdots\!44}{31\!\cdots\!91}a^{9}+\frac{51\!\cdots\!51}{31\!\cdots\!91}a^{8}+\frac{84\!\cdots\!27}{31\!\cdots\!91}a^{7}+\frac{20\!\cdots\!45}{31\!\cdots\!91}a^{6}-\frac{51\!\cdots\!54}{31\!\cdots\!91}a^{5}+\frac{85\!\cdots\!25}{31\!\cdots\!91}a^{4}-\frac{98\!\cdots\!29}{31\!\cdots\!91}a^{3}+\frac{24\!\cdots\!77}{31\!\cdots\!91}a^{2}-\frac{28\!\cdots\!38}{31\!\cdots\!91}a+\frac{33\!\cdots\!62}{31\!\cdots\!91}$, $\frac{48\!\cdots\!60}{31\!\cdots\!91}a^{35}+\frac{13\!\cdots\!28}{31\!\cdots\!91}a^{34}+\frac{41\!\cdots\!22}{31\!\cdots\!91}a^{33}-\frac{15\!\cdots\!34}{31\!\cdots\!91}a^{32}+\frac{24\!\cdots\!09}{31\!\cdots\!91}a^{31}-\frac{14\!\cdots\!02}{31\!\cdots\!91}a^{30}+\frac{12\!\cdots\!08}{31\!\cdots\!91}a^{29}-\frac{84\!\cdots\!88}{31\!\cdots\!91}a^{28}+\frac{56\!\cdots\!94}{31\!\cdots\!91}a^{27}-\frac{47\!\cdots\!71}{31\!\cdots\!91}a^{26}+\frac{19\!\cdots\!04}{31\!\cdots\!91}a^{25}-\frac{28\!\cdots\!82}{31\!\cdots\!91}a^{24}+\frac{56\!\cdots\!82}{31\!\cdots\!91}a^{23}-\frac{15\!\cdots\!24}{31\!\cdots\!91}a^{22}+\frac{14\!\cdots\!01}{31\!\cdots\!91}a^{21}-\frac{25\!\cdots\!58}{31\!\cdots\!91}a^{20}+\frac{33\!\cdots\!88}{31\!\cdots\!91}a^{19}+\frac{39\!\cdots\!84}{31\!\cdots\!91}a^{18}+\frac{52\!\cdots\!64}{31\!\cdots\!91}a^{17}+\frac{40\!\cdots\!29}{31\!\cdots\!91}a^{16}+\frac{75\!\cdots\!64}{31\!\cdots\!91}a^{15}+\frac{13\!\cdots\!66}{31\!\cdots\!91}a^{14}+\frac{94\!\cdots\!76}{31\!\cdots\!91}a^{13}+\frac{11\!\cdots\!12}{31\!\cdots\!91}a^{12}+\frac{86\!\cdots\!42}{31\!\cdots\!91}a^{11}+\frac{28\!\cdots\!82}{31\!\cdots\!91}a^{10}+\frac{20\!\cdots\!86}{31\!\cdots\!91}a^{9}-\frac{79\!\cdots\!60}{31\!\cdots\!91}a^{8}+\frac{86\!\cdots\!34}{31\!\cdots\!91}a^{7}-\frac{11\!\cdots\!97}{31\!\cdots\!91}a^{6}+\frac{26\!\cdots\!94}{31\!\cdots\!91}a^{5}-\frac{30\!\cdots\!52}{31\!\cdots\!91}a^{4}+\frac{35\!\cdots\!48}{31\!\cdots\!91}a^{3}-\frac{39\!\cdots\!34}{31\!\cdots\!91}a^{2}-\frac{78\!\cdots\!74}{31\!\cdots\!91}a+\frac{15\!\cdots\!92}{31\!\cdots\!91}$, $\frac{69\!\cdots\!17}{31\!\cdots\!91}a^{35}+\frac{44\!\cdots\!35}{31\!\cdots\!91}a^{34}+\frac{62\!\cdots\!53}{31\!\cdots\!91}a^{33}+\frac{37\!\cdots\!68}{31\!\cdots\!91}a^{31}-\frac{11\!\cdots\!50}{31\!\cdots\!91}a^{30}+\frac{19\!\cdots\!31}{31\!\cdots\!91}a^{29}-\frac{10\!\cdots\!50}{31\!\cdots\!91}a^{28}+\frac{87\!\cdots\!20}{31\!\cdots\!91}a^{27}-\frac{69\!\cdots\!17}{31\!\cdots\!91}a^{26}+\frac{30\!\cdots\!08}{31\!\cdots\!91}a^{25}-\frac{12\!\cdots\!46}{31\!\cdots\!91}a^{24}+\frac{90\!\cdots\!50}{31\!\cdots\!91}a^{23}-\frac{13\!\cdots\!63}{31\!\cdots\!91}a^{22}+\frac{24\!\cdots\!65}{31\!\cdots\!91}a^{21}-\frac{10\!\cdots\!01}{31\!\cdots\!91}a^{20}+\frac{55\!\cdots\!24}{31\!\cdots\!91}a^{19}+\frac{13\!\cdots\!45}{31\!\cdots\!91}a^{18}+\frac{93\!\cdots\!95}{31\!\cdots\!91}a^{17}+\frac{81\!\cdots\!88}{31\!\cdots\!91}a^{16}+\frac{13\!\cdots\!20}{31\!\cdots\!91}a^{15}+\frac{25\!\cdots\!09}{31\!\cdots\!91}a^{14}+\frac{17\!\cdots\!43}{31\!\cdots\!91}a^{13}+\frac{28\!\cdots\!94}{31\!\cdots\!91}a^{12}+\frac{17\!\cdots\!93}{31\!\cdots\!91}a^{11}+\frac{18\!\cdots\!33}{31\!\cdots\!91}a^{10}+\frac{85\!\cdots\!75}{31\!\cdots\!91}a^{9}+\frac{20\!\cdots\!92}{31\!\cdots\!91}a^{8}+\frac{38\!\cdots\!95}{31\!\cdots\!91}a^{7}-\frac{13\!\cdots\!89}{31\!\cdots\!91}a^{6}+\frac{15\!\cdots\!47}{31\!\cdots\!91}a^{5}-\frac{33\!\cdots\!25}{31\!\cdots\!91}a^{4}+\frac{41\!\cdots\!07}{31\!\cdots\!91}a^{3}-\frac{48\!\cdots\!83}{31\!\cdots\!91}a^{2}+\frac{56\!\cdots\!77}{31\!\cdots\!91}a-\frac{62\!\cdots\!03}{31\!\cdots\!91}$, $\frac{34\!\cdots\!80}{11\!\cdots\!21}a^{35}+\frac{11\!\cdots\!00}{31\!\cdots\!91}a^{34}+\frac{10\!\cdots\!50}{31\!\cdots\!91}a^{32}-\frac{39\!\cdots\!50}{31\!\cdots\!91}a^{31}+\frac{60\!\cdots\!15}{31\!\cdots\!91}a^{30}-\frac{35\!\cdots\!50}{31\!\cdots\!91}a^{29}+\frac{30\!\cdots\!00}{31\!\cdots\!91}a^{28}-\frac{21\!\cdots\!00}{31\!\cdots\!91}a^{27}+\frac{13\!\cdots\!50}{31\!\cdots\!91}a^{26}-\frac{22\!\cdots\!40}{31\!\cdots\!91}a^{25}+\frac{46\!\cdots\!00}{31\!\cdots\!91}a^{24}-\frac{69\!\cdots\!50}{31\!\cdots\!91}a^{23}+\frac{13\!\cdots\!50}{31\!\cdots\!91}a^{22}-\frac{38\!\cdots\!00}{31\!\cdots\!91}a^{21}+\frac{36\!\cdots\!91}{31\!\cdots\!91}a^{20}-\frac{62\!\cdots\!50}{31\!\cdots\!91}a^{19}+\frac{81\!\cdots\!00}{31\!\cdots\!91}a^{18}+\frac{96\!\cdots\!00}{31\!\cdots\!91}a^{17}+\frac{12\!\cdots\!00}{31\!\cdots\!91}a^{16}+\frac{11\!\cdots\!65}{31\!\cdots\!91}a^{15}+\frac{18\!\cdots\!00}{31\!\cdots\!91}a^{14}+\frac{33\!\cdots\!50}{31\!\cdots\!91}a^{13}+\frac{22\!\cdots\!00}{31\!\cdots\!91}a^{12}+\frac{27\!\cdots\!00}{31\!\cdots\!91}a^{11}+\frac{21\!\cdots\!11}{31\!\cdots\!91}a^{10}+\frac{68\!\cdots\!50}{31\!\cdots\!91}a^{9}+\frac{50\!\cdots\!50}{31\!\cdots\!91}a^{8}-\frac{19\!\cdots\!00}{31\!\cdots\!91}a^{7}+\frac{21\!\cdots\!50}{31\!\cdots\!91}a^{6}-\frac{49\!\cdots\!30}{31\!\cdots\!91}a^{5}+\frac{63\!\cdots\!50}{31\!\cdots\!91}a^{4}-\frac{73\!\cdots\!00}{31\!\cdots\!91}a^{3}+\frac{85\!\cdots\!00}{31\!\cdots\!91}a^{2}-\frac{94\!\cdots\!50}{31\!\cdots\!91}a+\frac{25\!\cdots\!02}{31\!\cdots\!91}$, $\frac{50\!\cdots\!90}{31\!\cdots\!91}a^{35}-\frac{35\!\cdots\!85}{31\!\cdots\!91}a^{34}+\frac{45\!\cdots\!40}{31\!\cdots\!91}a^{33}+\frac{27\!\cdots\!21}{31\!\cdots\!91}a^{31}-\frac{65\!\cdots\!01}{31\!\cdots\!91}a^{30}+\frac{13\!\cdots\!23}{31\!\cdots\!91}a^{29}-\frac{58\!\cdots\!09}{31\!\cdots\!91}a^{28}+\frac{64\!\cdots\!04}{31\!\cdots\!91}a^{27}-\frac{51\!\cdots\!44}{31\!\cdots\!91}a^{26}+\frac{22\!\cdots\!03}{31\!\cdots\!91}a^{25}-\frac{91\!\cdots\!48}{31\!\cdots\!91}a^{24}+\frac{66\!\cdots\!80}{31\!\cdots\!91}a^{23}-\frac{96\!\cdots\!01}{31\!\cdots\!91}a^{22}+\frac{17\!\cdots\!85}{31\!\cdots\!91}a^{21}-\frac{78\!\cdots\!52}{31\!\cdots\!91}a^{20}+\frac{40\!\cdots\!00}{31\!\cdots\!91}a^{19}+\frac{98\!\cdots\!11}{31\!\cdots\!91}a^{18}+\frac{67\!\cdots\!34}{31\!\cdots\!91}a^{17}+\frac{59\!\cdots\!92}{31\!\cdots\!91}a^{16}+\frac{10\!\cdots\!38}{31\!\cdots\!91}a^{15}+\frac{18\!\cdots\!04}{31\!\cdots\!91}a^{14}+\frac{13\!\cdots\!60}{31\!\cdots\!91}a^{13}+\frac{20\!\cdots\!29}{31\!\cdots\!91}a^{12}+\frac{13\!\cdots\!76}{31\!\cdots\!91}a^{11}+\frac{13\!\cdots\!82}{31\!\cdots\!91}a^{10}+\frac{62\!\cdots\!73}{31\!\cdots\!91}a^{9}+\frac{24\!\cdots\!43}{31\!\cdots\!91}a^{8}+\frac{28\!\cdots\!97}{31\!\cdots\!91}a^{7}-\frac{10\!\cdots\!20}{31\!\cdots\!91}a^{6}+\frac{11\!\cdots\!89}{31\!\cdots\!91}a^{5}-\frac{24\!\cdots\!14}{31\!\cdots\!91}a^{4}+\frac{30\!\cdots\!29}{31\!\cdots\!91}a^{3}-\frac{35\!\cdots\!02}{31\!\cdots\!91}a^{2}+\frac{41\!\cdots\!68}{31\!\cdots\!91}a-\frac{45\!\cdots\!53}{31\!\cdots\!91}$, $\frac{75\!\cdots\!32}{31\!\cdots\!91}a^{35}+\frac{84\!\cdots\!57}{31\!\cdots\!91}a^{34}+\frac{67\!\cdots\!33}{31\!\cdots\!91}a^{33}+\frac{75\!\cdots\!03}{31\!\cdots\!91}a^{32}+\frac{40\!\cdots\!28}{31\!\cdots\!91}a^{31}+\frac{45\!\cdots\!91}{31\!\cdots\!91}a^{30}+\frac{20\!\cdots\!03}{31\!\cdots\!91}a^{29}+\frac{22\!\cdots\!81}{31\!\cdots\!91}a^{28}+\frac{94\!\cdots\!23}{31\!\cdots\!91}a^{27}+\frac{10\!\cdots\!20}{31\!\cdots\!91}a^{26}+\frac{32\!\cdots\!29}{31\!\cdots\!91}a^{25}+\frac{35\!\cdots\!61}{31\!\cdots\!91}a^{24}+\frac{98\!\cdots\!09}{31\!\cdots\!91}a^{23}+\frac{95\!\cdots\!42}{31\!\cdots\!91}a^{22}+\frac{26\!\cdots\!64}{31\!\cdots\!91}a^{21}+\frac{27\!\cdots\!74}{31\!\cdots\!91}a^{20}+\frac{60\!\cdots\!93}{31\!\cdots\!91}a^{19}+\frac{81\!\cdots\!34}{31\!\cdots\!91}a^{18}+\frac{10\!\cdots\!28}{31\!\cdots\!91}a^{17}+\frac{19\!\cdots\!15}{31\!\cdots\!91}a^{16}+\frac{14\!\cdots\!24}{31\!\cdots\!91}a^{15}+\frac{44\!\cdots\!18}{31\!\cdots\!91}a^{14}+\frac{19\!\cdots\!93}{31\!\cdots\!91}a^{13}+\frac{52\!\cdots\!77}{31\!\cdots\!91}a^{12}+\frac{19\!\cdots\!69}{31\!\cdots\!91}a^{11}+\frac{41\!\cdots\!99}{31\!\cdots\!91}a^{10}+\frac{94\!\cdots\!65}{31\!\cdots\!91}a^{9}+\frac{10\!\cdots\!16}{31\!\cdots\!91}a^{8}+\frac{41\!\cdots\!01}{31\!\cdots\!91}a^{7}-\frac{10\!\cdots\!48}{31\!\cdots\!91}a^{6}+\frac{14\!\cdots\!73}{31\!\cdots\!91}a^{5}-\frac{17\!\cdots\!60}{31\!\cdots\!91}a^{4}+\frac{41\!\cdots\!71}{31\!\cdots\!91}a^{3}-\frac{22\!\cdots\!84}{31\!\cdots\!91}a^{2}+\frac{25\!\cdots\!13}{31\!\cdots\!91}a+\frac{11\!\cdots\!03}{31\!\cdots\!91}$, $\frac{12\!\cdots\!41}{31\!\cdots\!91}a^{35}+\frac{53\!\cdots\!86}{31\!\cdots\!91}a^{34}+\frac{67\!\cdots\!00}{31\!\cdots\!91}a^{33}+\frac{48\!\cdots\!30}{31\!\cdots\!91}a^{32}+\frac{57\!\cdots\!63}{31\!\cdots\!91}a^{31}+\frac{29\!\cdots\!44}{31\!\cdots\!91}a^{30}+\frac{33\!\cdots\!56}{31\!\cdots\!91}a^{29}+\frac{14\!\cdots\!29}{31\!\cdots\!91}a^{28}+\frac{17\!\cdots\!95}{31\!\cdots\!91}a^{27}+\frac{67\!\cdots\!78}{31\!\cdots\!91}a^{26}+\frac{76\!\cdots\!04}{31\!\cdots\!91}a^{25}+\frac{23\!\cdots\!49}{31\!\cdots\!91}a^{24}+\frac{25\!\cdots\!03}{31\!\cdots\!91}a^{23}+\frac{70\!\cdots\!04}{31\!\cdots\!91}a^{22}+\frac{68\!\cdots\!11}{31\!\cdots\!91}a^{21}+\frac{18\!\cdots\!79}{31\!\cdots\!91}a^{20}+\frac{20\!\cdots\!87}{31\!\cdots\!91}a^{19}+\frac{42\!\cdots\!84}{31\!\cdots\!91}a^{18}+\frac{58\!\cdots\!45}{31\!\cdots\!91}a^{17}+\frac{71\!\cdots\!08}{31\!\cdots\!91}a^{16}+\frac{14\!\cdots\!65}{31\!\cdots\!91}a^{15}+\frac{10\!\cdots\!84}{31\!\cdots\!91}a^{14}+\frac{31\!\cdots\!73}{31\!\cdots\!91}a^{13}+\frac{13\!\cdots\!25}{31\!\cdots\!91}a^{12}+\frac{37\!\cdots\!40}{31\!\cdots\!91}a^{11}+\frac{14\!\cdots\!96}{31\!\cdots\!91}a^{10}+\frac{29\!\cdots\!45}{31\!\cdots\!91}a^{9}+\frac{67\!\cdots\!21}{31\!\cdots\!91}a^{8}+\frac{76\!\cdots\!44}{31\!\cdots\!91}a^{7}+\frac{29\!\cdots\!66}{31\!\cdots\!91}a^{6}-\frac{73\!\cdots\!43}{31\!\cdots\!91}a^{5}+\frac{10\!\cdots\!57}{31\!\cdots\!91}a^{4}-\frac{12\!\cdots\!89}{31\!\cdots\!91}a^{3}+\frac{29\!\cdots\!71}{31\!\cdots\!91}a^{2}-\frac{16\!\cdots\!31}{31\!\cdots\!91}a+\frac{17\!\cdots\!20}{31\!\cdots\!91}$, $\frac{66\!\cdots\!89}{31\!\cdots\!91}a^{35}+\frac{34\!\cdots\!80}{11\!\cdots\!21}a^{34}+\frac{59\!\cdots\!01}{31\!\cdots\!91}a^{33}+\frac{35\!\cdots\!56}{31\!\cdots\!91}a^{31}-\frac{39\!\cdots\!50}{31\!\cdots\!91}a^{30}+\frac{18\!\cdots\!12}{31\!\cdots\!91}a^{29}-\frac{35\!\cdots\!50}{31\!\cdots\!91}a^{28}+\frac{83\!\cdots\!40}{31\!\cdots\!91}a^{27}-\frac{66\!\cdots\!89}{31\!\cdots\!91}a^{26}+\frac{28\!\cdots\!36}{31\!\cdots\!91}a^{25}-\frac{11\!\cdots\!42}{31\!\cdots\!91}a^{24}+\frac{86\!\cdots\!50}{31\!\cdots\!91}a^{23}-\frac{12\!\cdots\!71}{31\!\cdots\!91}a^{22}+\frac{23\!\cdots\!05}{31\!\cdots\!91}a^{21}-\frac{10\!\cdots\!17}{31\!\cdots\!91}a^{20}+\frac{52\!\cdots\!56}{31\!\cdots\!91}a^{19}+\frac{12\!\cdots\!65}{31\!\cdots\!91}a^{18}+\frac{88\!\cdots\!15}{31\!\cdots\!91}a^{17}+\frac{77\!\cdots\!96}{31\!\cdots\!91}a^{16}+\frac{13\!\cdots\!40}{31\!\cdots\!91}a^{15}+\frac{24\!\cdots\!53}{31\!\cdots\!91}a^{14}+\frac{16\!\cdots\!31}{31\!\cdots\!91}a^{13}+\frac{27\!\cdots\!98}{31\!\cdots\!91}a^{12}+\frac{17\!\cdots\!81}{31\!\cdots\!91}a^{11}+\frac{17\!\cdots\!61}{31\!\cdots\!91}a^{10}+\frac{80\!\cdots\!53}{31\!\cdots\!91}a^{9}+\frac{24\!\cdots\!64}{31\!\cdots\!91}a^{8}+\frac{36\!\cdots\!15}{31\!\cdots\!91}a^{7}-\frac{13\!\cdots\!13}{31\!\cdots\!91}a^{6}+\frac{14\!\cdots\!99}{31\!\cdots\!91}a^{5}-\frac{31\!\cdots\!58}{31\!\cdots\!91}a^{4}+\frac{39\!\cdots\!19}{31\!\cdots\!91}a^{3}-\frac{46\!\cdots\!11}{31\!\cdots\!91}a^{2}+\frac{53\!\cdots\!09}{31\!\cdots\!91}a-\frac{59\!\cdots\!51}{31\!\cdots\!91}$, $\frac{15\!\cdots\!65}{31\!\cdots\!91}a^{35}+\frac{71\!\cdots\!00}{31\!\cdots\!91}a^{34}+\frac{61\!\cdots\!80}{31\!\cdots\!91}a^{32}-\frac{23\!\cdots\!80}{31\!\cdots\!91}a^{31}+\frac{36\!\cdots\!10}{31\!\cdots\!91}a^{30}-\frac{21\!\cdots\!20}{31\!\cdots\!91}a^{29}+\frac{18\!\cdots\!20}{31\!\cdots\!91}a^{28}-\frac{12\!\cdots\!20}{31\!\cdots\!91}a^{27}+\frac{83\!\cdots\!40}{31\!\cdots\!91}a^{26}-\frac{73\!\cdots\!80}{31\!\cdots\!91}a^{25}+\frac{28\!\cdots\!00}{31\!\cdots\!91}a^{24}-\frac{42\!\cdots\!80}{31\!\cdots\!91}a^{23}+\frac{83\!\cdots\!00}{31\!\cdots\!91}a^{22}-\frac{23\!\cdots\!60}{31\!\cdots\!91}a^{21}+\frac{21\!\cdots\!80}{31\!\cdots\!91}a^{20}-\frac{37\!\cdots\!20}{31\!\cdots\!91}a^{19}+\frac{49\!\cdots\!20}{31\!\cdots\!91}a^{18}+\frac{58\!\cdots\!60}{31\!\cdots\!91}a^{17}+\frac{77\!\cdots\!40}{31\!\cdots\!91}a^{16}+\frac{60\!\cdots\!55}{31\!\cdots\!91}a^{15}+\frac{11\!\cdots\!00}{31\!\cdots\!91}a^{14}+\frac{20\!\cdots\!20}{31\!\cdots\!91}a^{13}+\frac{13\!\cdots\!80}{31\!\cdots\!91}a^{12}+\frac{16\!\cdots\!60}{31\!\cdots\!91}a^{11}+\frac{12\!\cdots\!86}{31\!\cdots\!91}a^{10}+\frac{41\!\cdots\!40}{31\!\cdots\!91}a^{9}+\frac{30\!\cdots\!60}{31\!\cdots\!91}a^{8}-\frac{11\!\cdots\!00}{31\!\cdots\!91}a^{7}+\frac{12\!\cdots\!20}{31\!\cdots\!91}a^{6}-\frac{17\!\cdots\!80}{31\!\cdots\!91}a^{5}+\frac{38\!\cdots\!20}{31\!\cdots\!91}a^{4}-\frac{44\!\cdots\!60}{31\!\cdots\!91}a^{3}+\frac{51\!\cdots\!40}{31\!\cdots\!91}a^{2}-\frac{57\!\cdots\!40}{31\!\cdots\!91}a-\frac{53\!\cdots\!97}{31\!\cdots\!91}$, $\frac{17\!\cdots\!77}{31\!\cdots\!91}a^{35}+\frac{14\!\cdots\!08}{31\!\cdots\!91}a^{34}+\frac{15\!\cdots\!48}{31\!\cdots\!91}a^{33}+\frac{13\!\cdots\!93}{31\!\cdots\!91}a^{32}+\frac{93\!\cdots\!99}{31\!\cdots\!91}a^{31}+\frac{78\!\cdots\!27}{31\!\cdots\!91}a^{30}+\frac{47\!\cdots\!84}{31\!\cdots\!91}a^{29}+\frac{39\!\cdots\!10}{31\!\cdots\!91}a^{28}+\frac{21\!\cdots\!13}{31\!\cdots\!91}a^{27}+\frac{18\!\cdots\!65}{31\!\cdots\!91}a^{26}+\frac{74\!\cdots\!52}{31\!\cdots\!91}a^{25}+\frac{63\!\cdots\!84}{31\!\cdots\!91}a^{24}+\frac{21\!\cdots\!84}{31\!\cdots\!91}a^{23}+\frac{18\!\cdots\!15}{31\!\cdots\!91}a^{22}+\frac{57\!\cdots\!93}{31\!\cdots\!91}a^{21}+\frac{49\!\cdots\!40}{31\!\cdots\!91}a^{20}+\frac{13\!\cdots\!82}{31\!\cdots\!91}a^{19}+\frac{11\!\cdots\!54}{31\!\cdots\!91}a^{18}+\frac{21\!\cdots\!93}{31\!\cdots\!91}a^{17}+\frac{21\!\cdots\!65}{31\!\cdots\!91}a^{16}+\frac{32\!\cdots\!94}{31\!\cdots\!91}a^{15}+\frac{34\!\cdots\!80}{31\!\cdots\!91}a^{14}+\frac{43\!\cdots\!12}{31\!\cdots\!91}a^{13}+\frac{42\!\cdots\!20}{31\!\cdots\!91}a^{12}+\frac{42\!\cdots\!46}{31\!\cdots\!91}a^{11}+\frac{38\!\cdots\!40}{31\!\cdots\!91}a^{10}+\frac{15\!\cdots\!58}{31\!\cdots\!91}a^{9}+\frac{14\!\cdots\!56}{31\!\cdots\!91}a^{8}+\frac{42\!\cdots\!82}{31\!\cdots\!91}a^{7}+\frac{29\!\cdots\!11}{31\!\cdots\!91}a^{6}-\frac{14\!\cdots\!82}{31\!\cdots\!91}a^{5}+\frac{26\!\cdots\!14}{31\!\cdots\!91}a^{4}-\frac{30\!\cdots\!24}{31\!\cdots\!91}a^{3}+\frac{85\!\cdots\!10}{31\!\cdots\!91}a^{2}-\frac{24\!\cdots\!30}{31\!\cdots\!91}a+\frac{11\!\cdots\!33}{31\!\cdots\!91}$, $\frac{37\!\cdots\!15}{31\!\cdots\!91}a^{35}+\frac{57\!\cdots\!31}{31\!\cdots\!91}a^{34}+\frac{33\!\cdots\!97}{31\!\cdots\!91}a^{33}+\frac{51\!\cdots\!15}{31\!\cdots\!91}a^{32}+\frac{20\!\cdots\!98}{31\!\cdots\!91}a^{31}+\frac{30\!\cdots\!58}{31\!\cdots\!91}a^{30}+\frac{10\!\cdots\!06}{31\!\cdots\!91}a^{29}+\frac{15\!\cdots\!88}{31\!\cdots\!91}a^{28}+\frac{46\!\cdots\!35}{31\!\cdots\!91}a^{27}+\frac{71\!\cdots\!43}{31\!\cdots\!91}a^{26}+\frac{16\!\cdots\!00}{31\!\cdots\!91}a^{25}+\frac{24\!\cdots\!01}{31\!\cdots\!91}a^{24}+\frac{48\!\cdots\!76}{31\!\cdots\!91}a^{23}+\frac{67\!\cdots\!29}{31\!\cdots\!91}a^{22}+\frac{12\!\cdots\!47}{31\!\cdots\!91}a^{21}+\frac{19\!\cdots\!49}{31\!\cdots\!91}a^{20}+\frac{29\!\cdots\!64}{31\!\cdots\!91}a^{19}+\frac{52\!\cdots\!53}{31\!\cdots\!91}a^{18}+\frac{50\!\cdots\!35}{31\!\cdots\!91}a^{17}+\frac{11\!\cdots\!88}{31\!\cdots\!91}a^{16}+\frac{74\!\cdots\!10}{31\!\cdots\!91}a^{15}+\frac{24\!\cdots\!05}{31\!\cdots\!91}a^{14}+\frac{98\!\cdots\!73}{31\!\cdots\!91}a^{13}+\frac{30\!\cdots\!49}{31\!\cdots\!91}a^{12}+\frac{99\!\cdots\!57}{31\!\cdots\!91}a^{11}+\frac{24\!\cdots\!69}{31\!\cdots\!91}a^{10}+\frac{48\!\cdots\!28}{31\!\cdots\!91}a^{9}+\frac{71\!\cdots\!50}{31\!\cdots\!91}a^{8}+\frac{21\!\cdots\!99}{31\!\cdots\!91}a^{7}-\frac{42\!\cdots\!89}{31\!\cdots\!91}a^{6}+\frac{73\!\cdots\!57}{31\!\cdots\!91}a^{5}-\frac{60\!\cdots\!00}{31\!\cdots\!91}a^{4}+\frac{20\!\cdots\!65}{31\!\cdots\!91}a^{3}+\frac{74\!\cdots\!73}{31\!\cdots\!91}a^{2}+\frac{28\!\cdots\!91}{31\!\cdots\!91}a+\frac{45\!\cdots\!44}{31\!\cdots\!91}$, $\frac{41\!\cdots\!37}{31\!\cdots\!91}a^{35}+\frac{48\!\cdots\!89}{31\!\cdots\!91}a^{34}+\frac{37\!\cdots\!51}{31\!\cdots\!91}a^{33}+\frac{43\!\cdots\!72}{31\!\cdots\!91}a^{32}+\frac{22\!\cdots\!72}{31\!\cdots\!91}a^{31}+\frac{26\!\cdots\!60}{31\!\cdots\!91}a^{30}+\frac{11\!\cdots\!10}{31\!\cdots\!91}a^{29}+\frac{13\!\cdots\!36}{31\!\cdots\!91}a^{28}+\frac{52\!\cdots\!04}{31\!\cdots\!91}a^{27}+\frac{60\!\cdots\!53}{31\!\cdots\!91}a^{26}+\frac{18\!\cdots\!98}{31\!\cdots\!91}a^{25}+\frac{20\!\cdots\!25}{31\!\cdots\!91}a^{24}+\frac{54\!\cdots\!05}{31\!\cdots\!91}a^{23}+\frac{55\!\cdots\!30}{31\!\cdots\!91}a^{22}+\frac{14\!\cdots\!65}{31\!\cdots\!91}a^{21}+\frac{16\!\cdots\!25}{31\!\cdots\!91}a^{20}+\frac{33\!\cdots\!70}{31\!\cdots\!91}a^{19}+\frac{46\!\cdots\!46}{31\!\cdots\!91}a^{18}+\frac{55\!\cdots\!14}{31\!\cdots\!91}a^{17}+\frac{11\!\cdots\!08}{31\!\cdots\!91}a^{16}+\frac{82\!\cdots\!00}{31\!\cdots\!91}a^{15}+\frac{24\!\cdots\!09}{31\!\cdots\!91}a^{14}+\frac{10\!\cdots\!26}{31\!\cdots\!91}a^{13}+\frac{29\!\cdots\!59}{31\!\cdots\!91}a^{12}+\frac{10\!\cdots\!87}{31\!\cdots\!91}a^{11}+\frac{23\!\cdots\!05}{31\!\cdots\!91}a^{10}+\frac{52\!\cdots\!04}{31\!\cdots\!91}a^{9}+\frac{60\!\cdots\!72}{31\!\cdots\!91}a^{8}+\frac{23\!\cdots\!49}{31\!\cdots\!91}a^{7}-\frac{55\!\cdots\!03}{31\!\cdots\!91}a^{6}+\frac{84\!\cdots\!87}{31\!\cdots\!91}a^{5}-\frac{97\!\cdots\!54}{31\!\cdots\!91}a^{4}+\frac{23\!\cdots\!99}{31\!\cdots\!91}a^{3}-\frac{12\!\cdots\!89}{31\!\cdots\!91}a^{2}+\frac{28\!\cdots\!73}{31\!\cdots\!91}a-\frac{31\!\cdots\!37}{31\!\cdots\!91}$, $\frac{58\!\cdots\!82}{31\!\cdots\!91}a^{35}+\frac{43\!\cdots\!13}{31\!\cdots\!91}a^{34}+\frac{52\!\cdots\!88}{31\!\cdots\!91}a^{33}+\frac{39\!\cdots\!38}{31\!\cdots\!91}a^{32}+\frac{31\!\cdots\!63}{31\!\cdots\!91}a^{31}+\frac{23\!\cdots\!18}{31\!\cdots\!91}a^{30}+\frac{15\!\cdots\!06}{31\!\cdots\!91}a^{29}+\frac{11\!\cdots\!71}{31\!\cdots\!91}a^{28}+\frac{73\!\cdots\!35}{31\!\cdots\!91}a^{27}+\frac{56\!\cdots\!88}{32\!\cdots\!21}a^{26}+\frac{25\!\cdots\!91}{31\!\cdots\!91}a^{25}+\frac{18\!\cdots\!69}{31\!\cdots\!91}a^{24}+\frac{76\!\cdots\!83}{31\!\cdots\!91}a^{23}+\frac{55\!\cdots\!17}{31\!\cdots\!91}a^{22}+\frac{20\!\cdots\!92}{31\!\cdots\!91}a^{21}+\frac{14\!\cdots\!99}{31\!\cdots\!91}a^{20}+\frac{46\!\cdots\!44}{31\!\cdots\!91}a^{19}+\frac{35\!\cdots\!50}{31\!\cdots\!91}a^{18}+\frac{78\!\cdots\!55}{31\!\cdots\!91}a^{17}+\frac{64\!\cdots\!88}{31\!\cdots\!91}a^{16}+\frac{12\!\cdots\!98}{31\!\cdots\!91}a^{15}+\frac{10\!\cdots\!53}{31\!\cdots\!91}a^{14}+\frac{16\!\cdots\!52}{31\!\cdots\!91}a^{13}+\frac{13\!\cdots\!17}{31\!\cdots\!91}a^{12}+\frac{16\!\cdots\!37}{31\!\cdots\!91}a^{11}+\frac{12\!\cdots\!16}{31\!\cdots\!91}a^{10}+\frac{82\!\cdots\!65}{31\!\cdots\!91}a^{9}+\frac{51\!\cdots\!58}{31\!\cdots\!91}a^{8}+\frac{32\!\cdots\!22}{31\!\cdots\!91}a^{7}+\frac{12\!\cdots\!51}{31\!\cdots\!91}a^{6}+\frac{42\!\cdots\!33}{31\!\cdots\!91}a^{5}+\frac{66\!\cdots\!56}{31\!\cdots\!91}a^{4}+\frac{15\!\cdots\!18}{31\!\cdots\!91}a^{3}+\frac{21\!\cdots\!09}{31\!\cdots\!91}a^{2}-\frac{25\!\cdots\!96}{31\!\cdots\!91}a-\frac{24\!\cdots\!92}{31\!\cdots\!91}$, $\frac{37\!\cdots\!38}{31\!\cdots\!91}a^{35}+\frac{83\!\cdots\!30}{31\!\cdots\!91}a^{34}+\frac{33\!\cdots\!46}{31\!\cdots\!91}a^{33}+\frac{74\!\cdots\!44}{31\!\cdots\!91}a^{32}+\frac{20\!\cdots\!28}{31\!\cdots\!91}a^{31}+\frac{44\!\cdots\!04}{31\!\cdots\!91}a^{30}+\frac{10\!\cdots\!20}{31\!\cdots\!91}a^{29}+\frac{22\!\cdots\!68}{31\!\cdots\!91}a^{28}+\frac{46\!\cdots\!07}{31\!\cdots\!91}a^{27}+\frac{10\!\cdots\!37}{31\!\cdots\!91}a^{26}+\frac{16\!\cdots\!11}{31\!\cdots\!91}a^{25}+\frac{35\!\cdots\!33}{31\!\cdots\!91}a^{24}+\frac{48\!\cdots\!73}{31\!\cdots\!91}a^{23}+\frac{10\!\cdots\!65}{31\!\cdots\!91}a^{22}+\frac{12\!\cdots\!81}{31\!\cdots\!91}a^{21}+\frac{28\!\cdots\!20}{31\!\cdots\!91}a^{20}+\frac{29\!\cdots\!07}{31\!\cdots\!91}a^{19}+\frac{73\!\cdots\!93}{31\!\cdots\!91}a^{18}+\frac{50\!\cdots\!52}{31\!\cdots\!91}a^{17}+\frac{15\!\cdots\!62}{31\!\cdots\!91}a^{16}+\frac{75\!\cdots\!44}{31\!\cdots\!91}a^{15}+\frac{30\!\cdots\!53}{31\!\cdots\!91}a^{14}+\frac{99\!\cdots\!04}{31\!\cdots\!91}a^{13}+\frac{36\!\cdots\!63}{31\!\cdots\!91}a^{12}+\frac{10\!\cdots\!60}{31\!\cdots\!91}a^{11}+\frac{31\!\cdots\!14}{31\!\cdots\!91}a^{10}+\frac{49\!\cdots\!93}{31\!\cdots\!91}a^{9}+\frac{10\!\cdots\!12}{31\!\cdots\!91}a^{8}+\frac{21\!\cdots\!45}{31\!\cdots\!91}a^{7}-\frac{27\!\cdots\!32}{31\!\cdots\!91}a^{6}+\frac{70\!\cdots\!96}{31\!\cdots\!91}a^{5}-\frac{93\!\cdots\!11}{32\!\cdots\!21}a^{4}+\frac{20\!\cdots\!75}{31\!\cdots\!91}a^{3}+\frac{18\!\cdots\!87}{31\!\cdots\!91}a^{2}+\frac{27\!\cdots\!68}{31\!\cdots\!91}a+\frac{13\!\cdots\!74}{31\!\cdots\!91}$, $\frac{37\!\cdots\!84}{31\!\cdots\!91}a^{35}+\frac{97\!\cdots\!99}{31\!\cdots\!91}a^{34}+\frac{33\!\cdots\!63}{31\!\cdots\!91}a^{33}+\frac{87\!\cdots\!18}{31\!\cdots\!91}a^{32}+\frac{20\!\cdots\!59}{31\!\cdots\!91}a^{31}+\frac{52\!\cdots\!30}{31\!\cdots\!91}a^{30}+\frac{10\!\cdots\!74}{31\!\cdots\!91}a^{29}+\frac{26\!\cdots\!11}{31\!\cdots\!91}a^{28}+\frac{47\!\cdots\!89}{31\!\cdots\!91}a^{27}+\frac{12\!\cdots\!85}{31\!\cdots\!91}a^{26}+\frac{16\!\cdots\!02}{31\!\cdots\!91}a^{25}+\frac{41\!\cdots\!32}{31\!\cdots\!91}a^{24}+\frac{48\!\cdots\!95}{31\!\cdots\!91}a^{23}+\frac{11\!\cdots\!00}{31\!\cdots\!91}a^{22}+\frac{12\!\cdots\!80}{31\!\cdots\!91}a^{21}+\frac{33\!\cdots\!16}{31\!\cdots\!91}a^{20}+\frac{29\!\cdots\!35}{31\!\cdots\!91}a^{19}+\frac{84\!\cdots\!01}{31\!\cdots\!91}a^{18}+\frac{50\!\cdots\!39}{31\!\cdots\!91}a^{17}+\frac{17\!\cdots\!60}{31\!\cdots\!91}a^{16}+\frac{75\!\cdots\!21}{31\!\cdots\!91}a^{15}+\frac{32\!\cdots\!25}{31\!\cdots\!91}a^{14}+\frac{10\!\cdots\!38}{31\!\cdots\!91}a^{13}+\frac{40\!\cdots\!72}{31\!\cdots\!91}a^{12}+\frac{10\!\cdots\!39}{31\!\cdots\!91}a^{11}+\frac{35\!\cdots\!71}{31\!\cdots\!91}a^{10}+\frac{50\!\cdots\!53}{31\!\cdots\!91}a^{9}+\frac{12\!\cdots\!05}{31\!\cdots\!91}a^{8}+\frac{21\!\cdots\!17}{31\!\cdots\!91}a^{7}-\frac{19\!\cdots\!89}{31\!\cdots\!91}a^{6}+\frac{68\!\cdots\!82}{31\!\cdots\!91}a^{5}+\frac{18\!\cdots\!45}{31\!\cdots\!91}a^{4}+\frac{20\!\cdots\!82}{31\!\cdots\!91}a^{3}+\frac{28\!\cdots\!62}{31\!\cdots\!91}a^{2}+\frac{27\!\cdots\!92}{31\!\cdots\!91}a+\frac{17\!\cdots\!04}{31\!\cdots\!91}$, $\frac{37\!\cdots\!03}{31\!\cdots\!91}a^{35}+\frac{13\!\cdots\!01}{31\!\cdots\!91}a^{34}+\frac{33\!\cdots\!55}{31\!\cdots\!91}a^{33}+\frac{11\!\cdots\!64}{31\!\cdots\!91}a^{32}+\frac{20\!\cdots\!04}{31\!\cdots\!91}a^{31}+\frac{70\!\cdots\!38}{31\!\cdots\!91}a^{30}+\frac{10\!\cdots\!46}{31\!\cdots\!91}a^{29}+\frac{35\!\cdots\!66}{31\!\cdots\!91}a^{28}+\frac{47\!\cdots\!02}{31\!\cdots\!91}a^{27}+\frac{16\!\cdots\!67}{31\!\cdots\!91}a^{26}+\frac{16\!\cdots\!30}{31\!\cdots\!91}a^{25}+\frac{56\!\cdots\!03}{31\!\cdots\!91}a^{24}+\frac{48\!\cdots\!76}{31\!\cdots\!91}a^{23}+\frac{16\!\cdots\!82}{31\!\cdots\!91}a^{22}+\frac{13\!\cdots\!37}{31\!\cdots\!91}a^{21}+\frac{44\!\cdots\!49}{31\!\cdots\!91}a^{20}+\frac{30\!\cdots\!24}{31\!\cdots\!91}a^{19}+\frac{11\!\cdots\!61}{31\!\cdots\!91}a^{18}+\frac{50\!\cdots\!52}{31\!\cdots\!91}a^{17}+\frac{21\!\cdots\!32}{31\!\cdots\!91}a^{16}+\frac{76\!\cdots\!68}{31\!\cdots\!91}a^{15}+\frac{39\!\cdots\!49}{31\!\cdots\!91}a^{14}+\frac{10\!\cdots\!41}{31\!\cdots\!91}a^{13}+\frac{49\!\cdots\!89}{31\!\cdots\!91}a^{12}+\frac{10\!\cdots\!63}{31\!\cdots\!91}a^{11}+\frac{43\!\cdots\!91}{31\!\cdots\!91}a^{10}+\frac{52\!\cdots\!48}{31\!\cdots\!91}a^{9}+\frac{16\!\cdots\!36}{31\!\cdots\!91}a^{8}+\frac{22\!\cdots\!49}{31\!\cdots\!91}a^{7}-\frac{14\!\cdots\!23}{31\!\cdots\!91}a^{6}+\frac{63\!\cdots\!89}{31\!\cdots\!91}a^{5}+\frac{84\!\cdots\!92}{31\!\cdots\!91}a^{4}+\frac{19\!\cdots\!47}{31\!\cdots\!91}a^{3}+\frac{48\!\cdots\!16}{31\!\cdots\!91}a^{2}+\frac{26\!\cdots\!23}{31\!\cdots\!91}a+\frac{28\!\cdots\!44}{31\!\cdots\!91}$, $\frac{37\!\cdots\!81}{31\!\cdots\!91}a^{35}-\frac{63\!\cdots\!34}{31\!\cdots\!91}a^{34}+\frac{33\!\cdots\!08}{31\!\cdots\!91}a^{33}-\frac{57\!\cdots\!37}{31\!\cdots\!91}a^{32}+\frac{19\!\cdots\!35}{31\!\cdots\!91}a^{31}-\frac{34\!\cdots\!52}{31\!\cdots\!91}a^{30}+\frac{10\!\cdots\!56}{31\!\cdots\!91}a^{29}-\frac{17\!\cdots\!31}{31\!\cdots\!91}a^{28}+\frac{46\!\cdots\!73}{31\!\cdots\!91}a^{27}-\frac{80\!\cdots\!89}{31\!\cdots\!91}a^{26}+\frac{16\!\cdots\!65}{31\!\cdots\!91}a^{25}-\frac{28\!\cdots\!10}{31\!\cdots\!91}a^{24}+\frac{48\!\cdots\!86}{31\!\cdots\!91}a^{23}-\frac{89\!\cdots\!68}{31\!\cdots\!91}a^{22}+\frac{12\!\cdots\!72}{31\!\cdots\!91}a^{21}-\frac{22\!\cdots\!71}{31\!\cdots\!91}a^{20}+\frac{29\!\cdots\!24}{31\!\cdots\!91}a^{19}-\frac{43\!\cdots\!31}{31\!\cdots\!91}a^{18}+\frac{48\!\cdots\!19}{31\!\cdots\!91}a^{17}-\frac{41\!\cdots\!54}{31\!\cdots\!91}a^{16}+\frac{71\!\cdots\!21}{31\!\cdots\!91}a^{15}+\frac{10\!\cdots\!33}{31\!\cdots\!91}a^{14}+\frac{91\!\cdots\!39}{31\!\cdots\!91}a^{13}-\frac{13\!\cdots\!19}{31\!\cdots\!91}a^{12}+\frac{90\!\cdots\!34}{31\!\cdots\!91}a^{11}-\frac{67\!\cdots\!12}{31\!\cdots\!91}a^{10}+\frac{41\!\cdots\!68}{31\!\cdots\!91}a^{9}-\frac{79\!\cdots\!43}{31\!\cdots\!91}a^{8}+\frac{19\!\cdots\!40}{31\!\cdots\!91}a^{7}-\frac{10\!\cdots\!72}{31\!\cdots\!91}a^{6}+\frac{90\!\cdots\!91}{31\!\cdots\!91}a^{5}-\frac{29\!\cdots\!46}{31\!\cdots\!91}a^{4}+\frac{23\!\cdots\!54}{31\!\cdots\!91}a^{3}-\frac{59\!\cdots\!73}{31\!\cdots\!91}a^{2}+\frac{31\!\cdots\!15}{31\!\cdots\!91}a-\frac{35\!\cdots\!06}{31\!\cdots\!91}$ (assuming GRH)
Regulator:		\( 6550249244897.024 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 6550249244897.024 \cdot 2053}{10\cdot\sqrt{7225377334561374804949923918873673793376691639423370361328125}}\cr\approx \mathstrut & 0.116534398629132 \end{aligned}\] (assuming GRH)

Galois group

$C_{36}$ (as 36T1):

A cyclic group of order 36

The 36 conjugacy class representatives for $C_{36}$

Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{5})\), 6.6.820125.1, \(\Q(\zeta_{27})^+\), 12.0.84075626953125.1, 18.18.1923380668327365689220703125.1

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	$36$	R	R	$36$	${\href{/padicField/11.9.0.1}{9} }^{4}$	$36$	${\href{/padicField/17.12.0.1}{12} }^{3}$	${\href{/padicField/19.6.0.1}{6} }^{6}$	$36$	$18^{2}$	${\href{/padicField/31.9.0.1}{9} }^{4}$	${\href{/padicField/37.12.0.1}{12} }^{3}$	${\href{/padicField/41.9.0.1}{9} }^{4}$	$36$	$36$	${\href{/padicField/53.4.0.1}{4} }^{9}$	$18^{2}$

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

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(3\)		Deg $36$	$9$	$4$	$88$
\(5\)		Deg $36$	$4$	$9$	$27$