Number field 36.0.619876750267203693326033178758188478035934269428253173828125.1

• Label: 36.0.619...125.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $6.199\times 10^{59}$
• Root discriminant: \(45.80\)
• Ramified primes: $5,19$
• Class number: $1417$ (GRH)
• Class group: [1417] (GRH)
• Galois group: $C_{36}$ (as 36T1)

Normalized defining polynomial

x^36 - x^35 + 9*x^34 - 10*x^33 + 54*x^32 - 49*x^31 + 257*x^30 - 206*x^29 + 1100*x^28 - 836*x^27 + 3655*x^26 - 2571*x^25 + 10339*x^24 - 5625*x^23 + 24829*x^22 - 12149*x^21 + 52298*x^20 - 24437*x^19 + 84375*x^18 - 31001*x^17 + 114806*x^16 - 20386*x^15 + 122457*x^14 - 26351*x^13 + 111183*x^12 - 31899*x^11 + 59771*x^10 - 9196*x^9 + 26744*x^8 + 4893*x^7 + 5542*x^6 + 509*x^5 + 1260*x^4 - 205*x^3 + 35*x^2 - 5*x + 1

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(619876750267203693326033178758188478035934269428253173828125\) \(\medspace = 5^{27}\cdot 19^{32}\)
Root discriminant:		\(45.80\)
Galois root discriminant:		$5^{3/4}19^{8/9}\approx 45.80336348303902$
Ramified primes:		\(5\), \(19\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$36$
This field is Galois and abelian over $\Q$.
Conductor:		\(95=5\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{95}(1,·)$, $\chi_{95}(4,·)$, $\chi_{95}(6,·)$, $\chi_{95}(7,·)$, $\chi_{95}(9,·)$, $\chi_{95}(11,·)$, $\chi_{95}(16,·)$, $\chi_{95}(17,·)$, $\chi_{95}(23,·)$, $\chi_{95}(24,·)$, $\chi_{95}(26,·)$, $\chi_{95}(28,·)$, $\chi_{95}(36,·)$, $\chi_{95}(39,·)$, $\chi_{95}(42,·)$, $\chi_{95}(43,·)$, $\chi_{95}(44,·)$, $\chi_{95}(47,·)$, $\chi_{95}(49,·)$, $\chi_{95}(54,·)$, $\chi_{95}(58,·)$, $\chi_{95}(61,·)$, $\chi_{95}(62,·)$, $\chi_{95}(63,·)$, $\chi_{95}(64,·)$, $\chi_{95}(66,·)$, $\chi_{95}(68,·)$, $\chi_{95}(73,·)$, $\chi_{95}(74,·)$, $\chi_{95}(77,·)$, $\chi_{95}(81,·)$, $\chi_{95}(82,·)$, $\chi_{95}(83,·)$, $\chi_{95}(87,·)$, $\chi_{95}(92,·)$, $\chi_{95}(93,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{69\!\cdots\!81}a^{33}+\frac{13\!\cdots\!95}{69\!\cdots\!81}a^{32}-\frac{26\!\cdots\!80}{69\!\cdots\!81}a^{31}-\frac{80\!\cdots\!15}{69\!\cdots\!81}a^{30}+\frac{25\!\cdots\!81}{69\!\cdots\!81}a^{29}+\frac{10\!\cdots\!02}{69\!\cdots\!81}a^{28}+\frac{34\!\cdots\!38}{69\!\cdots\!81}a^{27}-\frac{16\!\cdots\!11}{69\!\cdots\!81}a^{26}-\frac{27\!\cdots\!35}{69\!\cdots\!81}a^{25}+\frac{90\!\cdots\!71}{69\!\cdots\!81}a^{24}-\frac{29\!\cdots\!09}{69\!\cdots\!81}a^{23}-\frac{29\!\cdots\!60}{69\!\cdots\!81}a^{22}-\frac{10\!\cdots\!06}{69\!\cdots\!81}a^{21}-\frac{20\!\cdots\!91}{69\!\cdots\!81}a^{20}+\frac{29\!\cdots\!55}{69\!\cdots\!81}a^{19}-\frac{71\!\cdots\!96}{69\!\cdots\!81}a^{18}-\frac{84\!\cdots\!35}{69\!\cdots\!81}a^{17}-\frac{60\!\cdots\!28}{69\!\cdots\!81}a^{16}-\frac{25\!\cdots\!50}{69\!\cdots\!81}a^{15}+\frac{23\!\cdots\!30}{69\!\cdots\!81}a^{14}-\frac{23\!\cdots\!02}{69\!\cdots\!81}a^{13}-\frac{12\!\cdots\!22}{69\!\cdots\!81}a^{12}+\frac{65\!\cdots\!18}{69\!\cdots\!81}a^{11}+\frac{12\!\cdots\!13}{69\!\cdots\!81}a^{10}+\frac{85\!\cdots\!46}{69\!\cdots\!81}a^{9}+\frac{31\!\cdots\!18}{69\!\cdots\!81}a^{8}+\frac{23\!\cdots\!33}{69\!\cdots\!81}a^{7}+\frac{30\!\cdots\!24}{69\!\cdots\!81}a^{6}+\frac{48\!\cdots\!06}{69\!\cdots\!81}a^{5}-\frac{25\!\cdots\!85}{69\!\cdots\!81}a^{4}+\frac{13\!\cdots\!98}{69\!\cdots\!81}a^{3}+\frac{21\!\cdots\!53}{69\!\cdots\!81}a^{2}-\frac{23\!\cdots\!53}{69\!\cdots\!81}a+\frac{31\!\cdots\!98}{69\!\cdots\!81}$, $\frac{1}{69\!\cdots\!81}a^{34}-\frac{11\!\cdots\!27}{69\!\cdots\!81}a^{32}-\frac{24\!\cdots\!73}{69\!\cdots\!81}a^{31}-\frac{85\!\cdots\!83}{69\!\cdots\!81}a^{30}-\frac{74\!\cdots\!56}{69\!\cdots\!81}a^{29}+\frac{23\!\cdots\!12}{69\!\cdots\!81}a^{28}+\frac{32\!\cdots\!07}{69\!\cdots\!81}a^{27}-\frac{67\!\cdots\!52}{69\!\cdots\!81}a^{26}-\frac{27\!\cdots\!56}{69\!\cdots\!81}a^{25}-\frac{13\!\cdots\!54}{69\!\cdots\!81}a^{24}-\frac{24\!\cdots\!04}{69\!\cdots\!81}a^{23}+\frac{34\!\cdots\!30}{69\!\cdots\!81}a^{22}-\frac{17\!\cdots\!11}{69\!\cdots\!81}a^{21}+\frac{19\!\cdots\!94}{69\!\cdots\!81}a^{20}-\frac{66\!\cdots\!55}{69\!\cdots\!81}a^{19}-\frac{22\!\cdots\!65}{69\!\cdots\!81}a^{18}+\frac{81\!\cdots\!11}{69\!\cdots\!81}a^{17}-\frac{15\!\cdots\!22}{69\!\cdots\!81}a^{16}+\frac{25\!\cdots\!02}{69\!\cdots\!81}a^{15}+\frac{15\!\cdots\!58}{69\!\cdots\!81}a^{14}+\frac{12\!\cdots\!82}{69\!\cdots\!81}a^{13}+\frac{28\!\cdots\!53}{69\!\cdots\!81}a^{12}-\frac{12\!\cdots\!72}{69\!\cdots\!81}a^{11}-\frac{14\!\cdots\!61}{69\!\cdots\!81}a^{10}-\frac{29\!\cdots\!86}{69\!\cdots\!81}a^{9}-\frac{29\!\cdots\!26}{69\!\cdots\!81}a^{8}-\frac{32\!\cdots\!04}{69\!\cdots\!81}a^{7}+\frac{16\!\cdots\!82}{69\!\cdots\!81}a^{6}-\frac{14\!\cdots\!01}{69\!\cdots\!81}a^{5}+\frac{16\!\cdots\!46}{69\!\cdots\!81}a^{4}-\frac{18\!\cdots\!99}{69\!\cdots\!81}a^{3}+\frac{51\!\cdots\!29}{69\!\cdots\!81}a^{2}-\frac{15\!\cdots\!89}{69\!\cdots\!81}a-\frac{11\!\cdots\!31}{69\!\cdots\!81}$, $\frac{1}{69\!\cdots\!81}a^{35}-\frac{22\!\cdots\!28}{69\!\cdots\!81}a^{32}+\frac{18\!\cdots\!24}{69\!\cdots\!81}a^{31}+\frac{74\!\cdots\!05}{69\!\cdots\!81}a^{30}-\frac{17\!\cdots\!64}{69\!\cdots\!81}a^{29}+\frac{19\!\cdots\!60}{69\!\cdots\!81}a^{28}-\frac{26\!\cdots\!29}{69\!\cdots\!81}a^{27}-\frac{31\!\cdots\!75}{69\!\cdots\!81}a^{26}+\frac{21\!\cdots\!32}{69\!\cdots\!81}a^{25}-\frac{53\!\cdots\!88}{69\!\cdots\!81}a^{24}-\frac{18\!\cdots\!91}{69\!\cdots\!81}a^{23}+\frac{27\!\cdots\!68}{69\!\cdots\!81}a^{22}+\frac{33\!\cdots\!26}{69\!\cdots\!81}a^{21}+\frac{26\!\cdots\!57}{69\!\cdots\!81}a^{20}-\frac{84\!\cdots\!82}{69\!\cdots\!81}a^{19}-\frac{29\!\cdots\!07}{69\!\cdots\!81}a^{18}+\frac{75\!\cdots\!41}{69\!\cdots\!81}a^{17}-\frac{19\!\cdots\!07}{69\!\cdots\!81}a^{16}+\frac{28\!\cdots\!89}{69\!\cdots\!81}a^{15}+\frac{10\!\cdots\!29}{69\!\cdots\!81}a^{14}+\frac{23\!\cdots\!32}{69\!\cdots\!81}a^{13}-\frac{12\!\cdots\!15}{69\!\cdots\!81}a^{12}+\frac{80\!\cdots\!35}{69\!\cdots\!81}a^{11}-\frac{84\!\cdots\!20}{69\!\cdots\!81}a^{10}-\frac{48\!\cdots\!13}{69\!\cdots\!81}a^{9}-\frac{20\!\cdots\!59}{69\!\cdots\!81}a^{8}-\frac{44\!\cdots\!38}{69\!\cdots\!81}a^{7}+\frac{27\!\cdots\!45}{69\!\cdots\!81}a^{6}-\frac{17\!\cdots\!50}{69\!\cdots\!81}a^{5}-\frac{25\!\cdots\!42}{69\!\cdots\!81}a^{4}+\frac{27\!\cdots\!54}{69\!\cdots\!81}a^{3}-\frac{13\!\cdots\!62}{69\!\cdots\!81}a^{2}+\frac{17\!\cdots\!20}{12\!\cdots\!11}a+\frac{22\!\cdots\!98}{69\!\cdots\!81}$

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

Class group and class number

$C_{1417}$, which has order $1417$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( \frac{98018614603376248755638426887}{698211859637920482850137709081} a^{35} - \frac{77957495843925714929975481484}{698211859637920482850137709081} a^{34} + \frac{862411637926344248365493007986}{698211859637920482850137709081} a^{33} - \frac{800350758196794064491879104033}{698211859637920482850137709081} a^{32} + \frac{5095354822265598531637471164998}{698211859637920482850137709081} a^{31} - \frac{3726043831537884794738635959811}{698211859637920482850137709081} a^{30} + \frac{24226493356547394782218122554116}{698211859637920482850137709081} a^{29} - \frac{15070265556555738508634675258685}{698211859637920482850137709081} a^{28} + \frac{103774749969720191272627890150758}{698211859637920482850137709081} a^{27} - \frac{60029068712046560540659652121352}{698211859637920482850137709081} a^{26} + \frac{341851612176577391061422317593523}{698211859637920482850137709081} a^{25} - \frac{179320454596751802972625993660648}{698211859637920482850137709081} a^{24} + \frac{963058691935447486156170852747226}{698211859637920482850137709081} a^{23} - \frac{345944131698702167858601037752092}{698211859637920482850137709081} a^{22} + \frac{2324267885866910053028056340284198}{698211859637920482850137709081} a^{21} - \frac{697794779012632487408312034781440}{698211859637920482850137709081} a^{20} + \frac{4890058057468305769685686441449209}{698211859637920482850137709081} a^{19} - \frac{1357823927921002608813853637536163}{698211859637920482850137709081} a^{18} + \frac{7795633891756236022988070847970870}{698211859637920482850137709081} a^{17} - \frac{1369960091523807512821314650702732}{698211859637920482850137709081} a^{16} + \frac{10655621153439239851104224729587305}{698211859637920482850137709081} a^{15} + \frac{271057445897344586221999627297660}{698211859637920482850137709081} a^{14} + \frac{11623160278901377609099406021338138}{698211859637920482850137709081} a^{13} - \frac{164947979001725573592594889658194}{698211859637920482850137709081} a^{12} + \frac{10390321101751810372790451390091558}{698211859637920482850137709081} a^{11} - \frac{939070558154886093810413548555968}{698211859637920482850137709081} a^{10} + \frac{5238814762249825672048685538080554}{698211859637920482850137709081} a^{9} + \frac{259458766280017392596327653538553}{698211859637920482850137709081} a^{8} + \frac{2445369531531812359813664767472238}{698211859637920482850137709081} a^{7} + \frac{1006220166043971741659192684974359}{698211859637920482850137709081} a^{6} + \frac{640133813668034625377434461857485}{698211859637920482850137709081} a^{5} + \frac{158975791703915596210541900674723}{698211859637920482850137709081} a^{4} + \frac{128864394822851049184824277119801}{698211859637920482850137709081} a^{3} + \frac{4619392013553739456590457599211}{698211859637920482850137709081} a^{2} - \frac{589806502558576625488581748024}{698211859637920482850137709081} a + \frac{196486457262692508655909698442}{698211859637920482850137709081} \)  (order $10$)
Fundamental units:		$\frac{27\!\cdots\!40}{69\!\cdots\!81}a^{35}-\frac{41\!\cdots\!85}{69\!\cdots\!81}a^{34}+\frac{26\!\cdots\!36}{69\!\cdots\!81}a^{33}-\frac{40\!\cdots\!77}{69\!\cdots\!81}a^{32}+\frac{16\!\cdots\!14}{69\!\cdots\!81}a^{31}-\frac{20\!\cdots\!75}{69\!\cdots\!81}a^{30}+\frac{77\!\cdots\!84}{69\!\cdots\!81}a^{29}-\frac{92\!\cdots\!24}{69\!\cdots\!81}a^{28}+\frac{32\!\cdots\!62}{69\!\cdots\!81}a^{27}-\frac{38\!\cdots\!76}{69\!\cdots\!81}a^{26}+\frac{11\!\cdots\!20}{69\!\cdots\!81}a^{25}-\frac{12\!\cdots\!56}{69\!\cdots\!81}a^{24}+\frac{31\!\cdots\!60}{69\!\cdots\!81}a^{23}-\frac{29\!\cdots\!79}{69\!\cdots\!81}a^{22}+\frac{74\!\cdots\!19}{69\!\cdots\!81}a^{21}-\frac{67\!\cdots\!65}{69\!\cdots\!81}a^{20}+\frac{15\!\cdots\!74}{69\!\cdots\!81}a^{19}-\frac{13\!\cdots\!84}{69\!\cdots\!81}a^{18}+\frac{25\!\cdots\!50}{69\!\cdots\!81}a^{17}-\frac{19\!\cdots\!12}{69\!\cdots\!81}a^{16}+\frac{34\!\cdots\!85}{69\!\cdots\!81}a^{15}-\frac{20\!\cdots\!16}{69\!\cdots\!81}a^{14}+\frac{33\!\cdots\!36}{69\!\cdots\!81}a^{13}-\frac{23\!\cdots\!55}{69\!\cdots\!81}a^{12}+\frac{31\!\cdots\!32}{69\!\cdots\!81}a^{11}-\frac{23\!\cdots\!51}{69\!\cdots\!81}a^{10}+\frac{18\!\cdots\!68}{69\!\cdots\!81}a^{9}-\frac{10\!\cdots\!29}{69\!\cdots\!81}a^{8}+\frac{70\!\cdots\!18}{69\!\cdots\!81}a^{7}-\frac{21\!\cdots\!11}{69\!\cdots\!81}a^{6}+\frac{11\!\cdots\!09}{69\!\cdots\!81}a^{5}-\frac{76\!\cdots\!97}{69\!\cdots\!81}a^{4}+\frac{12\!\cdots\!72}{69\!\cdots\!81}a^{3}-\frac{27\!\cdots\!39}{69\!\cdots\!81}a^{2}+\frac{32\!\cdots\!10}{69\!\cdots\!81}a-\frac{75\!\cdots\!86}{69\!\cdots\!81}$, $\frac{37\!\cdots\!70}{69\!\cdots\!81}a^{35}-\frac{86\!\cdots\!40}{69\!\cdots\!81}a^{34}+\frac{36\!\cdots\!80}{69\!\cdots\!81}a^{33}-\frac{78\!\cdots\!60}{69\!\cdots\!81}a^{32}+\frac{22\!\cdots\!64}{69\!\cdots\!81}a^{31}-\frac{41\!\cdots\!85}{69\!\cdots\!81}a^{30}+\frac{10\!\cdots\!16}{69\!\cdots\!81}a^{29}-\frac{18\!\cdots\!20}{69\!\cdots\!81}a^{28}+\frac{44\!\cdots\!36}{69\!\cdots\!81}a^{27}-\frac{77\!\cdots\!76}{69\!\cdots\!81}a^{26}+\frac{14\!\cdots\!77}{69\!\cdots\!81}a^{25}-\frac{24\!\cdots\!44}{69\!\cdots\!81}a^{24}+\frac{41\!\cdots\!00}{69\!\cdots\!81}a^{23}-\frac{61\!\cdots\!24}{69\!\cdots\!81}a^{22}+\frac{92\!\cdots\!80}{69\!\cdots\!81}a^{21}-\frac{14\!\cdots\!38}{69\!\cdots\!81}a^{20}+\frac{18\!\cdots\!96}{69\!\cdots\!81}a^{19}-\frac{29\!\cdots\!20}{69\!\cdots\!81}a^{18}+\frac{29\!\cdots\!76}{69\!\cdots\!81}a^{17}-\frac{41\!\cdots\!16}{69\!\cdots\!81}a^{16}+\frac{35\!\cdots\!52}{69\!\cdots\!81}a^{15}-\frac{47\!\cdots\!48}{69\!\cdots\!81}a^{14}+\frac{25\!\cdots\!40}{69\!\cdots\!81}a^{13}-\frac{52\!\cdots\!64}{69\!\cdots\!81}a^{12}+\frac{24\!\cdots\!84}{69\!\cdots\!81}a^{11}-\frac{45\!\cdots\!79}{69\!\cdots\!81}a^{10}+\frac{10\!\cdots\!68}{69\!\cdots\!81}a^{9}-\frac{12\!\cdots\!24}{69\!\cdots\!81}a^{8}-\frac{15\!\cdots\!68}{69\!\cdots\!81}a^{7}-\frac{25\!\cdots\!76}{69\!\cdots\!81}a^{6}-\frac{70\!\cdots\!32}{69\!\cdots\!81}a^{5}-\frac{68\!\cdots\!44}{69\!\cdots\!81}a^{4}+\frac{11\!\cdots\!36}{69\!\cdots\!81}a^{3}-\frac{18\!\cdots\!48}{69\!\cdots\!81}a^{2}+\frac{27\!\cdots\!48}{69\!\cdots\!81}a+\frac{74\!\cdots\!25}{69\!\cdots\!81}$, $\frac{31\!\cdots\!40}{69\!\cdots\!81}a^{35}-\frac{73\!\cdots\!00}{69\!\cdots\!81}a^{34}+\frac{30\!\cdots\!00}{69\!\cdots\!81}a^{33}-\frac{66\!\cdots\!00}{69\!\cdots\!81}a^{32}+\frac{19\!\cdots\!60}{69\!\cdots\!81}a^{31}-\frac{35\!\cdots\!69}{69\!\cdots\!81}a^{30}+\frac{89\!\cdots\!40}{69\!\cdots\!81}a^{29}-\frac{15\!\cdots\!00}{69\!\cdots\!81}a^{28}+\frac{37\!\cdots\!40}{69\!\cdots\!81}a^{27}-\frac{65\!\cdots\!40}{69\!\cdots\!81}a^{26}+\frac{12\!\cdots\!20}{69\!\cdots\!81}a^{25}-\frac{20\!\cdots\!60}{69\!\cdots\!81}a^{24}+\frac{35\!\cdots\!00}{69\!\cdots\!81}a^{23}-\frac{52\!\cdots\!60}{69\!\cdots\!81}a^{22}+\frac{78\!\cdots\!00}{69\!\cdots\!81}a^{21}-\frac{12\!\cdots\!31}{69\!\cdots\!81}a^{20}+\frac{16\!\cdots\!40}{69\!\cdots\!81}a^{19}-\frac{24\!\cdots\!00}{69\!\cdots\!81}a^{18}+\frac{25\!\cdots\!40}{69\!\cdots\!81}a^{17}-\frac{34\!\cdots\!40}{69\!\cdots\!81}a^{16}+\frac{30\!\cdots\!35}{69\!\cdots\!81}a^{15}-\frac{39\!\cdots\!20}{69\!\cdots\!81}a^{14}+\frac{21\!\cdots\!00}{69\!\cdots\!81}a^{13}-\frac{44\!\cdots\!60}{69\!\cdots\!81}a^{12}+\frac{20\!\cdots\!60}{69\!\cdots\!81}a^{11}-\frac{38\!\cdots\!81}{69\!\cdots\!81}a^{10}+\frac{87\!\cdots\!20}{69\!\cdots\!81}a^{9}-\frac{10\!\cdots\!60}{69\!\cdots\!81}a^{8}-\frac{12\!\cdots\!20}{69\!\cdots\!81}a^{7}-\frac{21\!\cdots\!40}{69\!\cdots\!81}a^{6}-\frac{54\!\cdots\!85}{69\!\cdots\!81}a^{5}-\frac{58\!\cdots\!60}{69\!\cdots\!81}a^{4}+\frac{94\!\cdots\!40}{69\!\cdots\!81}a^{3}-\frac{16\!\cdots\!20}{69\!\cdots\!81}a^{2}+\frac{23\!\cdots\!20}{69\!\cdots\!81}a+\frac{17\!\cdots\!24}{69\!\cdots\!81}$, $\frac{30\!\cdots\!06}{69\!\cdots\!81}a^{35}-\frac{71\!\cdots\!90}{69\!\cdots\!81}a^{34}+\frac{29\!\cdots\!30}{69\!\cdots\!81}a^{33}-\frac{64\!\cdots\!10}{69\!\cdots\!81}a^{32}+\frac{18\!\cdots\!04}{69\!\cdots\!81}a^{31}-\frac{34\!\cdots\!34}{69\!\cdots\!81}a^{30}+\frac{86\!\cdots\!76}{69\!\cdots\!81}a^{29}-\frac{15\!\cdots\!20}{69\!\cdots\!81}a^{28}+\frac{36\!\cdots\!46}{69\!\cdots\!81}a^{27}-\frac{63\!\cdots\!36}{69\!\cdots\!81}a^{26}+\frac{12\!\cdots\!46}{69\!\cdots\!81}a^{25}-\frac{20\!\cdots\!34}{69\!\cdots\!81}a^{24}+\frac{34\!\cdots\!50}{69\!\cdots\!81}a^{23}-\frac{50\!\cdots\!64}{69\!\cdots\!81}a^{22}+\frac{76\!\cdots\!30}{69\!\cdots\!81}a^{21}-\frac{11\!\cdots\!38}{69\!\cdots\!81}a^{20}+\frac{15\!\cdots\!56}{69\!\cdots\!81}a^{19}-\frac{23\!\cdots\!70}{69\!\cdots\!81}a^{18}+\frac{24\!\cdots\!86}{69\!\cdots\!81}a^{17}-\frac{33\!\cdots\!76}{69\!\cdots\!81}a^{16}+\frac{29\!\cdots\!37}{69\!\cdots\!81}a^{15}-\frac{38\!\cdots\!28}{69\!\cdots\!81}a^{14}+\frac{20\!\cdots\!90}{69\!\cdots\!81}a^{13}-\frac{42\!\cdots\!04}{69\!\cdots\!81}a^{12}+\frac{20\!\cdots\!74}{69\!\cdots\!81}a^{11}-\frac{38\!\cdots\!08}{69\!\cdots\!81}a^{10}+\frac{84\!\cdots\!98}{69\!\cdots\!81}a^{9}-\frac{98\!\cdots\!64}{69\!\cdots\!81}a^{8}-\frac{12\!\cdots\!48}{69\!\cdots\!81}a^{7}-\frac{20\!\cdots\!86}{69\!\cdots\!81}a^{6}-\frac{48\!\cdots\!46}{69\!\cdots\!81}a^{5}-\frac{56\!\cdots\!34}{69\!\cdots\!81}a^{4}+\frac{92\!\cdots\!46}{69\!\cdots\!81}a^{3}-\frac{15\!\cdots\!28}{69\!\cdots\!81}a^{2}+\frac{22\!\cdots\!28}{69\!\cdots\!81}a+\frac{13\!\cdots\!59}{69\!\cdots\!81}$, $\frac{20\!\cdots\!45}{69\!\cdots\!81}a^{35}-\frac{46\!\cdots\!75}{69\!\cdots\!81}a^{34}+\frac{19\!\cdots\!25}{69\!\cdots\!81}a^{33}-\frac{42\!\cdots\!75}{69\!\cdots\!81}a^{32}+\frac{12\!\cdots\!60}{69\!\cdots\!81}a^{31}-\frac{22\!\cdots\!45}{69\!\cdots\!81}a^{30}+\frac{57\!\cdots\!90}{69\!\cdots\!81}a^{29}-\frac{10\!\cdots\!00}{69\!\cdots\!81}a^{28}+\frac{23\!\cdots\!15}{69\!\cdots\!81}a^{27}-\frac{41\!\cdots\!40}{69\!\cdots\!81}a^{26}+\frac{80\!\cdots\!80}{69\!\cdots\!81}a^{25}-\frac{13\!\cdots\!35}{69\!\cdots\!81}a^{24}+\frac{22\!\cdots\!75}{69\!\cdots\!81}a^{23}-\frac{33\!\cdots\!10}{69\!\cdots\!81}a^{22}+\frac{49\!\cdots\!25}{69\!\cdots\!81}a^{21}-\frac{76\!\cdots\!10}{69\!\cdots\!81}a^{20}+\frac{10\!\cdots\!90}{69\!\cdots\!81}a^{19}-\frac{15\!\cdots\!25}{69\!\cdots\!81}a^{18}+\frac{16\!\cdots\!65}{69\!\cdots\!81}a^{17}-\frac{22\!\cdots\!40}{69\!\cdots\!81}a^{16}+\frac{19\!\cdots\!54}{69\!\cdots\!81}a^{15}-\frac{25\!\cdots\!70}{69\!\cdots\!81}a^{14}+\frac{13\!\cdots\!25}{69\!\cdots\!81}a^{13}-\frac{28\!\cdots\!60}{69\!\cdots\!81}a^{12}+\frac{13\!\cdots\!35}{69\!\cdots\!81}a^{11}-\frac{24\!\cdots\!20}{69\!\cdots\!81}a^{10}+\frac{55\!\cdots\!45}{69\!\cdots\!81}a^{9}-\frac{65\!\cdots\!10}{69\!\cdots\!81}a^{8}-\frac{81\!\cdots\!70}{69\!\cdots\!81}a^{7}-\frac{13\!\cdots\!65}{69\!\cdots\!81}a^{6}-\frac{44\!\cdots\!83}{69\!\cdots\!81}a^{5}-\frac{37\!\cdots\!35}{69\!\cdots\!81}a^{4}+\frac{60\!\cdots\!65}{69\!\cdots\!81}a^{3}-\frac{10\!\cdots\!70}{69\!\cdots\!81}a^{2}+\frac{15\!\cdots\!20}{69\!\cdots\!81}a-\frac{76\!\cdots\!65}{69\!\cdots\!81}$, $\frac{31\!\cdots\!05}{69\!\cdots\!81}a^{35}-\frac{73\!\cdots\!75}{69\!\cdots\!81}a^{34}+\frac{30\!\cdots\!25}{69\!\cdots\!81}a^{33}-\frac{65\!\cdots\!75}{69\!\cdots\!81}a^{32}+\frac{19\!\cdots\!00}{69\!\cdots\!81}a^{31}-\frac{34\!\cdots\!00}{69\!\cdots\!81}a^{30}+\frac{88\!\cdots\!50}{69\!\cdots\!81}a^{29}-\frac{15\!\cdots\!00}{69\!\cdots\!81}a^{28}+\frac{37\!\cdots\!75}{69\!\cdots\!81}a^{27}-\frac{65\!\cdots\!00}{69\!\cdots\!81}a^{26}+\frac{12\!\cdots\!40}{69\!\cdots\!81}a^{25}-\frac{20\!\cdots\!75}{69\!\cdots\!81}a^{24}+\frac{34\!\cdots\!75}{69\!\cdots\!81}a^{23}-\frac{51\!\cdots\!50}{69\!\cdots\!81}a^{22}+\frac{77\!\cdots\!25}{69\!\cdots\!81}a^{21}-\frac{11\!\cdots\!90}{69\!\cdots\!81}a^{20}+\frac{15\!\cdots\!50}{69\!\cdots\!81}a^{19}-\frac{24\!\cdots\!25}{69\!\cdots\!81}a^{18}+\frac{24\!\cdots\!25}{69\!\cdots\!81}a^{17}-\frac{34\!\cdots\!00}{69\!\cdots\!81}a^{16}+\frac{29\!\cdots\!20}{69\!\cdots\!81}a^{15}-\frac{39\!\cdots\!50}{69\!\cdots\!81}a^{14}+\frac{21\!\cdots\!25}{69\!\cdots\!81}a^{13}-\frac{43\!\cdots\!00}{69\!\cdots\!81}a^{12}+\frac{20\!\cdots\!75}{69\!\cdots\!81}a^{11}-\frac{38\!\cdots\!40}{69\!\cdots\!81}a^{10}+\frac{86\!\cdots\!25}{69\!\cdots\!81}a^{9}-\frac{10\!\cdots\!50}{69\!\cdots\!81}a^{8}-\frac{12\!\cdots\!50}{69\!\cdots\!81}a^{7}-\frac{21\!\cdots\!25}{69\!\cdots\!81}a^{6}-\frac{62\!\cdots\!91}{69\!\cdots\!81}a^{5}-\frac{57\!\cdots\!75}{69\!\cdots\!81}a^{4}+\frac{94\!\cdots\!25}{69\!\cdots\!81}a^{3}-\frac{15\!\cdots\!50}{69\!\cdots\!81}a^{2}+\frac{23\!\cdots\!00}{69\!\cdots\!81}a+\frac{54\!\cdots\!70}{69\!\cdots\!81}$, $\frac{11\!\cdots\!60}{69\!\cdots\!81}a^{35}-\frac{26\!\cdots\!00}{69\!\cdots\!81}a^{34}+\frac{10\!\cdots\!00}{69\!\cdots\!81}a^{33}-\frac{23\!\cdots\!00}{69\!\cdots\!81}a^{32}+\frac{68\!\cdots\!40}{69\!\cdots\!81}a^{31}-\frac{12\!\cdots\!55}{69\!\cdots\!81}a^{30}+\frac{31\!\cdots\!60}{69\!\cdots\!81}a^{29}-\frac{56\!\cdots\!00}{69\!\cdots\!81}a^{28}+\frac{13\!\cdots\!60}{69\!\cdots\!81}a^{27}-\frac{23\!\cdots\!60}{69\!\cdots\!81}a^{26}+\frac{44\!\cdots\!60}{69\!\cdots\!81}a^{25}-\frac{73\!\cdots\!40}{69\!\cdots\!81}a^{24}+\frac{12\!\cdots\!00}{69\!\cdots\!81}a^{23}-\frac{18\!\cdots\!40}{69\!\cdots\!81}a^{22}+\frac{27\!\cdots\!00}{69\!\cdots\!81}a^{21}-\frac{42\!\cdots\!80}{69\!\cdots\!81}a^{20}+\frac{57\!\cdots\!60}{69\!\cdots\!81}a^{19}-\frac{87\!\cdots\!00}{69\!\cdots\!81}a^{18}+\frac{89\!\cdots\!60}{69\!\cdots\!81}a^{17}-\frac{12\!\cdots\!60}{69\!\cdots\!81}a^{16}+\frac{10\!\cdots\!66}{69\!\cdots\!81}a^{15}-\frac{14\!\cdots\!80}{69\!\cdots\!81}a^{14}+\frac{76\!\cdots\!00}{69\!\cdots\!81}a^{13}-\frac{15\!\cdots\!40}{69\!\cdots\!81}a^{12}+\frac{73\!\cdots\!40}{69\!\cdots\!81}a^{11}-\frac{14\!\cdots\!20}{69\!\cdots\!81}a^{10}+\frac{31\!\cdots\!80}{69\!\cdots\!81}a^{9}-\frac{36\!\cdots\!40}{69\!\cdots\!81}a^{8}-\frac{45\!\cdots\!80}{69\!\cdots\!81}a^{7}-\frac{76\!\cdots\!60}{69\!\cdots\!81}a^{6}-\frac{18\!\cdots\!08}{69\!\cdots\!81}a^{5}-\frac{20\!\cdots\!40}{69\!\cdots\!81}a^{4}+\frac{33\!\cdots\!60}{69\!\cdots\!81}a^{3}-\frac{57\!\cdots\!80}{69\!\cdots\!81}a^{2}+\frac{84\!\cdots\!80}{69\!\cdots\!81}a+\frac{13\!\cdots\!35}{69\!\cdots\!81}$, $\frac{68\!\cdots\!70}{69\!\cdots\!81}a^{35}-\frac{16\!\cdots\!75}{69\!\cdots\!81}a^{34}+\frac{66\!\cdots\!25}{69\!\cdots\!81}a^{33}-\frac{14\!\cdots\!75}{69\!\cdots\!81}a^{32}+\frac{41\!\cdots\!60}{69\!\cdots\!81}a^{31}-\frac{76\!\cdots\!75}{69\!\cdots\!81}a^{30}+\frac{19\!\cdots\!90}{69\!\cdots\!81}a^{29}-\frac{34\!\cdots\!00}{69\!\cdots\!81}a^{28}+\frac{81\!\cdots\!15}{69\!\cdots\!81}a^{27}-\frac{14\!\cdots\!40}{69\!\cdots\!81}a^{26}+\frac{27\!\cdots\!75}{69\!\cdots\!81}a^{25}-\frac{44\!\cdots\!35}{69\!\cdots\!81}a^{24}+\frac{76\!\cdots\!75}{69\!\cdots\!81}a^{23}-\frac{11\!\cdots\!10}{69\!\cdots\!81}a^{22}+\frac{17\!\cdots\!25}{69\!\cdots\!81}a^{21}-\frac{26\!\cdots\!20}{69\!\cdots\!81}a^{20}+\frac{34\!\cdots\!90}{69\!\cdots\!81}a^{19}-\frac{53\!\cdots\!25}{69\!\cdots\!81}a^{18}+\frac{54\!\cdots\!65}{69\!\cdots\!81}a^{17}-\frac{75\!\cdots\!40}{69\!\cdots\!81}a^{16}+\frac{65\!\cdots\!20}{69\!\cdots\!81}a^{15}-\frac{86\!\cdots\!70}{69\!\cdots\!81}a^{14}+\frac{46\!\cdots\!25}{69\!\cdots\!81}a^{13}-\frac{96\!\cdots\!60}{69\!\cdots\!81}a^{12}+\frac{44\!\cdots\!35}{69\!\cdots\!81}a^{11}-\frac{84\!\cdots\!01}{69\!\cdots\!81}a^{10}+\frac{18\!\cdots\!45}{69\!\cdots\!81}a^{9}-\frac{22\!\cdots\!10}{69\!\cdots\!81}a^{8}-\frac{27\!\cdots\!70}{69\!\cdots\!81}a^{7}-\frac{46\!\cdots\!65}{69\!\cdots\!81}a^{6}-\frac{12\!\cdots\!16}{69\!\cdots\!81}a^{5}-\frac{12\!\cdots\!35}{69\!\cdots\!81}a^{4}+\frac{20\!\cdots\!65}{69\!\cdots\!81}a^{3}-\frac{34\!\cdots\!70}{69\!\cdots\!81}a^{2}+\frac{51\!\cdots\!20}{69\!\cdots\!81}a+\frac{78\!\cdots\!16}{69\!\cdots\!81}$, $\frac{21\!\cdots\!75}{69\!\cdots\!81}a^{35}-\frac{51\!\cdots\!00}{69\!\cdots\!81}a^{34}+\frac{21\!\cdots\!00}{69\!\cdots\!81}a^{33}-\frac{45\!\cdots\!00}{69\!\cdots\!81}a^{32}+\frac{13\!\cdots\!00}{69\!\cdots\!81}a^{31}-\frac{24\!\cdots\!75}{69\!\cdots\!81}a^{30}+\frac{62\!\cdots\!00}{69\!\cdots\!81}a^{29}-\frac{10\!\cdots\!00}{69\!\cdots\!81}a^{28}+\frac{26\!\cdots\!00}{69\!\cdots\!81}a^{27}-\frac{45\!\cdots\!00}{69\!\cdots\!81}a^{26}+\frac{87\!\cdots\!69}{69\!\cdots\!81}a^{25}-\frac{14\!\cdots\!00}{69\!\cdots\!81}a^{24}+\frac{24\!\cdots\!00}{69\!\cdots\!81}a^{23}-\frac{36\!\cdots\!00}{69\!\cdots\!81}a^{22}+\frac{54\!\cdots\!00}{69\!\cdots\!81}a^{21}-\frac{83\!\cdots\!00}{69\!\cdots\!81}a^{20}+\frac{11\!\cdots\!00}{69\!\cdots\!81}a^{19}-\frac{17\!\cdots\!00}{69\!\cdots\!81}a^{18}+\frac{17\!\cdots\!00}{69\!\cdots\!81}a^{17}-\frac{24\!\cdots\!00}{69\!\cdots\!81}a^{16}+\frac{20\!\cdots\!80}{69\!\cdots\!81}a^{15}-\frac{27\!\cdots\!00}{69\!\cdots\!81}a^{14}+\frac{14\!\cdots\!00}{69\!\cdots\!81}a^{13}-\frac{30\!\cdots\!00}{69\!\cdots\!81}a^{12}+\frac{14\!\cdots\!00}{69\!\cdots\!81}a^{11}-\frac{27\!\cdots\!50}{69\!\cdots\!81}a^{10}+\frac{60\!\cdots\!00}{69\!\cdots\!81}a^{9}-\frac{70\!\cdots\!00}{69\!\cdots\!81}a^{8}-\frac{88\!\cdots\!00}{69\!\cdots\!81}a^{7}-\frac{14\!\cdots\!00}{69\!\cdots\!81}a^{6}-\frac{36\!\cdots\!05}{69\!\cdots\!81}a^{5}-\frac{40\!\cdots\!00}{69\!\cdots\!81}a^{4}+\frac{65\!\cdots\!00}{69\!\cdots\!81}a^{3}-\frac{11\!\cdots\!00}{69\!\cdots\!81}a^{2}+\frac{16\!\cdots\!00}{69\!\cdots\!81}a+\frac{12\!\cdots\!25}{69\!\cdots\!81}$, $\frac{10\!\cdots\!79}{69\!\cdots\!81}a^{35}-\frac{10\!\cdots\!06}{69\!\cdots\!81}a^{34}+\frac{89\!\cdots\!65}{69\!\cdots\!81}a^{33}-\frac{10\!\cdots\!55}{69\!\cdots\!81}a^{32}+\frac{53\!\cdots\!55}{69\!\cdots\!81}a^{31}-\frac{49\!\cdots\!49}{69\!\cdots\!81}a^{30}+\frac{25\!\cdots\!64}{69\!\cdots\!81}a^{29}-\frac{20\!\cdots\!26}{69\!\cdots\!81}a^{28}+\frac{10\!\cdots\!35}{69\!\cdots\!81}a^{27}-\frac{83\!\cdots\!36}{69\!\cdots\!81}a^{26}+\frac{36\!\cdots\!78}{69\!\cdots\!81}a^{25}-\frac{25\!\cdots\!12}{69\!\cdots\!81}a^{24}+\frac{10\!\cdots\!59}{69\!\cdots\!81}a^{23}-\frac{56\!\cdots\!76}{69\!\cdots\!81}a^{22}+\frac{24\!\cdots\!39}{69\!\cdots\!81}a^{21}-\frac{12\!\cdots\!48}{69\!\cdots\!81}a^{20}+\frac{51\!\cdots\!78}{69\!\cdots\!81}a^{19}-\frac{24\!\cdots\!51}{69\!\cdots\!81}a^{18}+\frac{83\!\cdots\!85}{69\!\cdots\!81}a^{17}-\frac{30\!\cdots\!17}{69\!\cdots\!81}a^{16}+\frac{11\!\cdots\!06}{69\!\cdots\!81}a^{15}-\frac{20\!\cdots\!44}{69\!\cdots\!81}a^{14}+\frac{11\!\cdots\!53}{69\!\cdots\!81}a^{13}-\frac{26\!\cdots\!06}{69\!\cdots\!81}a^{12}+\frac{10\!\cdots\!99}{69\!\cdots\!81}a^{11}-\frac{32\!\cdots\!42}{69\!\cdots\!81}a^{10}+\frac{57\!\cdots\!93}{69\!\cdots\!81}a^{9}-\frac{87\!\cdots\!18}{69\!\cdots\!81}a^{8}+\frac{25\!\cdots\!94}{69\!\cdots\!81}a^{7}+\frac{48\!\cdots\!71}{69\!\cdots\!81}a^{6}+\frac{48\!\cdots\!65}{69\!\cdots\!81}a^{5}+\frac{28\!\cdots\!12}{69\!\cdots\!81}a^{4}+\frac{11\!\cdots\!83}{69\!\cdots\!81}a^{3}-\frac{19\!\cdots\!78}{69\!\cdots\!81}a^{2}-\frac{65\!\cdots\!55}{69\!\cdots\!81}a-\frac{12\!\cdots\!29}{69\!\cdots\!81}$, $\frac{12\!\cdots\!72}{69\!\cdots\!81}a^{35}-\frac{12\!\cdots\!59}{69\!\cdots\!81}a^{34}+\frac{11\!\cdots\!52}{69\!\cdots\!81}a^{33}-\frac{12\!\cdots\!20}{69\!\cdots\!81}a^{32}+\frac{67\!\cdots\!56}{69\!\cdots\!81}a^{31}-\frac{58\!\cdots\!76}{69\!\cdots\!81}a^{30}+\frac{32\!\cdots\!36}{69\!\cdots\!81}a^{29}-\frac{24\!\cdots\!29}{69\!\cdots\!81}a^{28}+\frac{13\!\cdots\!26}{69\!\cdots\!81}a^{27}-\frac{98\!\cdots\!24}{69\!\cdots\!81}a^{26}+\frac{45\!\cdots\!40}{69\!\cdots\!81}a^{25}-\frac{30\!\cdots\!78}{69\!\cdots\!81}a^{24}+\frac{12\!\cdots\!36}{69\!\cdots\!81}a^{23}-\frac{64\!\cdots\!75}{69\!\cdots\!81}a^{22}+\frac{30\!\cdots\!47}{69\!\cdots\!81}a^{21}-\frac{13\!\cdots\!62}{69\!\cdots\!81}a^{20}+\frac{64\!\cdots\!99}{69\!\cdots\!81}a^{19}-\frac{27\!\cdots\!17}{69\!\cdots\!81}a^{18}+\frac{10\!\cdots\!66}{69\!\cdots\!81}a^{17}-\frac{33\!\cdots\!80}{69\!\cdots\!81}a^{16}+\frac{14\!\cdots\!08}{69\!\cdots\!81}a^{15}-\frac{18\!\cdots\!64}{69\!\cdots\!81}a^{14}+\frac{15\!\cdots\!64}{69\!\cdots\!81}a^{13}-\frac{26\!\cdots\!93}{69\!\cdots\!81}a^{12}+\frac{13\!\cdots\!04}{69\!\cdots\!81}a^{11}-\frac{33\!\cdots\!86}{69\!\cdots\!81}a^{10}+\frac{70\!\cdots\!00}{69\!\cdots\!81}a^{9}-\frac{75\!\cdots\!80}{69\!\cdots\!81}a^{8}+\frac{31\!\cdots\!28}{69\!\cdots\!81}a^{7}+\frac{79\!\cdots\!02}{69\!\cdots\!81}a^{6}+\frac{64\!\cdots\!59}{69\!\cdots\!81}a^{5}+\frac{81\!\cdots\!52}{69\!\cdots\!81}a^{4}+\frac{14\!\cdots\!79}{69\!\cdots\!81}a^{3}-\frac{23\!\cdots\!36}{69\!\cdots\!81}a^{2}-\frac{26\!\cdots\!06}{69\!\cdots\!81}a+\frac{46\!\cdots\!06}{69\!\cdots\!81}$, $\frac{12\!\cdots\!26}{69\!\cdots\!81}a^{35}-\frac{11\!\cdots\!94}{69\!\cdots\!81}a^{34}+\frac{11\!\cdots\!97}{69\!\cdots\!81}a^{33}-\frac{11\!\cdots\!35}{69\!\cdots\!81}a^{32}+\frac{65\!\cdots\!12}{69\!\cdots\!81}a^{31}-\frac{55\!\cdots\!36}{69\!\cdots\!81}a^{30}+\frac{31\!\cdots\!50}{69\!\cdots\!81}a^{29}-\frac{23\!\cdots\!09}{69\!\cdots\!81}a^{28}+\frac{13\!\cdots\!45}{69\!\cdots\!81}a^{27}-\frac{93\!\cdots\!28}{69\!\cdots\!81}a^{26}+\frac{44\!\cdots\!05}{69\!\cdots\!81}a^{25}-\frac{28\!\cdots\!29}{69\!\cdots\!81}a^{24}+\frac{12\!\cdots\!11}{69\!\cdots\!81}a^{23}-\frac{60\!\cdots\!21}{69\!\cdots\!81}a^{22}+\frac{30\!\cdots\!92}{69\!\cdots\!81}a^{21}-\frac{12\!\cdots\!65}{69\!\cdots\!81}a^{20}+\frac{63\!\cdots\!33}{69\!\cdots\!81}a^{19}-\frac{25\!\cdots\!22}{69\!\cdots\!81}a^{18}+\frac{10\!\cdots\!95}{69\!\cdots\!81}a^{17}-\frac{31\!\cdots\!44}{69\!\cdots\!81}a^{16}+\frac{13\!\cdots\!06}{69\!\cdots\!81}a^{15}-\frac{15\!\cdots\!06}{69\!\cdots\!81}a^{14}+\frac{14\!\cdots\!49}{69\!\cdots\!81}a^{13}-\frac{22\!\cdots\!49}{69\!\cdots\!81}a^{12}+\frac{13\!\cdots\!15}{69\!\cdots\!81}a^{11}-\frac{30\!\cdots\!69}{69\!\cdots\!81}a^{10}+\frac{69\!\cdots\!97}{69\!\cdots\!81}a^{9}-\frac{67\!\cdots\!26}{69\!\cdots\!81}a^{8}+\frac{31\!\cdots\!06}{69\!\cdots\!81}a^{7}+\frac{80\!\cdots\!73}{69\!\cdots\!81}a^{6}+\frac{69\!\cdots\!16}{69\!\cdots\!81}a^{5}+\frac{85\!\cdots\!01}{69\!\cdots\!81}a^{4}+\frac{14\!\cdots\!48}{69\!\cdots\!81}a^{3}-\frac{22\!\cdots\!78}{69\!\cdots\!81}a^{2}-\frac{28\!\cdots\!14}{69\!\cdots\!81}a-\frac{89\!\cdots\!24}{69\!\cdots\!81}$, $\frac{12\!\cdots\!02}{69\!\cdots\!81}a^{35}-\frac{12\!\cdots\!69}{69\!\cdots\!81}a^{34}+\frac{11\!\cdots\!22}{69\!\cdots\!81}a^{33}-\frac{12\!\cdots\!10}{69\!\cdots\!81}a^{32}+\frac{68\!\cdots\!12}{69\!\cdots\!81}a^{31}-\frac{60\!\cdots\!61}{69\!\cdots\!81}a^{30}+\frac{32\!\cdots\!00}{69\!\cdots\!81}a^{29}-\frac{25\!\cdots\!09}{69\!\cdots\!81}a^{28}+\frac{13\!\cdots\!20}{69\!\cdots\!81}a^{27}-\frac{10\!\cdots\!28}{69\!\cdots\!81}a^{26}+\frac{46\!\cdots\!12}{69\!\cdots\!81}a^{25}-\frac{31\!\cdots\!04}{69\!\cdots\!81}a^{24}+\frac{13\!\cdots\!86}{69\!\cdots\!81}a^{23}-\frac{67\!\cdots\!71}{69\!\cdots\!81}a^{22}+\frac{31\!\cdots\!17}{69\!\cdots\!81}a^{21}-\frac{14\!\cdots\!05}{69\!\cdots\!81}a^{20}+\frac{65\!\cdots\!83}{69\!\cdots\!81}a^{19}-\frac{29\!\cdots\!47}{69\!\cdots\!81}a^{18}+\frac{10\!\cdots\!20}{69\!\cdots\!81}a^{17}-\frac{35\!\cdots\!44}{69\!\cdots\!81}a^{16}+\frac{14\!\cdots\!70}{69\!\cdots\!81}a^{15}-\frac{20\!\cdots\!56}{69\!\cdots\!81}a^{14}+\frac{15\!\cdots\!74}{69\!\cdots\!81}a^{13}-\frac{28\!\cdots\!49}{69\!\cdots\!81}a^{12}+\frac{13\!\cdots\!90}{69\!\cdots\!81}a^{11}-\frac{35\!\cdots\!69}{69\!\cdots\!81}a^{10}+\frac{71\!\cdots\!22}{69\!\cdots\!81}a^{9}-\frac{81\!\cdots\!76}{69\!\cdots\!81}a^{8}+\frac{31\!\cdots\!56}{69\!\cdots\!81}a^{7}+\frac{78\!\cdots\!48}{69\!\cdots\!81}a^{6}+\frac{61\!\cdots\!89}{69\!\cdots\!81}a^{5}+\frac{78\!\cdots\!26}{69\!\cdots\!81}a^{4}+\frac{14\!\cdots\!73}{69\!\cdots\!81}a^{3}-\frac{23\!\cdots\!28}{69\!\cdots\!81}a^{2}-\frac{25\!\cdots\!14}{69\!\cdots\!81}a+\frac{66\!\cdots\!81}{69\!\cdots\!81}$, $\frac{12\!\cdots\!92}{69\!\cdots\!81}a^{35}-\frac{12\!\cdots\!69}{69\!\cdots\!81}a^{34}+\frac{11\!\cdots\!22}{69\!\cdots\!81}a^{33}-\frac{12\!\cdots\!10}{69\!\cdots\!81}a^{32}+\frac{69\!\cdots\!72}{69\!\cdots\!81}a^{31}-\frac{62\!\cdots\!61}{69\!\cdots\!81}a^{30}+\frac{32\!\cdots\!40}{69\!\cdots\!81}a^{29}-\frac{26\!\cdots\!09}{69\!\cdots\!81}a^{28}+\frac{14\!\cdots\!60}{69\!\cdots\!81}a^{27}-\frac{10\!\cdots\!68}{69\!\cdots\!81}a^{26}+\frac{46\!\cdots\!78}{69\!\cdots\!81}a^{25}-\frac{32\!\cdots\!64}{69\!\cdots\!81}a^{24}+\frac{13\!\cdots\!86}{69\!\cdots\!81}a^{23}-\frac{70\!\cdots\!31}{69\!\cdots\!81}a^{22}+\frac{31\!\cdots\!17}{69\!\cdots\!81}a^{21}-\frac{15\!\cdots\!35}{69\!\cdots\!81}a^{20}+\frac{66\!\cdots\!23}{69\!\cdots\!81}a^{19}-\frac{30\!\cdots\!47}{69\!\cdots\!81}a^{18}+\frac{10\!\cdots\!60}{69\!\cdots\!81}a^{17}-\frac{37\!\cdots\!84}{69\!\cdots\!81}a^{16}+\frac{14\!\cdots\!90}{69\!\cdots\!81}a^{15}-\frac{22\!\cdots\!76}{69\!\cdots\!81}a^{14}+\frac{15\!\cdots\!74}{69\!\cdots\!81}a^{13}-\frac{30\!\cdots\!09}{69\!\cdots\!81}a^{12}+\frac{13\!\cdots\!50}{69\!\cdots\!81}a^{11}-\frac{37\!\cdots\!80}{69\!\cdots\!81}a^{10}+\frac{71\!\cdots\!42}{69\!\cdots\!81}a^{9}-\frac{86\!\cdots\!36}{69\!\cdots\!81}a^{8}+\frac{31\!\cdots\!36}{69\!\cdots\!81}a^{7}+\frac{76\!\cdots\!08}{69\!\cdots\!81}a^{6}+\frac{58\!\cdots\!69}{69\!\cdots\!81}a^{5}+\frac{75\!\cdots\!66}{69\!\cdots\!81}a^{4}+\frac{14\!\cdots\!13}{69\!\cdots\!81}a^{3}-\frac{23\!\cdots\!48}{69\!\cdots\!81}a^{2}-\frac{24\!\cdots\!94}{69\!\cdots\!81}a+\frac{37\!\cdots\!83}{69\!\cdots\!81}$, $\frac{12\!\cdots\!22}{69\!\cdots\!81}a^{35}-\frac{11\!\cdots\!94}{69\!\cdots\!81}a^{34}+\frac{10\!\cdots\!97}{69\!\cdots\!81}a^{33}-\frac{11\!\cdots\!35}{69\!\cdots\!81}a^{32}+\frac{65\!\cdots\!12}{69\!\cdots\!81}a^{31}-\frac{54\!\cdots\!86}{69\!\cdots\!81}a^{30}+\frac{31\!\cdots\!50}{69\!\cdots\!81}a^{29}-\frac{22\!\cdots\!09}{69\!\cdots\!81}a^{28}+\frac{13\!\cdots\!45}{69\!\cdots\!81}a^{27}-\frac{91\!\cdots\!28}{69\!\cdots\!81}a^{26}+\frac{43\!\cdots\!03}{69\!\cdots\!81}a^{25}-\frac{27\!\cdots\!29}{69\!\cdots\!81}a^{24}+\frac{12\!\cdots\!11}{69\!\cdots\!81}a^{23}-\frac{58\!\cdots\!21}{69\!\cdots\!81}a^{22}+\frac{29\!\cdots\!92}{69\!\cdots\!81}a^{21}-\frac{12\!\cdots\!15}{69\!\cdots\!81}a^{20}+\frac{62\!\cdots\!33}{69\!\cdots\!81}a^{19}-\frac{24\!\cdots\!22}{69\!\cdots\!81}a^{18}+\frac{10\!\cdots\!95}{69\!\cdots\!81}a^{17}-\frac{29\!\cdots\!44}{69\!\cdots\!81}a^{16}+\frac{13\!\cdots\!70}{69\!\cdots\!81}a^{15}-\frac{14\!\cdots\!06}{69\!\cdots\!81}a^{14}+\frac{14\!\cdots\!49}{69\!\cdots\!81}a^{13}-\frac{21\!\cdots\!49}{69\!\cdots\!81}a^{12}+\frac{13\!\cdots\!15}{69\!\cdots\!81}a^{11}-\frac{29\!\cdots\!79}{69\!\cdots\!81}a^{10}+\frac{69\!\cdots\!97}{69\!\cdots\!81}a^{9}-\frac{64\!\cdots\!26}{69\!\cdots\!81}a^{8}+\frac{31\!\cdots\!06}{69\!\cdots\!81}a^{7}+\frac{81\!\cdots\!73}{69\!\cdots\!81}a^{6}+\frac{71\!\cdots\!85}{69\!\cdots\!81}a^{5}+\frac{87\!\cdots\!01}{69\!\cdots\!81}a^{4}+\frac{14\!\cdots\!48}{69\!\cdots\!81}a^{3}-\frac{22\!\cdots\!78}{69\!\cdots\!81}a^{2}-\frac{29\!\cdots\!14}{69\!\cdots\!81}a-\frac{40\!\cdots\!33}{69\!\cdots\!81}$, $\frac{12\!\cdots\!21}{69\!\cdots\!81}a^{35}-\frac{11\!\cdots\!79}{69\!\cdots\!81}a^{34}+\frac{11\!\cdots\!92}{69\!\cdots\!81}a^{33}-\frac{11\!\cdots\!00}{69\!\cdots\!81}a^{32}+\frac{66\!\cdots\!08}{69\!\cdots\!81}a^{31}-\frac{57\!\cdots\!52}{69\!\cdots\!81}a^{30}+\frac{31\!\cdots\!24}{69\!\cdots\!81}a^{29}-\frac{24\!\cdots\!89}{69\!\cdots\!81}a^{28}+\frac{13\!\cdots\!74}{69\!\cdots\!81}a^{27}-\frac{97\!\cdots\!92}{69\!\cdots\!81}a^{26}+\frac{45\!\cdots\!97}{69\!\cdots\!81}a^{25}-\frac{29\!\cdots\!70}{69\!\cdots\!81}a^{24}+\frac{12\!\cdots\!36}{69\!\cdots\!81}a^{23}-\frac{63\!\cdots\!07}{69\!\cdots\!81}a^{22}+\frac{30\!\cdots\!87}{69\!\cdots\!81}a^{21}-\frac{13\!\cdots\!67}{69\!\cdots\!81}a^{20}+\frac{64\!\cdots\!27}{69\!\cdots\!81}a^{19}-\frac{27\!\cdots\!77}{69\!\cdots\!81}a^{18}+\frac{10\!\cdots\!34}{69\!\cdots\!81}a^{17}-\frac{33\!\cdots\!68}{69\!\cdots\!81}a^{16}+\frac{14\!\cdots\!53}{69\!\cdots\!81}a^{15}-\frac{17\!\cdots\!28}{69\!\cdots\!81}a^{14}+\frac{14\!\cdots\!84}{69\!\cdots\!81}a^{13}-\frac{25\!\cdots\!45}{69\!\cdots\!81}a^{12}+\frac{13\!\cdots\!16}{69\!\cdots\!81}a^{11}-\frac{32\!\cdots\!11}{69\!\cdots\!81}a^{10}+\frac{70\!\cdots\!24}{69\!\cdots\!81}a^{9}-\frac{73\!\cdots\!12}{69\!\cdots\!81}a^{8}+\frac{31\!\cdots\!04}{69\!\cdots\!81}a^{7}+\frac{79\!\cdots\!34}{69\!\cdots\!81}a^{6}+\frac{65\!\cdots\!40}{69\!\cdots\!81}a^{5}+\frac{82\!\cdots\!60}{69\!\cdots\!81}a^{4}+\frac{14\!\cdots\!27}{69\!\cdots\!81}a^{3}-\frac{22\!\cdots\!00}{69\!\cdots\!81}a^{2}-\frac{27\!\cdots\!42}{69\!\cdots\!81}a-\frac{12\!\cdots\!22}{69\!\cdots\!81}$, $\frac{12\!\cdots\!97}{69\!\cdots\!81}a^{35}-\frac{12\!\cdots\!09}{69\!\cdots\!81}a^{34}+\frac{11\!\cdots\!02}{69\!\cdots\!81}a^{33}-\frac{12\!\cdots\!70}{69\!\cdots\!81}a^{32}+\frac{69\!\cdots\!76}{69\!\cdots\!81}a^{31}-\frac{62\!\cdots\!71}{69\!\cdots\!81}a^{30}+\frac{32\!\cdots\!16}{69\!\cdots\!81}a^{29}-\frac{26\!\cdots\!29}{69\!\cdots\!81}a^{28}+\frac{14\!\cdots\!56}{69\!\cdots\!81}a^{27}-\frac{10\!\cdots\!04}{69\!\cdots\!81}a^{26}+\frac{46\!\cdots\!20}{69\!\cdots\!81}a^{25}-\frac{32\!\cdots\!48}{69\!\cdots\!81}a^{24}+\frac{13\!\cdots\!86}{69\!\cdots\!81}a^{23}-\frac{70\!\cdots\!95}{69\!\cdots\!81}a^{22}+\frac{31\!\cdots\!97}{69\!\cdots\!81}a^{21}-\frac{15\!\cdots\!43}{69\!\cdots\!81}a^{20}+\frac{66\!\cdots\!79}{69\!\cdots\!81}a^{19}-\frac{30\!\cdots\!67}{69\!\cdots\!81}a^{18}+\frac{10\!\cdots\!96}{69\!\cdots\!81}a^{17}-\frac{37\!\cdots\!60}{69\!\cdots\!81}a^{16}+\frac{14\!\cdots\!42}{69\!\cdots\!81}a^{15}-\frac{22\!\cdots\!04}{69\!\cdots\!81}a^{14}+\frac{15\!\cdots\!14}{69\!\cdots\!81}a^{13}-\frac{30\!\cdots\!13}{69\!\cdots\!81}a^{12}+\frac{13\!\cdots\!74}{69\!\cdots\!81}a^{11}-\frac{37\!\cdots\!98}{69\!\cdots\!81}a^{10}+\frac{71\!\cdots\!90}{69\!\cdots\!81}a^{9}-\frac{86\!\cdots\!00}{69\!\cdots\!81}a^{8}+\frac{31\!\cdots\!88}{69\!\cdots\!81}a^{7}+\frac{76\!\cdots\!72}{69\!\cdots\!81}a^{6}+\frac{58\!\cdots\!62}{69\!\cdots\!81}a^{5}+\frac{75\!\cdots\!82}{69\!\cdots\!81}a^{4}+\frac{14\!\cdots\!09}{69\!\cdots\!81}a^{3}-\frac{23\!\cdots\!76}{69\!\cdots\!81}a^{2}-\frac{24\!\cdots\!66}{69\!\cdots\!81}a+\frac{89\!\cdots\!62}{69\!\cdots\!81}$ (assuming GRH)
Regulator:		\( 3431432369157.3267 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 3431432369157.3267 \cdot 1417}{10\cdot\sqrt{619876750267203693326033178758188478035934269428253173828125}}\cr\approx \mathstrut & 0.143856752017006 \end{aligned}\] (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 $C_{36}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\zeta_{5})\), 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.33171021564453125.1, 18.18.563362135874260093126953125.1

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	$36$	$36$	R	${\href{/padicField/7.12.0.1}{12} }^{3}$	${\href{/padicField/11.3.0.1}{3} }^{12}$	$36$	$36$	R	$36$	$18^{2}$	${\href{/padicField/31.3.0.1}{3} }^{12}$	${\href{/padicField/37.4.0.1}{4} }^{9}$	${\href{/padicField/41.9.0.1}{9} }^{4}$	$36$	$36$	$36$	$18^{2}$

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
\(5\)		Deg $36$	$4$	$9$	$27$
\(19\)	19.18.16.1	$x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$	$9$	$2$	$16$	$C_{18}$	$[\ ]_{9}^{2}$
	19.18.16.1	$x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$	$9$	$2$	$16$	$C_{18}$	$[\ ]_{9}^{2}$