Number field 36.0.122958312298624478991860638998897557972401876087680816650390625.1

• Label: 36.0.122...625.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $1.230\times 10^{62}$
• Root discriminant: \(53.05\)
• Ramified primes: $3,5,19$
• Class number: $14661$ (GRH)
• Class group: [9, 1629] (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

x^36 - x^35 + 26*x^34 - 15*x^33 + 413*x^32 - 173*x^31 + 4027*x^30 - 789*x^29 + 28017*x^28 - 2298*x^27 + 134511*x^26 + 7347*x^25 + 472642*x^24 + 48836*x^23 + 1180619*x^22 + 171615*x^21 + 2210278*x^20 + 335012*x^19 + 3064502*x^18 + 471070*x^17 + 3227734*x^16 + 424487*x^15 + 2498777*x^14 + 259210*x^13 + 1437594*x^12 + 75846*x^11 + 567447*x^10 - 2642*x^9 + 158238*x^8 - 9682*x^7 + 24598*x^6 - 3269*x^5 + 2670*x^4 - 215*x^3 + 80*x^2 + 5*x + 1

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(122958312298624478991860638998897557972401876087680816650390625\) \(\medspace = 3^{18}\cdot 5^{18}\cdot 19^{32}\)
Root discriminant:		\(53.05\)
Galois root discriminant:		$3^{1/2}5^{1/2}19^{8/9}\approx 53.05367798232217$
Ramified primes:		\(3\), \(5\), \(19\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$36$
This field is Galois and abelian over $\Q$.
Conductor:		\(285=3\cdot 5\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(131,·)$, $\chi_{285}(4,·)$, $\chi_{285}(134,·)$, $\chi_{285}(11,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(149,·)$, $\chi_{285}(26,·)$, $\chi_{285}(161,·)$, $\chi_{285}(169,·)$, $\chi_{285}(44,·)$, $\chi_{285}(176,·)$, $\chi_{285}(49,·)$, $\chi_{285}(61,·)$, $\chi_{285}(191,·)$, $\chi_{285}(64,·)$, $\chi_{285}(194,·)$, $\chi_{285}(139,·)$, $\chi_{285}(196,·)$, $\chi_{285}(199,·)$, $\chi_{285}(74,·)$, $\chi_{285}(206,·)$, $\chi_{285}(214,·)$, $\chi_{285}(101,·)$, $\chi_{285}(226,·)$, $\chi_{285}(229,·)$, $\chi_{285}(104,·)$, $\chi_{285}(106,·)$, $\chi_{285}(239,·)$, $\chi_{285}(244,·)$, $\chi_{285}(119,·)$, $\chi_{285}(121,·)$, $\chi_{285}(251,·)$, $\chi_{285}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{21}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{22}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{23}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{24}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{25}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{74}a^{32}-\frac{2}{37}a^{31}-\frac{15}{74}a^{30}+\frac{4}{37}a^{29}-\frac{5}{74}a^{28}-\frac{5}{74}a^{27}+\frac{11}{74}a^{26}+\frac{2}{37}a^{25}-\frac{3}{74}a^{24}+\frac{25}{74}a^{22}+\frac{29}{74}a^{21}-\frac{14}{37}a^{20}-\frac{9}{37}a^{19}+\frac{18}{37}a^{18}+\frac{12}{37}a^{17}-\frac{6}{37}a^{16}-\frac{17}{37}a^{15}+\frac{18}{37}a^{14}+\frac{17}{37}a^{13}-\frac{1}{37}a^{12}-\frac{33}{74}a^{11}-\frac{10}{37}a^{10}+\frac{19}{74}a^{9}+\frac{21}{74}a^{8}+\frac{27}{74}a^{7}-\frac{7}{37}a^{6}+\frac{1}{74}a^{5}+\frac{31}{74}a^{4}-\frac{14}{37}a^{3}-\frac{11}{37}a^{2}-\frac{35}{74}a+\frac{7}{74}$, $\frac{1}{74}a^{33}+\frac{3}{37}a^{31}-\frac{15}{74}a^{30}-\frac{5}{37}a^{29}+\frac{6}{37}a^{28}-\frac{9}{74}a^{27}-\frac{13}{37}a^{26}-\frac{12}{37}a^{25}+\frac{25}{74}a^{24}-\frac{6}{37}a^{23}+\frac{9}{37}a^{22}+\frac{7}{37}a^{21}+\frac{9}{37}a^{20}-\frac{18}{37}a^{19}+\frac{10}{37}a^{18}+\frac{5}{37}a^{17}-\frac{4}{37}a^{16}-\frac{13}{37}a^{15}+\frac{15}{37}a^{14}-\frac{7}{37}a^{13}+\frac{33}{74}a^{12}-\frac{2}{37}a^{11}-\frac{12}{37}a^{10}-\frac{7}{37}a^{9}+\frac{10}{37}a^{7}-\frac{9}{37}a^{6}-\frac{1}{37}a^{5}+\frac{11}{37}a^{4}-\frac{23}{74}a^{3}-\frac{6}{37}a^{2}-\frac{11}{37}a+\frac{14}{37}$, $\frac{1}{19\!\cdots\!18}a^{34}-\frac{17\!\cdots\!39}{26\!\cdots\!57}a^{33}+\frac{36\!\cdots\!01}{96\!\cdots\!09}a^{32}-\frac{18\!\cdots\!04}{96\!\cdots\!09}a^{31}+\frac{20\!\cdots\!62}{96\!\cdots\!09}a^{30}+\frac{11\!\cdots\!61}{96\!\cdots\!09}a^{29}-\frac{12\!\cdots\!90}{96\!\cdots\!09}a^{28}+\frac{22\!\cdots\!49}{19\!\cdots\!18}a^{27}+\frac{30\!\cdots\!98}{96\!\cdots\!09}a^{26}+\frac{39\!\cdots\!36}{96\!\cdots\!09}a^{25}-\frac{66\!\cdots\!83}{96\!\cdots\!09}a^{24}-\frac{44\!\cdots\!05}{96\!\cdots\!09}a^{23}+\frac{38\!\cdots\!55}{19\!\cdots\!18}a^{22}+\frac{68\!\cdots\!67}{19\!\cdots\!18}a^{21}-\frac{37\!\cdots\!38}{96\!\cdots\!09}a^{20}-\frac{29\!\cdots\!23}{96\!\cdots\!09}a^{19}-\frac{20\!\cdots\!50}{96\!\cdots\!09}a^{18}+\frac{29\!\cdots\!78}{96\!\cdots\!09}a^{17}-\frac{34\!\cdots\!62}{96\!\cdots\!09}a^{16}+\frac{91\!\cdots\!53}{96\!\cdots\!09}a^{15}-\frac{40\!\cdots\!42}{26\!\cdots\!57}a^{14}+\frac{87\!\cdots\!79}{19\!\cdots\!18}a^{13}+\frac{42\!\cdots\!46}{96\!\cdots\!09}a^{12}+\frac{15\!\cdots\!85}{96\!\cdots\!09}a^{11}+\frac{87\!\cdots\!83}{19\!\cdots\!18}a^{10}-\frac{35\!\cdots\!75}{96\!\cdots\!09}a^{9}-\frac{29\!\cdots\!03}{96\!\cdots\!09}a^{8}+\frac{32\!\cdots\!19}{96\!\cdots\!09}a^{7}-\frac{40\!\cdots\!35}{12\!\cdots\!18}a^{6}+\frac{34\!\cdots\!32}{96\!\cdots\!09}a^{5}+\frac{10\!\cdots\!75}{19\!\cdots\!18}a^{4}-\frac{83\!\cdots\!61}{19\!\cdots\!18}a^{3}-\frac{19\!\cdots\!90}{96\!\cdots\!09}a^{2}+\frac{57\!\cdots\!37}{52\!\cdots\!14}a+\frac{70\!\cdots\!07}{19\!\cdots\!18}$, $\frac{1}{31\!\cdots\!82}a^{35}-\frac{63\!\cdots\!63}{31\!\cdots\!82}a^{34}-\frac{27\!\cdots\!08}{15\!\cdots\!91}a^{33}-\frac{16\!\cdots\!55}{31\!\cdots\!82}a^{32}+\frac{13\!\cdots\!62}{15\!\cdots\!91}a^{31}+\frac{69\!\cdots\!15}{31\!\cdots\!82}a^{30}+\frac{47\!\cdots\!95}{31\!\cdots\!82}a^{29}+\frac{55\!\cdots\!21}{31\!\cdots\!82}a^{28}+\frac{85\!\cdots\!24}{15\!\cdots\!91}a^{27}+\frac{14\!\cdots\!75}{31\!\cdots\!82}a^{26}-\frac{66\!\cdots\!80}{15\!\cdots\!91}a^{25}+\frac{46\!\cdots\!79}{31\!\cdots\!82}a^{24}+\frac{19\!\cdots\!39}{15\!\cdots\!91}a^{23}-\frac{70\!\cdots\!00}{15\!\cdots\!91}a^{22}+\frac{70\!\cdots\!58}{15\!\cdots\!91}a^{21}-\frac{45\!\cdots\!35}{15\!\cdots\!91}a^{20}+\frac{56\!\cdots\!70}{15\!\cdots\!91}a^{19}-\frac{98\!\cdots\!64}{15\!\cdots\!91}a^{18}+\frac{53\!\cdots\!17}{15\!\cdots\!91}a^{17}+\frac{58\!\cdots\!88}{15\!\cdots\!91}a^{16}-\frac{13\!\cdots\!00}{15\!\cdots\!91}a^{15}-\frac{14\!\cdots\!37}{31\!\cdots\!82}a^{14}-\frac{89\!\cdots\!59}{31\!\cdots\!82}a^{13}-\frac{34\!\cdots\!02}{15\!\cdots\!91}a^{12}-\frac{15\!\cdots\!26}{42\!\cdots\!43}a^{11}+\frac{87\!\cdots\!77}{31\!\cdots\!82}a^{10}-\frac{18\!\cdots\!43}{31\!\cdots\!82}a^{9}-\frac{73\!\cdots\!40}{15\!\cdots\!91}a^{8}-\frac{28\!\cdots\!31}{31\!\cdots\!82}a^{7}-\frac{12\!\cdots\!67}{31\!\cdots\!82}a^{6}-\frac{69\!\cdots\!43}{31\!\cdots\!82}a^{5}+\frac{53\!\cdots\!93}{15\!\cdots\!91}a^{4}+\frac{83\!\cdots\!71}{31\!\cdots\!82}a^{3}+\frac{67\!\cdots\!71}{15\!\cdots\!91}a^{2}+\frac{49\!\cdots\!21}{15\!\cdots\!91}a+\frac{49\!\cdots\!33}{15\!\cdots\!91}$

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

Class group and class number

$C_{9}\times C_{1629}$, which has order $14661$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( \frac{6355253221129489500946454320832515795239537530518066091}{75890162852489880758613834906298611400603788341297007038} a^{35} - \frac{3770012955844936853836908891589672595247933467724199065}{37945081426244940379306917453149305700301894170648503519} a^{34} + \frac{83149155625423813594374889417142447996583870058115048117}{37945081426244940379306917453149305700301894170648503519} a^{33} - \frac{125960682773862194736481725374244992645495706767803681243}{75890162852489880758613834906298611400603788341297007038} a^{32} + \frac{1319622627216924824531971063335417027776041254408171986050}{37945081426244940379306917453149305700301894170648503519} a^{31} - \frac{792829009373877854787506487757191685194006672572950189488}{37945081426244940379306917453149305700301894170648503519} a^{30} + \frac{25746087862050570043047053993153508937285157833711753606517}{75890162852489880758613834906298611400603788341297007038} a^{29} - \frac{9744065977820939624630813345895279191716765256500816226269}{75890162852489880758613834906298611400603788341297007038} a^{28} + \frac{178487050153458216796936802525034284089765270565054934304879}{75890162852489880758613834906298611400603788341297007038} a^{27} - \frac{47507390064491807241947005517081277956144489055374612021561}{75890162852489880758613834906298611400603788341297007038} a^{26} + \frac{427049667171990026076534616706136472239933627374982773704883}{37945081426244940379306917453149305700301894170648503519} a^{25} - \frac{55526087310167660864164592307450938282670010347726676646021}{37945081426244940379306917453149305700301894170648503519} a^{24} + \frac{1489233281148114830252179235802283496215415441214473063192363}{37945081426244940379306917453149305700301894170648503519} a^{23} - \frac{244163139203608994889912121898659942285490230426072698835299}{75890162852489880758613834906298611400603788341297007038} a^{22} + \frac{7387827038259523949618329519053459161325984820508220652228741}{75890162852489880758613834906298611400603788341297007038} a^{21} - \frac{146042440210709256919920579457766598763138465773038392844826}{37945081426244940379306917453149305700301894170648503519} a^{20} + \frac{6850884054893245096022342927832138043283854261711155454936118}{37945081426244940379306917453149305700301894170648503519} a^{19} - \frac{228383083042706870005257540017638071169201830326063462104049}{37945081426244940379306917453149305700301894170648503519} a^{18} + \frac{9408856064975430034284217199906680443771549126674381083633571}{37945081426244940379306917453149305700301894170648503519} a^{17} - \frac{289694443682129423578953536701717088377216378478118269673865}{37945081426244940379306917453149305700301894170648503519} a^{16} + \frac{9799154351730193571901192739486326987955894563638565018738573}{37945081426244940379306917453149305700301894170648503519} a^{15} - \frac{1051630372599183794738169628090840217190322184938671232866063}{75890162852489880758613834906298611400603788341297007038} a^{14} + \frac{7504132365733125615987759479979619785890268820838675881680684}{37945081426244940379306917453149305700301894170648503519} a^{13} - \frac{615287389376944212501467290355839468907417713043677201932770}{37945081426244940379306917453149305700301894170648503519} a^{12} + \frac{4273425452850423843628626663086398099425041546193324541385337}{37945081426244940379306917453149305700301894170648503519} a^{11} - \frac{577280391959947983589959840713070328566503096658948988087249}{37945081426244940379306917453149305700301894170648503519} a^{10} + \frac{1677706071739338583956588715336999490501737337931141777693634}{37945081426244940379306917453149305700301894170648503519} a^{9} - \frac{322807770385142221194254713980599918360484467870170842423723}{37945081426244940379306917453149305700301894170648503519} a^{8} + \frac{944548021384264087683118185650095858067956441308271174372983}{75890162852489880758613834906298611400603788341297007038} a^{7} - \frac{230637200939241439860385280859455343528503604652113306558765}{75890162852489880758613834906298611400603788341297007038} a^{6} + \frac{149288361426748381931533306150693314017417206429108933541779}{75890162852489880758613834906298611400603788341297007038} a^{5} - \frac{44493602609240577206248095498277008610205704333539503937585}{75890162852489880758613834906298611400603788341297007038} a^{4} + \frac{17810301601453616584149889386511012026859424650139986645019}{75890162852489880758613834906298611400603788341297007038} a^{3} - \frac{1967038885408407577725075046420857343322998307991979101920}{37945081426244940379306917453149305700301894170648503519} a^{2} + \frac{404303415201056017939940620469411046013902096637029216781}{75890162852489880758613834906298611400603788341297007038} a + \frac{17862988947419591443044311891354748597506305695594109047}{75890162852489880758613834906298611400603788341297007038} \)  (order $6$)
Fundamental units:		$\frac{44\!\cdots\!29}{15\!\cdots\!91}a^{35}-\frac{41\!\cdots\!53}{15\!\cdots\!91}a^{34}+\frac{11\!\cdots\!19}{15\!\cdots\!91}a^{33}-\frac{59\!\cdots\!13}{15\!\cdots\!91}a^{32}+\frac{18\!\cdots\!71}{15\!\cdots\!91}a^{31}-\frac{66\!\cdots\!24}{15\!\cdots\!91}a^{30}+\frac{17\!\cdots\!81}{15\!\cdots\!91}a^{29}-\frac{24\!\cdots\!07}{15\!\cdots\!91}a^{28}+\frac{12\!\cdots\!14}{15\!\cdots\!91}a^{27}-\frac{33\!\cdots\!75}{15\!\cdots\!91}a^{26}+\frac{58\!\cdots\!60}{15\!\cdots\!91}a^{25}+\frac{64\!\cdots\!02}{15\!\cdots\!91}a^{24}+\frac{20\!\cdots\!16}{15\!\cdots\!91}a^{23}+\frac{31\!\cdots\!46}{15\!\cdots\!91}a^{22}+\frac{50\!\cdots\!61}{15\!\cdots\!91}a^{21}+\frac{99\!\cdots\!36}{15\!\cdots\!91}a^{20}+\frac{94\!\cdots\!49}{15\!\cdots\!91}a^{19}+\frac{97\!\cdots\!06}{82\!\cdots\!01}a^{18}+\frac{12\!\cdots\!84}{15\!\cdots\!91}a^{17}+\frac{25\!\cdots\!06}{15\!\cdots\!91}a^{16}+\frac{13\!\cdots\!42}{15\!\cdots\!91}a^{15}+\frac{22\!\cdots\!73}{15\!\cdots\!91}a^{14}+\frac{27\!\cdots\!38}{42\!\cdots\!43}a^{13}+\frac{12\!\cdots\!53}{15\!\cdots\!91}a^{12}+\frac{55\!\cdots\!64}{15\!\cdots\!91}a^{11}+\frac{34\!\cdots\!60}{15\!\cdots\!91}a^{10}+\frac{20\!\cdots\!19}{15\!\cdots\!91}a^{9}-\frac{38\!\cdots\!54}{15\!\cdots\!91}a^{8}+\frac{50\!\cdots\!96}{15\!\cdots\!91}a^{7}-\frac{55\!\cdots\!12}{15\!\cdots\!91}a^{6}+\frac{55\!\cdots\!31}{15\!\cdots\!91}a^{5}-\frac{18\!\cdots\!82}{15\!\cdots\!91}a^{4}+\frac{43\!\cdots\!17}{15\!\cdots\!91}a^{3}-\frac{69\!\cdots\!16}{15\!\cdots\!91}a^{2}-\frac{14\!\cdots\!27}{15\!\cdots\!91}a-\frac{11\!\cdots\!57}{15\!\cdots\!91}$, $\frac{49\!\cdots\!55}{15\!\cdots\!91}a^{35}-\frac{51\!\cdots\!45}{15\!\cdots\!91}a^{34}+\frac{12\!\cdots\!85}{15\!\cdots\!91}a^{33}-\frac{15\!\cdots\!35}{31\!\cdots\!82}a^{32}+\frac{40\!\cdots\!05}{31\!\cdots\!82}a^{31}-\frac{18\!\cdots\!75}{31\!\cdots\!82}a^{30}+\frac{19\!\cdots\!70}{15\!\cdots\!91}a^{29}-\frac{94\!\cdots\!35}{31\!\cdots\!82}a^{28}+\frac{27\!\cdots\!05}{31\!\cdots\!82}a^{27}-\frac{34\!\cdots\!75}{31\!\cdots\!82}a^{26}+\frac{13\!\cdots\!31}{31\!\cdots\!82}a^{25}+\frac{14\!\cdots\!15}{31\!\cdots\!82}a^{24}+\frac{23\!\cdots\!80}{15\!\cdots\!91}a^{23}+\frac{26\!\cdots\!05}{31\!\cdots\!82}a^{22}+\frac{11\!\cdots\!75}{31\!\cdots\!82}a^{21}+\frac{55\!\cdots\!25}{15\!\cdots\!91}a^{20}+\frac{10\!\cdots\!25}{15\!\cdots\!91}a^{19}+\frac{56\!\cdots\!00}{82\!\cdots\!01}a^{18}+\frac{39\!\cdots\!85}{42\!\cdots\!43}a^{17}+\frac{14\!\cdots\!00}{15\!\cdots\!91}a^{16}+\frac{15\!\cdots\!05}{15\!\cdots\!91}a^{15}+\frac{10\!\cdots\!80}{15\!\cdots\!91}a^{14}+\frac{11\!\cdots\!80}{15\!\cdots\!91}a^{13}+\frac{42\!\cdots\!65}{15\!\cdots\!91}a^{12}+\frac{12\!\cdots\!65}{31\!\cdots\!82}a^{11}-\frac{32\!\cdots\!45}{31\!\cdots\!82}a^{10}+\frac{48\!\cdots\!45}{31\!\cdots\!82}a^{9}-\frac{49\!\cdots\!65}{31\!\cdots\!82}a^{8}+\frac{64\!\cdots\!95}{15\!\cdots\!91}a^{7}-\frac{11\!\cdots\!40}{15\!\cdots\!91}a^{6}+\frac{17\!\cdots\!35}{31\!\cdots\!82}a^{5}-\frac{28\!\cdots\!50}{15\!\cdots\!91}a^{4}+\frac{93\!\cdots\!35}{15\!\cdots\!91}a^{3}-\frac{18\!\cdots\!85}{15\!\cdots\!91}a^{2}+\frac{22\!\cdots\!45}{31\!\cdots\!82}a-\frac{57\!\cdots\!15}{31\!\cdots\!82}$, $\frac{55\!\cdots\!49}{31\!\cdots\!82}a^{35}-\frac{88\!\cdots\!67}{31\!\cdots\!82}a^{34}+\frac{14\!\cdots\!35}{31\!\cdots\!82}a^{33}-\frac{83\!\cdots\!79}{15\!\cdots\!91}a^{32}+\frac{23\!\cdots\!33}{31\!\cdots\!82}a^{31}-\frac{22\!\cdots\!35}{31\!\cdots\!82}a^{30}+\frac{22\!\cdots\!65}{31\!\cdots\!82}a^{29}-\frac{86\!\cdots\!55}{15\!\cdots\!91}a^{28}+\frac{15\!\cdots\!65}{31\!\cdots\!82}a^{27}-\frac{27\!\cdots\!32}{82\!\cdots\!01}a^{26}+\frac{72\!\cdots\!15}{31\!\cdots\!82}a^{25}-\frac{39\!\cdots\!77}{31\!\cdots\!82}a^{24}+\frac{12\!\cdots\!55}{15\!\cdots\!91}a^{23}-\frac{12\!\cdots\!27}{31\!\cdots\!82}a^{22}+\frac{29\!\cdots\!68}{15\!\cdots\!91}a^{21}-\frac{14\!\cdots\!21}{15\!\cdots\!91}a^{20}+\frac{53\!\cdots\!26}{15\!\cdots\!91}a^{19}-\frac{27\!\cdots\!73}{15\!\cdots\!91}a^{18}+\frac{18\!\cdots\!71}{42\!\cdots\!43}a^{17}-\frac{38\!\cdots\!58}{15\!\cdots\!91}a^{16}+\frac{67\!\cdots\!68}{15\!\cdots\!91}a^{15}-\frac{86\!\cdots\!81}{31\!\cdots\!82}a^{14}+\frac{95\!\cdots\!89}{31\!\cdots\!82}a^{13}-\frac{70\!\cdots\!89}{31\!\cdots\!82}a^{12}+\frac{49\!\cdots\!37}{31\!\cdots\!82}a^{11}-\frac{22\!\cdots\!95}{15\!\cdots\!91}a^{10}+\frac{22\!\cdots\!59}{41\!\cdots\!29}a^{9}-\frac{19\!\cdots\!13}{31\!\cdots\!82}a^{8}+\frac{43\!\cdots\!37}{31\!\cdots\!82}a^{7}-\frac{27\!\cdots\!88}{15\!\cdots\!91}a^{6}+\frac{26\!\cdots\!43}{15\!\cdots\!91}a^{5}-\frac{41\!\cdots\!16}{15\!\cdots\!91}a^{4}+\frac{11\!\cdots\!97}{31\!\cdots\!82}a^{3}-\frac{26\!\cdots\!04}{15\!\cdots\!91}a^{2}-\frac{36\!\cdots\!39}{31\!\cdots\!82}a-\frac{12\!\cdots\!61}{15\!\cdots\!91}$, $\frac{42\!\cdots\!88}{15\!\cdots\!91}a^{35}-\frac{26\!\cdots\!92}{15\!\cdots\!91}a^{34}+\frac{10\!\cdots\!48}{15\!\cdots\!91}a^{33}-\frac{21\!\cdots\!48}{15\!\cdots\!91}a^{32}+\frac{17\!\cdots\!04}{15\!\cdots\!91}a^{31}-\frac{62\!\cdots\!87}{15\!\cdots\!91}a^{30}+\frac{17\!\cdots\!86}{15\!\cdots\!91}a^{29}+\frac{32\!\cdots\!48}{15\!\cdots\!91}a^{28}+\frac{11\!\cdots\!18}{15\!\cdots\!91}a^{27}+\frac{35\!\cdots\!56}{15\!\cdots\!91}a^{26}+\frac{57\!\cdots\!80}{15\!\cdots\!91}a^{25}+\frac{25\!\cdots\!20}{15\!\cdots\!91}a^{24}+\frac{20\!\cdots\!62}{15\!\cdots\!91}a^{23}+\frac{97\!\cdots\!56}{15\!\cdots\!91}a^{22}+\frac{51\!\cdots\!36}{15\!\cdots\!91}a^{21}+\frac{26\!\cdots\!00}{15\!\cdots\!91}a^{20}+\frac{98\!\cdots\!72}{15\!\cdots\!91}a^{19}+\frac{49\!\cdots\!28}{15\!\cdots\!91}a^{18}+\frac{13\!\cdots\!92}{15\!\cdots\!91}a^{17}+\frac{68\!\cdots\!00}{15\!\cdots\!91}a^{16}+\frac{14\!\cdots\!10}{15\!\cdots\!91}a^{15}+\frac{68\!\cdots\!92}{15\!\cdots\!91}a^{14}+\frac{11\!\cdots\!36}{15\!\cdots\!91}a^{13}+\frac{48\!\cdots\!13}{15\!\cdots\!91}a^{12}+\frac{18\!\cdots\!24}{42\!\cdots\!43}a^{11}+\frac{24\!\cdots\!24}{15\!\cdots\!91}a^{10}+\frac{25\!\cdots\!80}{15\!\cdots\!91}a^{9}+\frac{74\!\cdots\!18}{15\!\cdots\!91}a^{8}+\frac{66\!\cdots\!20}{15\!\cdots\!91}a^{7}+\frac{39\!\cdots\!68}{42\!\cdots\!43}a^{6}+\frac{85\!\cdots\!98}{15\!\cdots\!91}a^{5}+\frac{67\!\cdots\!44}{15\!\cdots\!91}a^{4}+\frac{45\!\cdots\!06}{15\!\cdots\!91}a^{3}+\frac{74\!\cdots\!34}{15\!\cdots\!91}a^{2}+\frac{77\!\cdots\!08}{15\!\cdots\!91}a-\frac{29\!\cdots\!78}{15\!\cdots\!91}$, $\frac{24\!\cdots\!25}{15\!\cdots\!91}a^{35}-\frac{15\!\cdots\!46}{15\!\cdots\!91}a^{34}+\frac{63\!\cdots\!71}{15\!\cdots\!91}a^{33}-\frac{12\!\cdots\!25}{15\!\cdots\!91}a^{32}+\frac{10\!\cdots\!61}{15\!\cdots\!91}a^{31}-\frac{36\!\cdots\!00}{15\!\cdots\!91}a^{30}+\frac{99\!\cdots\!95}{15\!\cdots\!91}a^{29}+\frac{18\!\cdots\!87}{15\!\cdots\!91}a^{28}+\frac{69\!\cdots\!40}{15\!\cdots\!91}a^{27}+\frac{20\!\cdots\!09}{15\!\cdots\!91}a^{26}+\frac{33\!\cdots\!69}{15\!\cdots\!91}a^{25}+\frac{14\!\cdots\!81}{15\!\cdots\!91}a^{24}+\frac{11\!\cdots\!58}{15\!\cdots\!91}a^{23}+\frac{56\!\cdots\!89}{15\!\cdots\!91}a^{22}+\frac{30\!\cdots\!49}{15\!\cdots\!91}a^{21}+\frac{15\!\cdots\!66}{15\!\cdots\!91}a^{20}+\frac{57\!\cdots\!32}{15\!\cdots\!91}a^{19}+\frac{28\!\cdots\!32}{15\!\cdots\!91}a^{18}+\frac{80\!\cdots\!50}{15\!\cdots\!91}a^{17}+\frac{39\!\cdots\!30}{15\!\cdots\!91}a^{16}+\frac{85\!\cdots\!15}{15\!\cdots\!91}a^{15}+\frac{39\!\cdots\!31}{15\!\cdots\!91}a^{14}+\frac{66\!\cdots\!90}{15\!\cdots\!91}a^{13}+\frac{28\!\cdots\!35}{15\!\cdots\!91}a^{12}+\frac{10\!\cdots\!30}{42\!\cdots\!43}a^{11}+\frac{13\!\cdots\!41}{15\!\cdots\!91}a^{10}+\frac{14\!\cdots\!46}{15\!\cdots\!91}a^{9}+\frac{43\!\cdots\!84}{15\!\cdots\!91}a^{8}+\frac{38\!\cdots\!46}{15\!\cdots\!91}a^{7}+\frac{23\!\cdots\!21}{42\!\cdots\!43}a^{6}+\frac{49\!\cdots\!47}{15\!\cdots\!91}a^{5}+\frac{39\!\cdots\!58}{15\!\cdots\!91}a^{4}+\frac{26\!\cdots\!69}{15\!\cdots\!91}a^{3}+\frac{42\!\cdots\!46}{15\!\cdots\!91}a^{2}+\frac{44\!\cdots\!27}{15\!\cdots\!91}a-\frac{17\!\cdots\!72}{15\!\cdots\!91}$, $\frac{39\!\cdots\!37}{15\!\cdots\!91}a^{35}-\frac{39\!\cdots\!52}{15\!\cdots\!91}a^{34}+\frac{20\!\cdots\!43}{31\!\cdots\!82}a^{33}-\frac{11\!\cdots\!57}{31\!\cdots\!82}a^{32}+\frac{32\!\cdots\!15}{31\!\cdots\!82}a^{31}-\frac{68\!\cdots\!34}{15\!\cdots\!91}a^{30}+\frac{31\!\cdots\!73}{31\!\cdots\!82}a^{29}-\frac{31\!\cdots\!53}{15\!\cdots\!91}a^{28}+\frac{22\!\cdots\!81}{31\!\cdots\!82}a^{27}-\frac{18\!\cdots\!37}{31\!\cdots\!82}a^{26}+\frac{10\!\cdots\!97}{31\!\cdots\!82}a^{25}+\frac{28\!\cdots\!15}{15\!\cdots\!91}a^{24}+\frac{37\!\cdots\!39}{31\!\cdots\!82}a^{23}+\frac{18\!\cdots\!03}{15\!\cdots\!91}a^{22}+\frac{45\!\cdots\!89}{15\!\cdots\!91}a^{21}+\frac{64\!\cdots\!97}{15\!\cdots\!91}a^{20}+\frac{23\!\cdots\!44}{42\!\cdots\!43}a^{19}+\frac{64\!\cdots\!67}{82\!\cdots\!01}a^{18}+\frac{11\!\cdots\!78}{15\!\cdots\!91}a^{17}+\frac{16\!\cdots\!46}{15\!\cdots\!91}a^{16}+\frac{12\!\cdots\!79}{15\!\cdots\!91}a^{15}+\frac{14\!\cdots\!36}{15\!\cdots\!91}a^{14}+\frac{91\!\cdots\!22}{15\!\cdots\!91}a^{13}+\frac{14\!\cdots\!55}{31\!\cdots\!82}a^{12}+\frac{10\!\cdots\!57}{31\!\cdots\!82}a^{11}+\frac{19\!\cdots\!13}{31\!\cdots\!82}a^{10}+\frac{37\!\cdots\!11}{31\!\cdots\!82}a^{9}-\frac{10\!\cdots\!13}{15\!\cdots\!91}a^{8}+\frac{95\!\cdots\!91}{31\!\cdots\!82}a^{7}-\frac{13\!\cdots\!15}{31\!\cdots\!82}a^{6}+\frac{57\!\cdots\!10}{15\!\cdots\!91}a^{5}-\frac{36\!\cdots\!11}{31\!\cdots\!82}a^{4}+\frac{51\!\cdots\!22}{15\!\cdots\!91}a^{3}-\frac{50\!\cdots\!43}{84\!\cdots\!86}a^{2}+\frac{55\!\cdots\!85}{15\!\cdots\!91}a-\frac{14\!\cdots\!96}{15\!\cdots\!91}$, $\frac{98\!\cdots\!14}{15\!\cdots\!91}a^{35}+\frac{14\!\cdots\!10}{15\!\cdots\!91}a^{34}+\frac{22\!\cdots\!48}{15\!\cdots\!91}a^{33}+\frac{47\!\cdots\!14}{15\!\cdots\!91}a^{32}+\frac{71\!\cdots\!85}{31\!\cdots\!82}a^{31}+\frac{81\!\cdots\!23}{15\!\cdots\!91}a^{30}+\frac{66\!\cdots\!35}{31\!\cdots\!82}a^{29}+\frac{17\!\cdots\!33}{31\!\cdots\!82}a^{28}+\frac{23\!\cdots\!48}{15\!\cdots\!91}a^{27}+\frac{63\!\cdots\!76}{15\!\cdots\!91}a^{26}+\frac{22\!\cdots\!81}{31\!\cdots\!82}a^{25}+\frac{31\!\cdots\!17}{15\!\cdots\!91}a^{24}+\frac{81\!\cdots\!11}{31\!\cdots\!82}a^{23}+\frac{22\!\cdots\!97}{31\!\cdots\!82}a^{22}+\frac{10\!\cdots\!75}{15\!\cdots\!91}a^{21}+\frac{28\!\cdots\!06}{15\!\cdots\!91}a^{20}+\frac{18\!\cdots\!71}{15\!\cdots\!91}a^{19}+\frac{26\!\cdots\!62}{82\!\cdots\!01}a^{18}+\frac{24\!\cdots\!55}{15\!\cdots\!91}a^{17}+\frac{69\!\cdots\!65}{15\!\cdots\!91}a^{16}+\frac{23\!\cdots\!85}{15\!\cdots\!91}a^{15}+\frac{69\!\cdots\!28}{15\!\cdots\!91}a^{14}+\frac{12\!\cdots\!94}{15\!\cdots\!91}a^{13}+\frac{50\!\cdots\!94}{15\!\cdots\!91}a^{12}+\frac{28\!\cdots\!00}{15\!\cdots\!91}a^{11}+\frac{53\!\cdots\!17}{31\!\cdots\!82}a^{10}-\frac{29\!\cdots\!41}{15\!\cdots\!91}a^{9}+\frac{18\!\cdots\!97}{31\!\cdots\!82}a^{8}-\frac{26\!\cdots\!84}{15\!\cdots\!91}a^{7}+\frac{23\!\cdots\!60}{15\!\cdots\!91}a^{6}-\frac{21\!\cdots\!11}{31\!\cdots\!82}a^{5}+\frac{26\!\cdots\!09}{15\!\cdots\!91}a^{4}-\frac{18\!\cdots\!43}{15\!\cdots\!91}a^{3}+\frac{97\!\cdots\!91}{31\!\cdots\!82}a^{2}-\frac{16\!\cdots\!45}{31\!\cdots\!82}a+\frac{69\!\cdots\!71}{15\!\cdots\!91}$, $\frac{15\!\cdots\!57}{31\!\cdots\!82}a^{35}-\frac{14\!\cdots\!83}{31\!\cdots\!82}a^{34}+\frac{40\!\cdots\!91}{31\!\cdots\!82}a^{33}-\frac{20\!\cdots\!77}{31\!\cdots\!82}a^{32}+\frac{64\!\cdots\!05}{31\!\cdots\!82}a^{31}-\frac{23\!\cdots\!11}{31\!\cdots\!82}a^{30}+\frac{31\!\cdots\!18}{15\!\cdots\!91}a^{29}-\frac{41\!\cdots\!11}{15\!\cdots\!91}a^{28}+\frac{43\!\cdots\!51}{31\!\cdots\!82}a^{27}-\frac{84\!\cdots\!15}{31\!\cdots\!82}a^{26}+\frac{20\!\cdots\!59}{31\!\cdots\!82}a^{25}+\frac{24\!\cdots\!71}{31\!\cdots\!82}a^{24}+\frac{73\!\cdots\!13}{31\!\cdots\!82}a^{23}+\frac{11\!\cdots\!33}{31\!\cdots\!82}a^{22}+\frac{90\!\cdots\!90}{15\!\cdots\!91}a^{21}+\frac{18\!\cdots\!72}{15\!\cdots\!91}a^{20}+\frac{16\!\cdots\!64}{15\!\cdots\!91}a^{19}+\frac{17\!\cdots\!26}{82\!\cdots\!01}a^{18}+\frac{23\!\cdots\!20}{15\!\cdots\!91}a^{17}+\frac{46\!\cdots\!70}{15\!\cdots\!91}a^{16}+\frac{23\!\cdots\!31}{15\!\cdots\!91}a^{15}+\frac{82\!\cdots\!47}{31\!\cdots\!82}a^{14}+\frac{35\!\cdots\!89}{31\!\cdots\!82}a^{13}+\frac{13\!\cdots\!37}{84\!\cdots\!86}a^{12}+\frac{98\!\cdots\!06}{15\!\cdots\!91}a^{11}+\frac{70\!\cdots\!49}{15\!\cdots\!91}a^{10}+\frac{35\!\cdots\!88}{15\!\cdots\!91}a^{9}-\frac{77\!\cdots\!49}{31\!\cdots\!82}a^{8}+\frac{17\!\cdots\!45}{31\!\cdots\!82}a^{7}-\frac{89\!\cdots\!31}{15\!\cdots\!91}a^{6}+\frac{94\!\cdots\!26}{15\!\cdots\!91}a^{5}-\frac{31\!\cdots\!86}{15\!\cdots\!91}a^{4}+\frac{14\!\cdots\!95}{31\!\cdots\!82}a^{3}-\frac{20\!\cdots\!13}{31\!\cdots\!82}a^{2}-\frac{24\!\cdots\!27}{31\!\cdots\!82}a-\frac{17\!\cdots\!10}{15\!\cdots\!91}$, $\frac{99\!\cdots\!23}{31\!\cdots\!82}a^{35}-\frac{83\!\cdots\!61}{15\!\cdots\!91}a^{34}+\frac{13\!\cdots\!93}{15\!\cdots\!91}a^{33}-\frac{32\!\cdots\!25}{31\!\cdots\!82}a^{32}+\frac{41\!\cdots\!23}{31\!\cdots\!82}a^{31}-\frac{22\!\cdots\!72}{15\!\cdots\!91}a^{30}+\frac{20\!\cdots\!89}{15\!\cdots\!91}a^{29}-\frac{11\!\cdots\!03}{10\!\cdots\!41}a^{28}+\frac{27\!\cdots\!61}{31\!\cdots\!82}a^{27}-\frac{10\!\cdots\!33}{16\!\cdots\!02}a^{26}+\frac{12\!\cdots\!69}{31\!\cdots\!82}a^{25}-\frac{41\!\cdots\!30}{15\!\cdots\!91}a^{24}+\frac{43\!\cdots\!79}{31\!\cdots\!82}a^{23}-\frac{13\!\cdots\!24}{15\!\cdots\!91}a^{22}+\frac{10\!\cdots\!15}{31\!\cdots\!82}a^{21}-\frac{31\!\cdots\!54}{15\!\cdots\!91}a^{20}+\frac{90\!\cdots\!53}{15\!\cdots\!91}a^{19}-\frac{58\!\cdots\!44}{15\!\cdots\!91}a^{18}+\frac{11\!\cdots\!26}{15\!\cdots\!91}a^{17}-\frac{81\!\cdots\!08}{15\!\cdots\!91}a^{16}+\frac{10\!\cdots\!60}{15\!\cdots\!91}a^{15}-\frac{17\!\cdots\!83}{31\!\cdots\!82}a^{14}+\frac{73\!\cdots\!88}{15\!\cdots\!91}a^{13}-\frac{72\!\cdots\!52}{15\!\cdots\!91}a^{12}+\frac{34\!\cdots\!64}{15\!\cdots\!91}a^{11}-\frac{90\!\cdots\!87}{31\!\cdots\!82}a^{10}+\frac{27\!\cdots\!08}{41\!\cdots\!29}a^{9}-\frac{37\!\cdots\!37}{31\!\cdots\!82}a^{8}+\frac{45\!\cdots\!99}{31\!\cdots\!82}a^{7}-\frac{10\!\cdots\!51}{31\!\cdots\!82}a^{6}+\frac{12\!\cdots\!36}{15\!\cdots\!91}a^{5}-\frac{15\!\cdots\!87}{31\!\cdots\!82}a^{4}+\frac{42\!\cdots\!49}{84\!\cdots\!86}a^{3}-\frac{95\!\cdots\!83}{31\!\cdots\!82}a^{2}-\frac{14\!\cdots\!55}{15\!\cdots\!91}a-\frac{62\!\cdots\!59}{31\!\cdots\!82}$, $\frac{31\!\cdots\!75}{31\!\cdots\!82}a^{35}-\frac{49\!\cdots\!81}{15\!\cdots\!91}a^{34}+\frac{42\!\cdots\!72}{15\!\cdots\!91}a^{33}-\frac{22\!\cdots\!89}{31\!\cdots\!82}a^{32}+\frac{66\!\cdots\!51}{15\!\cdots\!91}a^{31}-\frac{16\!\cdots\!64}{15\!\cdots\!91}a^{30}+\frac{12\!\cdots\!03}{31\!\cdots\!82}a^{29}-\frac{29\!\cdots\!15}{31\!\cdots\!82}a^{28}+\frac{42\!\cdots\!11}{15\!\cdots\!91}a^{27}-\frac{10\!\cdots\!95}{16\!\cdots\!02}a^{26}+\frac{19\!\cdots\!15}{15\!\cdots\!91}a^{25}-\frac{44\!\cdots\!65}{15\!\cdots\!91}a^{24}+\frac{58\!\cdots\!87}{15\!\cdots\!91}a^{23}-\frac{30\!\cdots\!73}{31\!\cdots\!82}a^{22}+\frac{12\!\cdots\!55}{15\!\cdots\!91}a^{21}-\frac{37\!\cdots\!91}{15\!\cdots\!91}a^{20}+\frac{17\!\cdots\!48}{15\!\cdots\!91}a^{19}-\frac{71\!\cdots\!01}{15\!\cdots\!91}a^{18}+\frac{16\!\cdots\!88}{15\!\cdots\!91}a^{17}-\frac{10\!\cdots\!19}{15\!\cdots\!91}a^{16}+\frac{67\!\cdots\!34}{15\!\cdots\!91}a^{15}-\frac{21\!\cdots\!47}{31\!\cdots\!82}a^{14}-\frac{39\!\cdots\!83}{15\!\cdots\!91}a^{13}-\frac{85\!\cdots\!17}{15\!\cdots\!91}a^{12}-\frac{78\!\cdots\!53}{15\!\cdots\!91}a^{11}-\frac{50\!\cdots\!68}{15\!\cdots\!91}a^{10}-\frac{14\!\cdots\!35}{41\!\cdots\!29}a^{9}-\frac{19\!\cdots\!42}{15\!\cdots\!91}a^{8}-\frac{36\!\cdots\!57}{31\!\cdots\!82}a^{7}-\frac{53\!\cdots\!46}{15\!\cdots\!91}a^{6}-\frac{81\!\cdots\!93}{31\!\cdots\!82}a^{5}-\frac{14\!\cdots\!17}{31\!\cdots\!82}a^{4}+\frac{19\!\cdots\!13}{15\!\cdots\!91}a^{3}-\frac{50\!\cdots\!43}{15\!\cdots\!91}a^{2}-\frac{70\!\cdots\!35}{31\!\cdots\!82}a-\frac{45\!\cdots\!12}{15\!\cdots\!91}$, $\frac{14\!\cdots\!07}{31\!\cdots\!82}a^{35}+\frac{33\!\cdots\!68}{42\!\cdots\!43}a^{34}+\frac{17\!\cdots\!54}{15\!\cdots\!91}a^{33}+\frac{39\!\cdots\!65}{15\!\cdots\!91}a^{32}+\frac{29\!\cdots\!22}{15\!\cdots\!91}a^{31}+\frac{13\!\cdots\!87}{31\!\cdots\!82}a^{30}+\frac{28\!\cdots\!95}{15\!\cdots\!91}a^{29}+\frac{14\!\cdots\!85}{31\!\cdots\!82}a^{28}+\frac{21\!\cdots\!36}{15\!\cdots\!91}a^{27}+\frac{27\!\cdots\!41}{82\!\cdots\!01}a^{26}+\frac{11\!\cdots\!18}{15\!\cdots\!91}a^{25}+\frac{54\!\cdots\!37}{31\!\cdots\!82}a^{24}+\frac{89\!\cdots\!89}{31\!\cdots\!82}a^{23}+\frac{19\!\cdots\!87}{31\!\cdots\!82}a^{22}+\frac{12\!\cdots\!66}{15\!\cdots\!91}a^{21}+\frac{24\!\cdots\!79}{15\!\cdots\!91}a^{20}+\frac{26\!\cdots\!18}{15\!\cdots\!91}a^{19}+\frac{47\!\cdots\!22}{15\!\cdots\!91}a^{18}+\frac{42\!\cdots\!03}{15\!\cdots\!91}a^{17}+\frac{11\!\cdots\!80}{27\!\cdots\!21}a^{16}+\frac{51\!\cdots\!52}{15\!\cdots\!91}a^{15}+\frac{13\!\cdots\!27}{31\!\cdots\!82}a^{14}+\frac{45\!\cdots\!70}{15\!\cdots\!91}a^{13}+\frac{53\!\cdots\!97}{15\!\cdots\!91}a^{12}+\frac{59\!\cdots\!79}{31\!\cdots\!82}a^{11}+\frac{30\!\cdots\!90}{15\!\cdots\!91}a^{10}+\frac{71\!\cdots\!05}{82\!\cdots\!58}a^{9}+\frac{11\!\cdots\!75}{15\!\cdots\!91}a^{8}+\frac{21\!\cdots\!75}{84\!\cdots\!86}a^{7}+\frac{57\!\cdots\!55}{31\!\cdots\!82}a^{6}+\frac{14\!\cdots\!53}{31\!\cdots\!82}a^{5}+\frac{72\!\cdots\!75}{31\!\cdots\!82}a^{4}+\frac{80\!\cdots\!63}{31\!\cdots\!82}a^{3}+\frac{48\!\cdots\!81}{31\!\cdots\!82}a^{2}+\frac{71\!\cdots\!97}{31\!\cdots\!82}a+\frac{35\!\cdots\!87}{15\!\cdots\!91}$, $\frac{53\!\cdots\!69}{31\!\cdots\!82}a^{35}-\frac{57\!\cdots\!49}{31\!\cdots\!82}a^{34}+\frac{13\!\cdots\!49}{31\!\cdots\!82}a^{33}-\frac{45\!\cdots\!86}{15\!\cdots\!91}a^{32}+\frac{11\!\cdots\!20}{15\!\cdots\!91}a^{31}-\frac{54\!\cdots\!54}{15\!\cdots\!91}a^{30}+\frac{10\!\cdots\!32}{15\!\cdots\!91}a^{29}-\frac{28\!\cdots\!98}{15\!\cdots\!91}a^{28}+\frac{14\!\cdots\!83}{31\!\cdots\!82}a^{27}-\frac{60\!\cdots\!72}{82\!\cdots\!01}a^{26}+\frac{35\!\cdots\!25}{15\!\cdots\!91}a^{25}-\frac{65\!\cdots\!35}{15\!\cdots\!91}a^{24}+\frac{24\!\cdots\!03}{31\!\cdots\!82}a^{23}+\frac{75\!\cdots\!35}{31\!\cdots\!82}a^{22}+\frac{30\!\cdots\!16}{15\!\cdots\!91}a^{21}+\frac{22\!\cdots\!81}{15\!\cdots\!91}a^{20}+\frac{57\!\cdots\!01}{15\!\cdots\!91}a^{19}+\frac{46\!\cdots\!64}{15\!\cdots\!91}a^{18}+\frac{78\!\cdots\!49}{15\!\cdots\!91}a^{17}+\frac{65\!\cdots\!03}{15\!\cdots\!91}a^{16}+\frac{81\!\cdots\!07}{15\!\cdots\!91}a^{15}+\frac{10\!\cdots\!57}{31\!\cdots\!82}a^{14}+\frac{12\!\cdots\!91}{31\!\cdots\!82}a^{13}+\frac{42\!\cdots\!79}{31\!\cdots\!82}a^{12}+\frac{70\!\cdots\!93}{31\!\cdots\!82}a^{11}-\frac{12\!\cdots\!37}{31\!\cdots\!82}a^{10}+\frac{71\!\cdots\!49}{82\!\cdots\!58}a^{9}-\frac{10\!\cdots\!68}{15\!\cdots\!91}a^{8}+\frac{36\!\cdots\!21}{15\!\cdots\!91}a^{7}-\frac{98\!\cdots\!91}{31\!\cdots\!82}a^{6}+\frac{10\!\cdots\!03}{31\!\cdots\!82}a^{5}-\frac{22\!\cdots\!59}{31\!\cdots\!82}a^{4}+\frac{62\!\cdots\!02}{15\!\cdots\!91}a^{3}-\frac{13\!\cdots\!23}{31\!\cdots\!82}a^{2}+\frac{33\!\cdots\!15}{31\!\cdots\!82}a+\frac{93\!\cdots\!28}{15\!\cdots\!91}$, $\frac{99\!\cdots\!01}{15\!\cdots\!91}a^{35}-\frac{18\!\cdots\!11}{31\!\cdots\!82}a^{34}+\frac{51\!\cdots\!49}{31\!\cdots\!82}a^{33}-\frac{27\!\cdots\!73}{31\!\cdots\!82}a^{32}+\frac{81\!\cdots\!07}{31\!\cdots\!82}a^{31}-\frac{30\!\cdots\!47}{31\!\cdots\!82}a^{30}+\frac{39\!\cdots\!10}{15\!\cdots\!91}a^{29}-\frac{11\!\cdots\!75}{31\!\cdots\!82}a^{28}+\frac{27\!\cdots\!27}{15\!\cdots\!91}a^{27}-\frac{18\!\cdots\!89}{31\!\cdots\!82}a^{26}+\frac{26\!\cdots\!09}{31\!\cdots\!82}a^{25}+\frac{27\!\cdots\!71}{31\!\cdots\!82}a^{24}+\frac{46\!\cdots\!18}{15\!\cdots\!91}a^{23}+\frac{68\!\cdots\!98}{15\!\cdots\!91}a^{22}+\frac{22\!\cdots\!91}{31\!\cdots\!82}a^{21}+\frac{21\!\cdots\!77}{15\!\cdots\!91}a^{20}+\frac{21\!\cdots\!91}{15\!\cdots\!91}a^{19}+\frac{21\!\cdots\!10}{82\!\cdots\!01}a^{18}+\frac{29\!\cdots\!88}{15\!\cdots\!91}a^{17}+\frac{54\!\cdots\!71}{15\!\cdots\!91}a^{16}+\frac{19\!\cdots\!99}{10\!\cdots\!41}a^{15}+\frac{12\!\cdots\!21}{42\!\cdots\!43}a^{14}+\frac{45\!\cdots\!09}{31\!\cdots\!82}a^{13}+\frac{54\!\cdots\!81}{31\!\cdots\!82}a^{12}+\frac{25\!\cdots\!09}{31\!\cdots\!82}a^{11}+\frac{69\!\cdots\!70}{15\!\cdots\!91}a^{10}+\frac{46\!\cdots\!45}{15\!\cdots\!91}a^{9}-\frac{25\!\cdots\!37}{31\!\cdots\!82}a^{8}+\frac{11\!\cdots\!87}{15\!\cdots\!91}a^{7}-\frac{27\!\cdots\!69}{31\!\cdots\!82}a^{6}+\frac{26\!\cdots\!73}{31\!\cdots\!82}a^{5}-\frac{87\!\cdots\!99}{31\!\cdots\!82}a^{4}+\frac{10\!\cdots\!40}{15\!\cdots\!91}a^{3}-\frac{18\!\cdots\!36}{15\!\cdots\!91}a^{2}-\frac{16\!\cdots\!95}{15\!\cdots\!91}a-\frac{58\!\cdots\!33}{31\!\cdots\!82}$, $\frac{94\!\cdots\!77}{31\!\cdots\!82}a^{35}-\frac{34\!\cdots\!23}{15\!\cdots\!91}a^{34}+\frac{12\!\cdots\!52}{15\!\cdots\!91}a^{33}-\frac{75\!\cdots\!95}{31\!\cdots\!82}a^{32}+\frac{19\!\cdots\!60}{15\!\cdots\!91}a^{31}-\frac{29\!\cdots\!53}{15\!\cdots\!91}a^{30}+\frac{37\!\cdots\!33}{31\!\cdots\!82}a^{29}+\frac{27\!\cdots\!65}{31\!\cdots\!82}a^{28}+\frac{26\!\cdots\!85}{31\!\cdots\!82}a^{27}+\frac{49\!\cdots\!67}{31\!\cdots\!82}a^{26}+\frac{63\!\cdots\!08}{15\!\cdots\!91}a^{25}+\frac{20\!\cdots\!20}{15\!\cdots\!91}a^{24}+\frac{22\!\cdots\!95}{15\!\cdots\!91}a^{23}+\frac{16\!\cdots\!19}{31\!\cdots\!82}a^{22}+\frac{11\!\cdots\!69}{31\!\cdots\!82}a^{21}+\frac{22\!\cdots\!19}{15\!\cdots\!91}a^{20}+\frac{10\!\cdots\!37}{15\!\cdots\!91}a^{19}+\frac{28\!\cdots\!47}{10\!\cdots\!41}a^{18}+\frac{14\!\cdots\!26}{15\!\cdots\!91}a^{17}+\frac{59\!\cdots\!41}{15\!\cdots\!91}a^{16}+\frac{15\!\cdots\!54}{15\!\cdots\!91}a^{15}+\frac{11\!\cdots\!11}{31\!\cdots\!82}a^{14}+\frac{12\!\cdots\!57}{15\!\cdots\!91}a^{13}+\frac{41\!\cdots\!39}{15\!\cdots\!91}a^{12}+\frac{70\!\cdots\!88}{15\!\cdots\!91}a^{11}+\frac{19\!\cdots\!95}{15\!\cdots\!91}a^{10}+\frac{27\!\cdots\!80}{15\!\cdots\!91}a^{9}+\frac{59\!\cdots\!35}{15\!\cdots\!91}a^{8}+\frac{14\!\cdots\!97}{31\!\cdots\!82}a^{7}+\frac{21\!\cdots\!39}{31\!\cdots\!82}a^{6}+\frac{18\!\cdots\!59}{31\!\cdots\!82}a^{5}+\frac{61\!\cdots\!95}{31\!\cdots\!82}a^{4}+\frac{13\!\cdots\!81}{31\!\cdots\!82}a^{3}+\frac{65\!\cdots\!07}{15\!\cdots\!91}a^{2}+\frac{14\!\cdots\!61}{31\!\cdots\!82}a+\frac{82\!\cdots\!37}{31\!\cdots\!82}$, $\frac{11\!\cdots\!61}{31\!\cdots\!82}a^{35}-\frac{42\!\cdots\!79}{15\!\cdots\!91}a^{34}+\frac{15\!\cdots\!34}{15\!\cdots\!91}a^{33}-\frac{88\!\cdots\!07}{31\!\cdots\!82}a^{32}+\frac{24\!\cdots\!52}{15\!\cdots\!91}a^{31}-\frac{31\!\cdots\!25}{15\!\cdots\!91}a^{30}+\frac{47\!\cdots\!91}{31\!\cdots\!82}a^{29}+\frac{45\!\cdots\!73}{31\!\cdots\!82}a^{28}+\frac{33\!\cdots\!27}{31\!\cdots\!82}a^{27}+\frac{45\!\cdots\!17}{20\!\cdots\!82}a^{26}+\frac{79\!\cdots\!18}{15\!\cdots\!91}a^{25}+\frac{27\!\cdots\!95}{15\!\cdots\!91}a^{24}+\frac{28\!\cdots\!68}{15\!\cdots\!91}a^{23}+\frac{21\!\cdots\!11}{31\!\cdots\!82}a^{22}+\frac{14\!\cdots\!53}{31\!\cdots\!82}a^{21}+\frac{30\!\cdots\!23}{15\!\cdots\!91}a^{20}+\frac{13\!\cdots\!35}{15\!\cdots\!91}a^{19}+\frac{57\!\cdots\!40}{15\!\cdots\!91}a^{18}+\frac{18\!\cdots\!36}{15\!\cdots\!91}a^{17}+\frac{78\!\cdots\!57}{15\!\cdots\!91}a^{16}+\frac{20\!\cdots\!21}{15\!\cdots\!91}a^{15}+\frac{15\!\cdots\!55}{31\!\cdots\!82}a^{14}+\frac{15\!\cdots\!51}{15\!\cdots\!91}a^{13}+\frac{55\!\cdots\!71}{15\!\cdots\!91}a^{12}+\frac{89\!\cdots\!22}{15\!\cdots\!91}a^{11}+\frac{26\!\cdots\!45}{15\!\cdots\!91}a^{10}+\frac{34\!\cdots\!90}{15\!\cdots\!91}a^{9}+\frac{79\!\cdots\!58}{15\!\cdots\!91}a^{8}+\frac{18\!\cdots\!89}{31\!\cdots\!82}a^{7}+\frac{29\!\cdots\!31}{31\!\cdots\!82}a^{6}+\frac{23\!\cdots\!69}{31\!\cdots\!82}a^{5}+\frac{97\!\cdots\!27}{31\!\cdots\!82}a^{4}+\frac{42\!\cdots\!55}{84\!\cdots\!86}a^{3}+\frac{86\!\cdots\!82}{15\!\cdots\!91}a^{2}+\frac{19\!\cdots\!65}{31\!\cdots\!82}a+\frac{48\!\cdots\!51}{31\!\cdots\!82}$, $\frac{56\!\cdots\!97}{31\!\cdots\!82}a^{35}-\frac{23\!\cdots\!15}{15\!\cdots\!91}a^{34}+\frac{73\!\cdots\!00}{15\!\cdots\!91}a^{33}-\frac{60\!\cdots\!67}{31\!\cdots\!82}a^{32}+\frac{11\!\cdots\!96}{15\!\cdots\!91}a^{31}-\frac{29\!\cdots\!83}{15\!\cdots\!91}a^{30}+\frac{22\!\cdots\!97}{31\!\cdots\!82}a^{29}-\frac{67\!\cdots\!59}{31\!\cdots\!82}a^{28}+\frac{15\!\cdots\!13}{31\!\cdots\!82}a^{27}+\frac{13\!\cdots\!11}{31\!\cdots\!82}a^{26}+\frac{37\!\cdots\!68}{15\!\cdots\!91}a^{25}+\frac{83\!\cdots\!20}{15\!\cdots\!91}a^{24}+\frac{13\!\cdots\!67}{15\!\cdots\!91}a^{23}+\frac{71\!\cdots\!55}{31\!\cdots\!82}a^{22}+\frac{67\!\cdots\!77}{31\!\cdots\!82}a^{21}+\frac{10\!\cdots\!63}{15\!\cdots\!91}a^{20}+\frac{63\!\cdots\!89}{15\!\cdots\!91}a^{19}+\frac{19\!\cdots\!49}{15\!\cdots\!91}a^{18}+\frac{87\!\cdots\!74}{15\!\cdots\!91}a^{17}+\frac{27\!\cdots\!49}{15\!\cdots\!91}a^{16}+\frac{92\!\cdots\!42}{15\!\cdots\!91}a^{15}+\frac{53\!\cdots\!31}{31\!\cdots\!82}a^{14}+\frac{71\!\cdots\!61}{15\!\cdots\!91}a^{13}+\frac{18\!\cdots\!51}{15\!\cdots\!91}a^{12}+\frac{41\!\cdots\!92}{15\!\cdots\!91}a^{11}+\frac{84\!\cdots\!47}{15\!\cdots\!91}a^{10}+\frac{15\!\cdots\!04}{15\!\cdots\!91}a^{9}+\frac{23\!\cdots\!59}{15\!\cdots\!91}a^{8}+\frac{84\!\cdots\!25}{31\!\cdots\!82}a^{7}+\frac{71\!\cdots\!45}{31\!\cdots\!82}a^{6}+\frac{11\!\cdots\!27}{31\!\cdots\!82}a^{5}-\frac{11\!\cdots\!13}{31\!\cdots\!82}a^{4}+\frac{93\!\cdots\!53}{31\!\cdots\!82}a^{3}+\frac{31\!\cdots\!51}{15\!\cdots\!91}a^{2}+\frac{76\!\cdots\!09}{31\!\cdots\!82}a+\frac{72\!\cdots\!93}{31\!\cdots\!82}$, $\frac{20\!\cdots\!01}{15\!\cdots\!91}a^{35}-\frac{80\!\cdots\!43}{31\!\cdots\!82}a^{34}+\frac{10\!\cdots\!95}{31\!\cdots\!82}a^{33}-\frac{81\!\cdots\!91}{15\!\cdots\!91}a^{32}+\frac{17\!\cdots\!51}{31\!\cdots\!82}a^{31}-\frac{11\!\cdots\!72}{15\!\cdots\!91}a^{30}+\frac{83\!\cdots\!91}{15\!\cdots\!91}a^{29}-\frac{94\!\cdots\!81}{15\!\cdots\!91}a^{28}+\frac{11\!\cdots\!51}{31\!\cdots\!82}a^{27}-\frac{31\!\cdots\!23}{82\!\cdots\!01}a^{26}+\frac{53\!\cdots\!69}{31\!\cdots\!82}a^{25}-\frac{24\!\cdots\!65}{15\!\cdots\!91}a^{24}+\frac{88\!\cdots\!01}{15\!\cdots\!91}a^{23}-\frac{16\!\cdots\!41}{31\!\cdots\!82}a^{22}+\frac{20\!\cdots\!19}{15\!\cdots\!91}a^{21}-\frac{20\!\cdots\!01}{15\!\cdots\!91}a^{20}+\frac{36\!\cdots\!04}{15\!\cdots\!91}a^{19}-\frac{37\!\cdots\!22}{15\!\cdots\!91}a^{18}+\frac{45\!\cdots\!00}{15\!\cdots\!91}a^{17}-\frac{52\!\cdots\!33}{15\!\cdots\!91}a^{16}+\frac{42\!\cdots\!48}{15\!\cdots\!91}a^{15}-\frac{57\!\cdots\!62}{15\!\cdots\!91}a^{14}+\frac{55\!\cdots\!21}{31\!\cdots\!82}a^{13}-\frac{92\!\cdots\!89}{31\!\cdots\!82}a^{12}+\frac{13\!\cdots\!19}{15\!\cdots\!91}a^{11}-\frac{28\!\cdots\!76}{15\!\cdots\!91}a^{10}+\frac{20\!\cdots\!37}{82\!\cdots\!58}a^{9}-\frac{11\!\cdots\!59}{15\!\cdots\!91}a^{8}+\frac{19\!\cdots\!05}{31\!\cdots\!82}a^{7}-\frac{64\!\cdots\!01}{31\!\cdots\!82}a^{6}+\frac{97\!\cdots\!14}{15\!\cdots\!91}a^{5}-\frac{47\!\cdots\!09}{15\!\cdots\!91}a^{4}+\frac{47\!\cdots\!29}{15\!\cdots\!91}a^{3}-\frac{14\!\cdots\!73}{68\!\cdots\!79}a^{2}-\frac{19\!\cdots\!21}{31\!\cdots\!82}a-\frac{19\!\cdots\!60}{15\!\cdots\!91}$ (assuming GRH)
Regulator:		\( 3431432369157.3267 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 3431432369157.3267 \cdot 14661}{6\cdot\sqrt{122958312298624478991860638998897557972401876087680816650390625}}\cr\approx \mathstrut & 0.176135327309686 \end{aligned}\] (assuming GRH)

Galois group

$C_2\times C_{18}$ (as 36T2):

An abelian group of order 36

The 36 conjugacy class representatives for $C_2\times C_{18}$

Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.3518667.1, 6.0.439833375.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.193453397763890625.1, 18.0.5677392343251487443465123.1, 18.0.11088656920413061413017818359375.2, 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	$18^{2}$	R	R	${\href{/padicField/7.6.0.1}{6} }^{6}$	${\href{/padicField/11.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	R	$18^{2}$	$18^{2}$	${\href{/padicField/31.3.0.1}{3} }^{12}$	${\href{/padicField/37.2.0.1}{2} }^{18}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$

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 $36$	$2$	$18$	$18$
\(5\)		Deg $36$	$2$	$18$	$18$
\(19\)	19.9.8.8	$x^{9} + 19$	$9$	$1$	$8$	$C_9$	$[\ ]_{9}$
	19.9.8.8	$x^{9} + 19$	$9$	$1$	$8$	$C_9$	$[\ ]_{9}$
	19.9.8.8	$x^{9} + 19$	$9$	$1$	$8$	$C_9$	$[\ ]_{9}$
	19.9.8.8	$x^{9} + 19$	$9$	$1$	$8$	$C_9$	$[\ ]_{9}$