Number field 32.0.7231362775399344187879888625220455562201364498198121976692736.1

• Label: 32.0.723...736.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $7.231\times 10^{60}$
• Root discriminant: \(79.77\)
• Ramified primes: $2,3,13$
• Class number: $1800$ (GRH)
• Class group: [15, 120] (GRH)
• Galois group: $C_2\times C_4^2$ (as 32T36)

Normalized defining polynomial

x^32 - 4*x^31 - 30*x^30 + 88*x^29 + 531*x^28 - 1028*x^27 - 5254*x^26 + 7696*x^25 + 29657*x^24 - 43172*x^23 - 106514*x^22 + 247944*x^21 + 198866*x^20 - 1131864*x^19 + 508272*x^18 + 1288480*x^17 - 2969721*x^16 + 12413912*x^15 + 372598*x^14 - 68988532*x^13 + 48726322*x^12 + 138776892*x^11 - 154416048*x^10 - 119345528*x^9 + 162618454*x^8 + 101306868*x^7 - 76133580*x^6 - 190278160*x^5 + 211310100*x^4 - 84510352*x^3 + 27524944*x^2 - 6403360*x + 796849

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(7231362775399344187879888625220455562201364498198121976692736\) \(\medspace = 2^{88}\cdot 3^{16}\cdot 13^{24}\)
Root discriminant:		\(79.77\)
Galois root discriminant:		$2^{11/4}3^{1/2}13^{3/4}\approx 79.77202738406362$
Ramified primes:		\(2\), \(3\), \(13\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$32$
This field is Galois and abelian over $\Q$.
Conductor:		\(624=2^{4}\cdot 3\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{624}(1,·)$, $\chi_{624}(5,·)$, $\chi_{624}(385,·)$, $\chi_{624}(521,·)$, $\chi_{624}(209,·)$, $\chi_{624}(493,·)$, $\chi_{624}(25,·)$, $\chi_{624}(281,·)$, $\chi_{624}(157,·)$, $\chi_{624}(389,·)$, $\chi_{624}(545,·)$, $\chi_{624}(421,·)$, $\chi_{624}(541,·)$, $\chi_{624}(53,·)$, $\chi_{624}(265,·)$, $\chi_{624}(313,·)$, $\chi_{624}(317,·)$, $\chi_{624}(437,·)$, $\chi_{624}(577,·)$, $\chi_{624}(181,·)$, $\chi_{624}(161,·)$, $\chi_{624}(73,·)$, $\chi_{624}(77,·)$, $\chi_{624}(109,·)$, $\chi_{624}(593,·)$, $\chi_{624}(469,·)$, $\chi_{624}(473,·)$, $\chi_{624}(229,·)$, $\chi_{624}(337,·)$, $\chi_{624}(233,·)$, $\chi_{624}(365,·)$, $\chi_{624}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3}a^{20}-\frac{1}{3}a^{18}+\frac{1}{3}a^{16}-\frac{1}{3}a^{12}+\frac{1}{3}a^{8}-\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{19}+\frac{1}{3}a^{17}-\frac{1}{3}a^{13}+\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{3}a^{22}+\frac{1}{3}a^{16}-\frac{1}{3}a^{14}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{23}+\frac{1}{3}a^{17}-\frac{1}{3}a^{15}-\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{24}+\frac{1}{3}a^{18}-\frac{1}{3}a^{16}-\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{25}+\frac{1}{3}a^{19}-\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{26}+\frac{1}{3}a^{16}+\frac{1}{3}a^{14}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{27}+\frac{1}{3}a^{17}+\frac{1}{3}a^{15}-\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{216855}a^{28}+\frac{11416}{216855}a^{27}+\frac{637}{72285}a^{26}-\frac{2338}{14457}a^{25}+\frac{4534}{43371}a^{24}+\frac{1588}{14457}a^{23}-\frac{7427}{216855}a^{22}+\frac{17081}{216855}a^{21}+\frac{1079}{14457}a^{20}+\frac{25597}{216855}a^{19}-\frac{38}{24095}a^{18}-\frac{35324}{72285}a^{17}+\frac{7354}{24095}a^{16}+\frac{103816}{216855}a^{15}+\frac{108362}{216855}a^{14}-\frac{6814}{72285}a^{13}-\frac{22267}{72285}a^{12}-\frac{40801}{216855}a^{11}+\frac{34159}{216855}a^{10}-\frac{3602}{24095}a^{9}-\frac{37652}{216855}a^{8}-\frac{12199}{216855}a^{7}+\frac{528}{4819}a^{6}+\frac{14636}{43371}a^{5}+\frac{5167}{72285}a^{4}+\frac{5638}{14457}a^{3}-\frac{53714}{216855}a^{2}+\frac{23096}{216855}a-\frac{89422}{216855}$, $\frac{1}{216855}a^{29}+\frac{22}{711}a^{27}-\frac{6992}{72285}a^{26}-\frac{965}{43371}a^{25}+\frac{680}{43371}a^{24}-\frac{377}{216855}a^{23}+\frac{13408}{216855}a^{22}-\frac{27866}{216855}a^{21}+\frac{18097}{216855}a^{20}+\frac{32561}{216855}a^{19}-\frac{729}{4819}a^{18}+\frac{26926}{72285}a^{17}-\frac{19462}{43371}a^{16}-\frac{87089}{216855}a^{15}-\frac{67829}{216855}a^{14}+\frac{35887}{72285}a^{13}+\frac{4990}{43371}a^{12}-\frac{11690}{43371}a^{11}+\frac{58298}{216855}a^{10}-\frac{52964}{216855}a^{9}+\frac{88708}{216855}a^{8}-\frac{77936}{216855}a^{7}-\frac{20446}{43371}a^{6}-\frac{9634}{216855}a^{5}+\frac{2183}{72285}a^{4}+\frac{5911}{216855}a^{3}+\frac{20150}{43371}a^{2}-\frac{19226}{72285}a+\frac{32782}{216855}$, $\frac{1}{13\!\cdots\!15}a^{30}+\frac{14\!\cdots\!42}{13\!\cdots\!15}a^{29}-\frac{27\!\cdots\!52}{13\!\cdots\!15}a^{28}-\frac{12\!\cdots\!23}{13\!\cdots\!15}a^{27}-\frac{37\!\cdots\!79}{13\!\cdots\!15}a^{26}-\frac{12\!\cdots\!95}{26\!\cdots\!03}a^{25}-\frac{66\!\cdots\!09}{44\!\cdots\!05}a^{24}-\frac{47\!\cdots\!32}{44\!\cdots\!05}a^{23}-\frac{11\!\cdots\!19}{14\!\cdots\!35}a^{22}+\frac{13\!\cdots\!71}{44\!\cdots\!05}a^{21}-\frac{36\!\cdots\!63}{26\!\cdots\!03}a^{20}-\frac{59\!\cdots\!77}{13\!\cdots\!15}a^{19}-\frac{86\!\cdots\!36}{44\!\cdots\!05}a^{18}-\frac{66\!\cdots\!67}{26\!\cdots\!03}a^{17}-\frac{20\!\cdots\!12}{44\!\cdots\!05}a^{16}+\frac{47\!\cdots\!09}{14\!\cdots\!35}a^{15}-\frac{18\!\cdots\!06}{13\!\cdots\!15}a^{14}+\frac{57\!\cdots\!66}{13\!\cdots\!15}a^{13}+\frac{21\!\cdots\!54}{44\!\cdots\!05}a^{12}+\frac{10\!\cdots\!33}{29\!\cdots\!67}a^{11}-\frac{62\!\cdots\!06}{13\!\cdots\!15}a^{10}-\frac{39\!\cdots\!26}{14\!\cdots\!35}a^{9}+\frac{57\!\cdots\!86}{14\!\cdots\!35}a^{8}+\frac{15\!\cdots\!12}{44\!\cdots\!05}a^{7}-\frac{84\!\cdots\!99}{13\!\cdots\!15}a^{6}+\frac{27\!\cdots\!21}{13\!\cdots\!15}a^{5}-\frac{31\!\cdots\!58}{13\!\cdots\!15}a^{4}+\frac{19\!\cdots\!49}{44\!\cdots\!05}a^{3}+\frac{74\!\cdots\!00}{26\!\cdots\!03}a^{2}+\frac{52\!\cdots\!79}{13\!\cdots\!15}a+\frac{61\!\cdots\!98}{13\!\cdots\!15}$, $\frac{1}{15\!\cdots\!35}a^{31}+\frac{23\!\cdots\!13}{13\!\cdots\!35}a^{30}+\frac{45\!\cdots\!00}{30\!\cdots\!67}a^{29}+\frac{60\!\cdots\!48}{51\!\cdots\!45}a^{28}-\frac{14\!\cdots\!97}{17\!\cdots\!15}a^{27}+\frac{17\!\cdots\!59}{30\!\cdots\!67}a^{26}+\frac{24\!\cdots\!67}{19\!\cdots\!65}a^{25}-\frac{49\!\cdots\!86}{51\!\cdots\!45}a^{24}+\frac{23\!\cdots\!92}{17\!\cdots\!15}a^{23}-\frac{58\!\cdots\!12}{51\!\cdots\!45}a^{22}-\frac{17\!\cdots\!28}{15\!\cdots\!35}a^{21}-\frac{11\!\cdots\!52}{15\!\cdots\!35}a^{20}+\frac{99\!\cdots\!53}{15\!\cdots\!35}a^{19}-\frac{62\!\cdots\!16}{15\!\cdots\!35}a^{18}+\frac{62\!\cdots\!71}{15\!\cdots\!35}a^{17}+\frac{41\!\cdots\!56}{17\!\cdots\!15}a^{16}+\frac{25\!\cdots\!24}{15\!\cdots\!35}a^{15}-\frac{19\!\cdots\!82}{15\!\cdots\!35}a^{14}-\frac{44\!\cdots\!68}{30\!\cdots\!67}a^{13}-\frac{28\!\cdots\!53}{17\!\cdots\!15}a^{12}+\frac{29\!\cdots\!77}{30\!\cdots\!67}a^{11}-\frac{10\!\cdots\!79}{15\!\cdots\!35}a^{10}+\frac{66\!\cdots\!11}{51\!\cdots\!45}a^{9}+\frac{22\!\cdots\!44}{51\!\cdots\!45}a^{8}+\frac{40\!\cdots\!76}{15\!\cdots\!35}a^{7}-\frac{25\!\cdots\!98}{15\!\cdots\!35}a^{6}-\frac{44\!\cdots\!32}{15\!\cdots\!35}a^{5}+\frac{60\!\cdots\!98}{15\!\cdots\!35}a^{4}+\frac{26\!\cdots\!22}{15\!\cdots\!35}a^{3}+\frac{17\!\cdots\!19}{65\!\cdots\!55}a^{2}+\frac{20\!\cdots\!77}{51\!\cdots\!45}a+\frac{12\!\cdots\!84}{30\!\cdots\!67}$

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

Class group and class number

$C_{15}\times C_{120}$, which has order $1800$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( -\frac{394931953917134471018985621754596480880270185032}{707014925521692241255833606606329961249677767873323063} a^{31} + \frac{1202503181232027639235957204426270644517865662}{566064792251154716778089356770480353282368108785687} a^{30} + \frac{12171037358062045622622600923532671545903673405336}{707014925521692241255833606606329961249677767873323063} a^{29} - \frac{32468335536414472550605966495562563050467158322905}{707014925521692241255833606606329961249677767873323063} a^{28} - \frac{216943262096275415023580124554801077502557937836768}{707014925521692241255833606606329961249677767873323063} a^{27} + \frac{121935121220408892106850451666620532777616541320350}{235671641840564080418611202202109987083225922624441021} a^{26} + \frac{27372493670038980141943963933912177047430526766728}{8949556019261927104504222868434556471514908454092697} a^{25} - \frac{881176350870431951924083928918539599983220907074097}{235671641840564080418611202202109987083225922624441021} a^{24} - \frac{4131304382283306269351399638749131092963915453923612}{235671641840564080418611202202109987083225922624441021} a^{23} + \frac{4941749198105775971913767957184702532816770811103008}{235671641840564080418611202202109987083225922624441021} a^{22} + \frac{45956031319053034996239053781849696313338057711177568}{707014925521692241255833606606329961249677767873323063} a^{21} - \frac{29996109987318600883181864242348858822739104492669685}{235671641840564080418611202202109987083225922624441021} a^{20} - \frac{99952256039024233099112440275051796407403196471198356}{707014925521692241255833606606329961249677767873323063} a^{19} + \frac{433749721393063353744134871334300464675870020984768962}{707014925521692241255833606606329961249677767873323063} a^{18} - \frac{107327147532189871235795271136037947981064228766034592}{707014925521692241255833606606329961249677767873323063} a^{17} - \frac{187410529812650265529556404321821231983919246724461085}{235671641840564080418611202202109987083225922624441021} a^{16} + \frac{355702755628589043646861347875360679667126936485111704}{235671641840564080418611202202109987083225922624441021} a^{15} - \frac{1548302605163060198788778084153422086473125331724223192}{235671641840564080418611202202109987083225922624441021} a^{14} - \frac{1120935340725684873616892704334960085594709221173468884}{707014925521692241255833606606329961249677767873323063} a^{13} + \frac{9122078554758496873134855416454312953924692038346229278}{235671641840564080418611202202109987083225922624441021} a^{12} - \frac{13813838172804736219825367679651486958832635678910186324}{707014925521692241255833606606329961249677767873323063} a^{11} - \frac{59688493966438965831757139493941296952080033473856660112}{707014925521692241255833606606329961249677767873323063} a^{10} + \frac{50055294510368730563625176532646277305430225606381379568}{707014925521692241255833606606329961249677767873323063} a^{9} + \frac{20603283315404308689321892422945806963597369119744611895}{235671641840564080418611202202109987083225922624441021} a^{8} - \frac{55029120212153812537212147384465877515821698196912790700}{707014925521692241255833606606329961249677767873323063} a^{7} - \frac{18748793023204757032718154657777010559465988936376541822}{235671641840564080418611202202109987083225922624441021} a^{6} + \frac{21918527007010355821977532448617028220498688556762729228}{707014925521692241255833606606329961249677767873323063} a^{5} + \frac{84602803633821426934747138335595128204043262458232426888}{707014925521692241255833606606329961249677767873323063} a^{4} - \frac{22316227705752167050689870825950473730367972613933424260}{235671641840564080418611202202109987083225922624441021} a^{3} + \frac{173598640749765875491073058328177992666803920937386292}{8949556019261927104504222868434556471514908454092697} a^{2} - \frac{4401305290129025200875071423657427885875775032012334208}{707014925521692241255833606606329961249677767873323063} a + \frac{1052743357728579016735921236295813722094266882000265873}{707014925521692241255833606606329961249677767873323063} \)  (order $6$)
Fundamental units:		$\frac{64\!\cdots\!72}{49\!\cdots\!85}a^{31}-\frac{12\!\cdots\!79}{23\!\cdots\!59}a^{30}-\frac{58\!\cdots\!12}{14\!\cdots\!55}a^{29}+\frac{17\!\cdots\!11}{14\!\cdots\!55}a^{28}+\frac{34\!\cdots\!24}{49\!\cdots\!85}a^{27}-\frac{68\!\cdots\!27}{49\!\cdots\!85}a^{26}-\frac{42\!\cdots\!76}{62\!\cdots\!15}a^{25}+\frac{15\!\cdots\!78}{14\!\cdots\!55}a^{24}+\frac{57\!\cdots\!28}{14\!\cdots\!55}a^{23}-\frac{30\!\cdots\!21}{49\!\cdots\!85}a^{22}-\frac{13\!\cdots\!56}{98\!\cdots\!97}a^{21}+\frac{84\!\cdots\!31}{24\!\cdots\!55}a^{20}+\frac{76\!\cdots\!40}{29\!\cdots\!91}a^{19}-\frac{23\!\cdots\!09}{14\!\cdots\!55}a^{18}+\frac{36\!\cdots\!62}{49\!\cdots\!85}a^{17}+\frac{28\!\cdots\!23}{14\!\cdots\!55}a^{16}-\frac{21\!\cdots\!88}{49\!\cdots\!85}a^{15}+\frac{23\!\cdots\!22}{14\!\cdots\!55}a^{14}+\frac{26\!\cdots\!38}{49\!\cdots\!85}a^{13}-\frac{13\!\cdots\!64}{14\!\cdots\!55}a^{12}+\frac{35\!\cdots\!68}{49\!\cdots\!85}a^{11}+\frac{92\!\cdots\!77}{48\!\cdots\!31}a^{10}-\frac{35\!\cdots\!84}{14\!\cdots\!55}a^{9}-\frac{17\!\cdots\!47}{98\!\cdots\!97}a^{8}+\frac{85\!\cdots\!80}{29\!\cdots\!91}a^{7}+\frac{36\!\cdots\!14}{24\!\cdots\!55}a^{6}-\frac{26\!\cdots\!96}{14\!\cdots\!55}a^{5}-\frac{41\!\cdots\!96}{14\!\cdots\!55}a^{4}+\frac{10\!\cdots\!96}{29\!\cdots\!91}a^{3}-\frac{11\!\cdots\!94}{18\!\cdots\!45}a^{2}-\frac{53\!\cdots\!22}{14\!\cdots\!55}a+\frac{36\!\cdots\!69}{49\!\cdots\!85}$, $\frac{14\!\cdots\!24}{56\!\cdots\!83}a^{31}-\frac{23\!\cdots\!68}{22\!\cdots\!35}a^{30}-\frac{20\!\cdots\!68}{28\!\cdots\!15}a^{29}+\frac{66\!\cdots\!58}{28\!\cdots\!15}a^{28}+\frac{69\!\cdots\!40}{56\!\cdots\!83}a^{27}-\frac{81\!\cdots\!84}{28\!\cdots\!15}a^{26}-\frac{85\!\cdots\!64}{71\!\cdots\!77}a^{25}+\frac{64\!\cdots\!84}{28\!\cdots\!15}a^{24}+\frac{36\!\cdots\!56}{56\!\cdots\!83}a^{23}-\frac{37\!\cdots\!48}{28\!\cdots\!15}a^{22}-\frac{60\!\cdots\!28}{28\!\cdots\!15}a^{21}+\frac{20\!\cdots\!72}{28\!\cdots\!15}a^{20}+\frac{73\!\cdots\!48}{28\!\cdots\!15}a^{19}-\frac{83\!\cdots\!32}{28\!\cdots\!15}a^{18}+\frac{12\!\cdots\!36}{56\!\cdots\!83}a^{17}+\frac{76\!\cdots\!51}{28\!\cdots\!15}a^{16}-\frac{25\!\cdots\!32}{28\!\cdots\!15}a^{15}+\frac{92\!\cdots\!36}{28\!\cdots\!15}a^{14}-\frac{21\!\cdots\!84}{28\!\cdots\!15}a^{13}-\frac{46\!\cdots\!08}{28\!\cdots\!15}a^{12}+\frac{51\!\cdots\!96}{28\!\cdots\!15}a^{11}+\frac{82\!\cdots\!16}{28\!\cdots\!15}a^{10}-\frac{15\!\cdots\!08}{28\!\cdots\!15}a^{9}-\frac{42\!\cdots\!08}{28\!\cdots\!15}a^{8}+\frac{16\!\cdots\!56}{28\!\cdots\!15}a^{7}+\frac{23\!\cdots\!08}{28\!\cdots\!15}a^{6}-\frac{10\!\cdots\!56}{28\!\cdots\!15}a^{5}-\frac{11\!\cdots\!56}{28\!\cdots\!15}a^{4}+\frac{21\!\cdots\!92}{28\!\cdots\!15}a^{3}-\frac{12\!\cdots\!24}{35\!\cdots\!85}a^{2}+\frac{18\!\cdots\!23}{28\!\cdots\!15}a-\frac{93\!\cdots\!07}{28\!\cdots\!15}$, $\frac{76\!\cdots\!14}{28\!\cdots\!15}a^{31}-\frac{22\!\cdots\!94}{22\!\cdots\!35}a^{30}-\frac{23\!\cdots\!21}{28\!\cdots\!15}a^{29}+\frac{57\!\cdots\!92}{28\!\cdots\!15}a^{28}+\frac{42\!\cdots\!02}{28\!\cdots\!15}a^{27}-\frac{61\!\cdots\!28}{28\!\cdots\!15}a^{26}-\frac{52\!\cdots\!37}{35\!\cdots\!85}a^{25}+\frac{41\!\cdots\!74}{28\!\cdots\!15}a^{24}+\frac{46\!\cdots\!90}{56\!\cdots\!83}a^{23}-\frac{23\!\cdots\!16}{28\!\cdots\!15}a^{22}-\frac{83\!\cdots\!06}{28\!\cdots\!15}a^{21}+\frac{15\!\cdots\!07}{28\!\cdots\!15}a^{20}+\frac{18\!\cdots\!28}{28\!\cdots\!15}a^{19}-\frac{15\!\cdots\!73}{56\!\cdots\!83}a^{18}+\frac{16\!\cdots\!59}{28\!\cdots\!15}a^{17}+\frac{88\!\cdots\!74}{28\!\cdots\!15}a^{16}-\frac{20\!\cdots\!42}{28\!\cdots\!15}a^{15}+\frac{17\!\cdots\!38}{56\!\cdots\!83}a^{14}+\frac{34\!\cdots\!42}{28\!\cdots\!15}a^{13}-\frac{49\!\cdots\!59}{28\!\cdots\!15}a^{12}+\frac{41\!\cdots\!00}{56\!\cdots\!83}a^{11}+\frac{10\!\cdots\!22}{28\!\cdots\!15}a^{10}-\frac{82\!\cdots\!06}{28\!\cdots\!15}a^{9}-\frac{93\!\cdots\!44}{28\!\cdots\!15}a^{8}+\frac{86\!\cdots\!56}{28\!\cdots\!15}a^{7}+\frac{14\!\cdots\!80}{56\!\cdots\!83}a^{6}-\frac{28\!\cdots\!56}{28\!\cdots\!15}a^{5}-\frac{12\!\cdots\!49}{28\!\cdots\!15}a^{4}+\frac{12\!\cdots\!97}{28\!\cdots\!15}a^{3}-\frac{66\!\cdots\!22}{35\!\cdots\!85}a^{2}+\frac{10\!\cdots\!62}{28\!\cdots\!15}a+\frac{40\!\cdots\!01}{28\!\cdots\!15}$, $\frac{53\!\cdots\!32}{15\!\cdots\!35}a^{31}-\frac{14\!\cdots\!28}{24\!\cdots\!83}a^{30}+\frac{10\!\cdots\!84}{15\!\cdots\!35}a^{29}+\frac{28\!\cdots\!56}{17\!\cdots\!15}a^{28}-\frac{31\!\cdots\!24}{15\!\cdots\!35}a^{27}-\frac{14\!\cdots\!56}{51\!\cdots\!45}a^{26}+\frac{17\!\cdots\!08}{65\!\cdots\!55}a^{25}+\frac{14\!\cdots\!22}{51\!\cdots\!45}a^{24}-\frac{69\!\cdots\!36}{30\!\cdots\!67}a^{23}-\frac{25\!\cdots\!46}{15\!\cdots\!35}a^{22}+\frac{21\!\cdots\!44}{15\!\cdots\!35}a^{21}+\frac{10\!\cdots\!08}{15\!\cdots\!35}a^{20}-\frac{15\!\cdots\!28}{15\!\cdots\!35}a^{19}-\frac{26\!\cdots\!42}{15\!\cdots\!35}a^{18}+\frac{82\!\cdots\!72}{15\!\cdots\!35}a^{17}-\frac{45\!\cdots\!12}{15\!\cdots\!35}a^{16}-\frac{13\!\cdots\!48}{15\!\cdots\!35}a^{15}+\frac{77\!\cdots\!12}{51\!\cdots\!45}a^{14}-\frac{80\!\cdots\!28}{15\!\cdots\!35}a^{13}-\frac{10\!\cdots\!79}{30\!\cdots\!67}a^{12}+\frac{18\!\cdots\!52}{51\!\cdots\!45}a^{11}-\frac{14\!\cdots\!58}{15\!\cdots\!35}a^{10}-\frac{12\!\cdots\!76}{15\!\cdots\!35}a^{9}+\frac{14\!\cdots\!16}{30\!\cdots\!67}a^{8}+\frac{14\!\cdots\!64}{15\!\cdots\!35}a^{7}-\frac{77\!\cdots\!42}{15\!\cdots\!35}a^{6}-\frac{15\!\cdots\!36}{17\!\cdots\!15}a^{5}+\frac{14\!\cdots\!07}{15\!\cdots\!35}a^{4}+\frac{20\!\cdots\!32}{17\!\cdots\!15}a^{3}-\frac{22\!\cdots\!58}{39\!\cdots\!73}a^{2}+\frac{15\!\cdots\!48}{51\!\cdots\!45}a-\frac{87\!\cdots\!59}{15\!\cdots\!35}$, $\frac{32\!\cdots\!48}{51\!\cdots\!45}a^{31}-\frac{19\!\cdots\!80}{82\!\cdots\!61}a^{30}-\frac{32\!\cdots\!68}{17\!\cdots\!15}a^{29}+\frac{25\!\cdots\!51}{51\!\cdots\!45}a^{28}+\frac{16\!\cdots\!04}{51\!\cdots\!45}a^{27}-\frac{28\!\cdots\!32}{51\!\cdots\!45}a^{26}-\frac{20\!\cdots\!64}{65\!\cdots\!55}a^{25}+\frac{70\!\cdots\!78}{17\!\cdots\!15}a^{24}+\frac{59\!\cdots\!96}{34\!\cdots\!63}a^{23}-\frac{40\!\cdots\!28}{17\!\cdots\!15}a^{22}-\frac{31\!\cdots\!24}{51\!\cdots\!45}a^{21}+\frac{73\!\cdots\!17}{51\!\cdots\!45}a^{20}+\frac{19\!\cdots\!96}{17\!\cdots\!15}a^{19}-\frac{11\!\cdots\!56}{17\!\cdots\!15}a^{18}+\frac{15\!\cdots\!48}{51\!\cdots\!45}a^{17}+\frac{10\!\cdots\!34}{17\!\cdots\!15}a^{16}-\frac{96\!\cdots\!32}{51\!\cdots\!45}a^{15}+\frac{39\!\cdots\!64}{51\!\cdots\!45}a^{14}+\frac{57\!\cdots\!48}{51\!\cdots\!45}a^{13}-\frac{13\!\cdots\!92}{34\!\cdots\!63}a^{12}+\frac{46\!\cdots\!88}{17\!\cdots\!15}a^{11}+\frac{38\!\cdots\!88}{51\!\cdots\!45}a^{10}-\frac{45\!\cdots\!04}{51\!\cdots\!45}a^{9}-\frac{18\!\cdots\!00}{34\!\cdots\!63}a^{8}+\frac{15\!\cdots\!52}{17\!\cdots\!15}a^{7}+\frac{18\!\cdots\!22}{51\!\cdots\!45}a^{6}-\frac{22\!\cdots\!56}{51\!\cdots\!45}a^{5}-\frac{47\!\cdots\!47}{51\!\cdots\!45}a^{4}+\frac{68\!\cdots\!72}{51\!\cdots\!45}a^{3}-\frac{87\!\cdots\!98}{13\!\cdots\!91}a^{2}+\frac{90\!\cdots\!16}{51\!\cdots\!45}a-\frac{12\!\cdots\!01}{51\!\cdots\!45}$, $\frac{22\!\cdots\!28}{60\!\cdots\!09}a^{31}-\frac{62\!\cdots\!36}{48\!\cdots\!41}a^{30}-\frac{69\!\cdots\!68}{60\!\cdots\!09}a^{29}+\frac{51\!\cdots\!38}{20\!\cdots\!03}a^{28}+\frac{12\!\cdots\!84}{60\!\cdots\!09}a^{27}-\frac{17\!\cdots\!96}{67\!\cdots\!01}a^{26}-\frac{16\!\cdots\!08}{84\!\cdots\!19}a^{25}+\frac{33\!\cdots\!78}{20\!\cdots\!03}a^{24}+\frac{62\!\cdots\!72}{60\!\cdots\!09}a^{23}-\frac{58\!\cdots\!86}{60\!\cdots\!09}a^{22}-\frac{21\!\cdots\!24}{60\!\cdots\!09}a^{21}+\frac{40\!\cdots\!79}{60\!\cdots\!09}a^{20}+\frac{46\!\cdots\!44}{60\!\cdots\!09}a^{19}-\frac{20\!\cdots\!54}{60\!\cdots\!09}a^{18}+\frac{57\!\cdots\!00}{60\!\cdots\!09}a^{17}+\frac{17\!\cdots\!52}{60\!\cdots\!09}a^{16}-\frac{59\!\cdots\!84}{60\!\cdots\!09}a^{15}+\frac{87\!\cdots\!42}{20\!\cdots\!03}a^{14}+\frac{10\!\cdots\!24}{60\!\cdots\!09}a^{13}-\frac{13\!\cdots\!68}{60\!\cdots\!09}a^{12}+\frac{18\!\cdots\!04}{20\!\cdots\!03}a^{11}+\frac{23\!\cdots\!96}{60\!\cdots\!09}a^{10}-\frac{21\!\cdots\!00}{60\!\cdots\!09}a^{9}-\frac{15\!\cdots\!94}{60\!\cdots\!09}a^{8}+\frac{19\!\cdots\!28}{60\!\cdots\!09}a^{7}+\frac{89\!\cdots\!82}{60\!\cdots\!09}a^{6}-\frac{14\!\cdots\!52}{20\!\cdots\!03}a^{5}-\frac{24\!\cdots\!88}{60\!\cdots\!09}a^{4}+\frac{12\!\cdots\!32}{20\!\cdots\!03}a^{3}-\frac{33\!\cdots\!76}{76\!\cdots\!71}a^{2}+\frac{28\!\cdots\!12}{20\!\cdots\!03}a-\frac{12\!\cdots\!78}{60\!\cdots\!09}$, $\frac{50\!\cdots\!64}{23\!\cdots\!45}a^{31}-\frac{30\!\cdots\!20}{47\!\cdots\!89}a^{30}-\frac{16\!\cdots\!68}{23\!\cdots\!45}a^{29}+\frac{17\!\cdots\!93}{15\!\cdots\!63}a^{28}+\frac{28\!\cdots\!84}{23\!\cdots\!45}a^{27}-\frac{70\!\cdots\!36}{78\!\cdots\!15}a^{26}-\frac{37\!\cdots\!68}{33\!\cdots\!95}a^{25}+\frac{33\!\cdots\!79}{78\!\cdots\!15}a^{24}+\frac{14\!\cdots\!12}{23\!\cdots\!45}a^{23}-\frac{67\!\cdots\!84}{23\!\cdots\!45}a^{22}-\frac{49\!\cdots\!04}{23\!\cdots\!45}a^{21}+\frac{71\!\cdots\!29}{23\!\cdots\!45}a^{20}+\frac{13\!\cdots\!32}{23\!\cdots\!45}a^{19}-\frac{39\!\cdots\!88}{23\!\cdots\!45}a^{18}-\frac{29\!\cdots\!98}{23\!\cdots\!45}a^{17}+\frac{34\!\cdots\!18}{23\!\cdots\!45}a^{16}-\frac{25\!\cdots\!28}{47\!\cdots\!89}a^{15}+\frac{11\!\cdots\!52}{52\!\cdots\!21}a^{14}+\frac{50\!\cdots\!44}{23\!\cdots\!45}a^{13}-\frac{26\!\cdots\!22}{23\!\cdots\!45}a^{12}+\frac{11\!\cdots\!96}{15\!\cdots\!63}a^{11}+\frac{51\!\cdots\!04}{23\!\cdots\!45}a^{10}-\frac{29\!\cdots\!44}{23\!\cdots\!45}a^{9}-\frac{44\!\cdots\!12}{23\!\cdots\!45}a^{8}+\frac{29\!\cdots\!36}{23\!\cdots\!45}a^{7}+\frac{28\!\cdots\!86}{23\!\cdots\!45}a^{6}-\frac{13\!\cdots\!68}{78\!\cdots\!15}a^{5}-\frac{56\!\cdots\!13}{23\!\cdots\!45}a^{4}+\frac{43\!\cdots\!00}{15\!\cdots\!63}a^{3}-\frac{35\!\cdots\!87}{29\!\cdots\!55}a^{2}+\frac{62\!\cdots\!32}{26\!\cdots\!05}a-\frac{25\!\cdots\!29}{23\!\cdots\!45}$, $\frac{99\!\cdots\!31}{15\!\cdots\!35}a^{31}-\frac{34\!\cdots\!79}{12\!\cdots\!15}a^{30}-\frac{32\!\cdots\!63}{17\!\cdots\!15}a^{29}+\frac{19\!\cdots\!63}{30\!\cdots\!67}a^{28}+\frac{16\!\cdots\!39}{51\!\cdots\!45}a^{27}-\frac{40\!\cdots\!89}{51\!\cdots\!45}a^{26}-\frac{20\!\cdots\!28}{62\!\cdots\!15}a^{25}+\frac{19\!\cdots\!13}{30\!\cdots\!67}a^{24}+\frac{18\!\cdots\!53}{10\!\cdots\!89}a^{23}-\frac{11\!\cdots\!35}{30\!\cdots\!67}a^{22}-\frac{10\!\cdots\!43}{15\!\cdots\!35}a^{21}+\frac{19\!\cdots\!69}{10\!\cdots\!89}a^{20}+\frac{15\!\cdots\!33}{15\!\cdots\!35}a^{19}-\frac{25\!\cdots\!47}{30\!\cdots\!67}a^{18}+\frac{80\!\cdots\!83}{15\!\cdots\!35}a^{17}+\frac{17\!\cdots\!79}{17\!\cdots\!15}a^{16}-\frac{36\!\cdots\!06}{15\!\cdots\!35}a^{15}+\frac{12\!\cdots\!19}{15\!\cdots\!35}a^{14}-\frac{23\!\cdots\!74}{15\!\cdots\!35}a^{13}-\frac{23\!\cdots\!96}{51\!\cdots\!45}a^{12}+\frac{23\!\cdots\!89}{51\!\cdots\!45}a^{11}+\frac{97\!\cdots\!71}{10\!\cdots\!89}a^{10}-\frac{73\!\cdots\!03}{51\!\cdots\!45}a^{9}-\frac{12\!\cdots\!27}{15\!\cdots\!35}a^{8}+\frac{27\!\cdots\!34}{15\!\cdots\!35}a^{7}+\frac{34\!\cdots\!61}{51\!\cdots\!45}a^{6}-\frac{37\!\cdots\!36}{30\!\cdots\!67}a^{5}-\frac{22\!\cdots\!48}{15\!\cdots\!35}a^{4}+\frac{62\!\cdots\!90}{30\!\cdots\!67}a^{3}-\frac{64\!\cdots\!31}{19\!\cdots\!65}a^{2}-\frac{84\!\cdots\!01}{30\!\cdots\!67}a+\frac{81\!\cdots\!59}{15\!\cdots\!35}$, $\frac{30\!\cdots\!36}{15\!\cdots\!35}a^{31}-\frac{93\!\cdots\!42}{12\!\cdots\!15}a^{30}-\frac{94\!\cdots\!27}{15\!\cdots\!35}a^{29}+\frac{50\!\cdots\!71}{30\!\cdots\!67}a^{28}+\frac{16\!\cdots\!04}{15\!\cdots\!35}a^{27}-\frac{31\!\cdots\!18}{17\!\cdots\!15}a^{26}-\frac{21\!\cdots\!33}{19\!\cdots\!65}a^{25}+\frac{20\!\cdots\!34}{15\!\cdots\!35}a^{24}+\frac{96\!\cdots\!14}{15\!\cdots\!35}a^{23}-\frac{11\!\cdots\!58}{15\!\cdots\!35}a^{22}-\frac{11\!\cdots\!99}{51\!\cdots\!45}a^{21}+\frac{70\!\cdots\!83}{15\!\cdots\!35}a^{20}+\frac{85\!\cdots\!52}{17\!\cdots\!15}a^{19}-\frac{33\!\cdots\!11}{15\!\cdots\!35}a^{18}+\frac{84\!\cdots\!03}{15\!\cdots\!35}a^{17}+\frac{43\!\cdots\!49}{15\!\cdots\!35}a^{16}-\frac{83\!\cdots\!08}{15\!\cdots\!35}a^{15}+\frac{36\!\cdots\!22}{15\!\cdots\!35}a^{14}+\frac{88\!\cdots\!46}{15\!\cdots\!35}a^{13}-\frac{21\!\cdots\!68}{15\!\cdots\!35}a^{12}+\frac{35\!\cdots\!82}{51\!\cdots\!45}a^{11}+\frac{46\!\cdots\!98}{15\!\cdots\!35}a^{10}-\frac{78\!\cdots\!55}{30\!\cdots\!67}a^{9}-\frac{95\!\cdots\!33}{30\!\cdots\!67}a^{8}+\frac{49\!\cdots\!06}{17\!\cdots\!15}a^{7}+\frac{42\!\cdots\!01}{15\!\cdots\!35}a^{6}-\frac{18\!\cdots\!68}{15\!\cdots\!35}a^{5}-\frac{12\!\cdots\!27}{30\!\cdots\!67}a^{4}+\frac{10\!\cdots\!08}{30\!\cdots\!67}a^{3}-\frac{13\!\cdots\!62}{19\!\cdots\!65}a^{2}+\frac{13\!\cdots\!86}{15\!\cdots\!35}a+\frac{17\!\cdots\!38}{51\!\cdots\!45}$, $\frac{14\!\cdots\!48}{15\!\cdots\!35}a^{31}-\frac{43\!\cdots\!62}{12\!\cdots\!15}a^{30}-\frac{43\!\cdots\!24}{15\!\cdots\!35}a^{29}+\frac{11\!\cdots\!13}{15\!\cdots\!35}a^{28}+\frac{78\!\cdots\!84}{15\!\cdots\!35}a^{27}-\frac{29\!\cdots\!17}{34\!\cdots\!63}a^{26}-\frac{98\!\cdots\!94}{19\!\cdots\!65}a^{25}+\frac{18\!\cdots\!91}{30\!\cdots\!67}a^{24}+\frac{44\!\cdots\!56}{15\!\cdots\!35}a^{23}-\frac{10\!\cdots\!76}{30\!\cdots\!67}a^{22}-\frac{54\!\cdots\!96}{51\!\cdots\!45}a^{21}+\frac{32\!\cdots\!74}{15\!\cdots\!35}a^{20}+\frac{11\!\cdots\!44}{51\!\cdots\!45}a^{19}-\frac{15\!\cdots\!31}{15\!\cdots\!35}a^{18}+\frac{38\!\cdots\!44}{15\!\cdots\!35}a^{17}+\frac{19\!\cdots\!56}{15\!\cdots\!35}a^{16}-\frac{38\!\cdots\!48}{15\!\cdots\!35}a^{15}+\frac{33\!\cdots\!64}{30\!\cdots\!67}a^{14}+\frac{41\!\cdots\!04}{15\!\cdots\!35}a^{13}-\frac{97\!\cdots\!72}{15\!\cdots\!35}a^{12}+\frac{16\!\cdots\!16}{51\!\cdots\!45}a^{11}+\frac{21\!\cdots\!33}{15\!\cdots\!35}a^{10}-\frac{17\!\cdots\!14}{15\!\cdots\!35}a^{9}-\frac{21\!\cdots\!86}{15\!\cdots\!35}a^{8}+\frac{21\!\cdots\!52}{17\!\cdots\!15}a^{7}+\frac{20\!\cdots\!89}{15\!\cdots\!35}a^{6}-\frac{68\!\cdots\!38}{15\!\cdots\!35}a^{5}-\frac{30\!\cdots\!58}{15\!\cdots\!35}a^{4}+\frac{23\!\cdots\!52}{15\!\cdots\!35}a^{3}-\frac{61\!\cdots\!91}{19\!\cdots\!65}a^{2}+\frac{22\!\cdots\!22}{15\!\cdots\!35}a-\frac{15\!\cdots\!11}{10\!\cdots\!89}$, $\frac{75\!\cdots\!44}{15\!\cdots\!35}a^{31}-\frac{22\!\cdots\!13}{12\!\cdots\!15}a^{30}-\frac{77\!\cdots\!86}{51\!\cdots\!45}a^{29}+\frac{12\!\cdots\!61}{30\!\cdots\!67}a^{28}+\frac{13\!\cdots\!12}{51\!\cdots\!45}a^{27}-\frac{74\!\cdots\!82}{17\!\cdots\!15}a^{26}-\frac{52\!\cdots\!02}{19\!\cdots\!65}a^{25}+\frac{47\!\cdots\!41}{15\!\cdots\!35}a^{24}+\frac{78\!\cdots\!32}{51\!\cdots\!45}a^{23}-\frac{26\!\cdots\!92}{15\!\cdots\!35}a^{22}-\frac{86\!\cdots\!58}{15\!\cdots\!35}a^{21}+\frac{55\!\cdots\!84}{51\!\cdots\!45}a^{20}+\frac{18\!\cdots\!72}{15\!\cdots\!35}a^{19}-\frac{13\!\cdots\!94}{25\!\cdots\!35}a^{18}+\frac{19\!\cdots\!52}{15\!\cdots\!35}a^{17}+\frac{33\!\cdots\!67}{51\!\cdots\!45}a^{16}-\frac{20\!\cdots\!12}{15\!\cdots\!35}a^{15}+\frac{89\!\cdots\!93}{15\!\cdots\!35}a^{14}+\frac{23\!\cdots\!94}{15\!\cdots\!35}a^{13}-\frac{56\!\cdots\!98}{17\!\cdots\!15}a^{12}+\frac{81\!\cdots\!28}{51\!\cdots\!45}a^{11}+\frac{12\!\cdots\!83}{17\!\cdots\!15}a^{10}-\frac{19\!\cdots\!62}{34\!\cdots\!63}a^{9}-\frac{21\!\cdots\!25}{30\!\cdots\!67}a^{8}+\frac{93\!\cdots\!36}{15\!\cdots\!35}a^{7}+\frac{31\!\cdots\!53}{51\!\cdots\!45}a^{6}-\frac{31\!\cdots\!52}{15\!\cdots\!35}a^{5}-\frac{29\!\cdots\!27}{30\!\cdots\!67}a^{4}+\frac{24\!\cdots\!08}{30\!\cdots\!67}a^{3}-\frac{49\!\cdots\!93}{19\!\cdots\!65}a^{2}+\frac{15\!\cdots\!94}{15\!\cdots\!35}a-\frac{24\!\cdots\!09}{15\!\cdots\!35}$, $\frac{25\!\cdots\!09}{51\!\cdots\!45}a^{31}-\frac{14\!\cdots\!53}{82\!\cdots\!61}a^{30}-\frac{23\!\cdots\!26}{15\!\cdots\!35}a^{29}+\frac{58\!\cdots\!29}{15\!\cdots\!35}a^{28}+\frac{42\!\cdots\!66}{15\!\cdots\!35}a^{27}-\frac{20\!\cdots\!06}{51\!\cdots\!45}a^{26}-\frac{53\!\cdots\!16}{19\!\cdots\!65}a^{25}+\frac{42\!\cdots\!91}{15\!\cdots\!35}a^{24}+\frac{48\!\cdots\!10}{30\!\cdots\!67}a^{23}-\frac{26\!\cdots\!89}{17\!\cdots\!15}a^{22}-\frac{90\!\cdots\!56}{15\!\cdots\!35}a^{21}+\frac{15\!\cdots\!33}{15\!\cdots\!35}a^{20}+\frac{21\!\cdots\!77}{15\!\cdots\!35}a^{19}-\frac{42\!\cdots\!19}{84\!\cdots\!45}a^{18}+\frac{29\!\cdots\!14}{51\!\cdots\!45}a^{17}+\frac{16\!\cdots\!78}{25\!\cdots\!35}a^{16}-\frac{20\!\cdots\!62}{17\!\cdots\!15}a^{15}+\frac{87\!\cdots\!61}{15\!\cdots\!35}a^{14}+\frac{12\!\cdots\!89}{51\!\cdots\!45}a^{13}-\frac{10\!\cdots\!39}{30\!\cdots\!67}a^{12}+\frac{58\!\cdots\!92}{51\!\cdots\!45}a^{11}+\frac{11\!\cdots\!62}{15\!\cdots\!35}a^{10}-\frac{75\!\cdots\!46}{15\!\cdots\!35}a^{9}-\frac{80\!\cdots\!69}{10\!\cdots\!89}a^{8}+\frac{79\!\cdots\!89}{15\!\cdots\!35}a^{7}+\frac{10\!\cdots\!08}{15\!\cdots\!35}a^{6}-\frac{18\!\cdots\!49}{15\!\cdots\!35}a^{5}-\frac{50\!\cdots\!66}{51\!\cdots\!45}a^{4}+\frac{10\!\cdots\!03}{15\!\cdots\!35}a^{3}-\frac{66\!\cdots\!13}{43\!\cdots\!97}a^{2}+\frac{98\!\cdots\!94}{15\!\cdots\!35}a-\frac{10\!\cdots\!44}{15\!\cdots\!35}$, $\frac{59\!\cdots\!15}{30\!\cdots\!67}a^{31}-\frac{89\!\cdots\!28}{12\!\cdots\!15}a^{30}-\frac{10\!\cdots\!28}{17\!\cdots\!15}a^{29}+\frac{24\!\cdots\!18}{15\!\cdots\!35}a^{28}+\frac{10\!\cdots\!63}{10\!\cdots\!89}a^{27}-\frac{18\!\cdots\!82}{10\!\cdots\!89}a^{26}-\frac{40\!\cdots\!83}{39\!\cdots\!73}a^{25}+\frac{19\!\cdots\!69}{15\!\cdots\!35}a^{24}+\frac{10\!\cdots\!27}{17\!\cdots\!15}a^{23}-\frac{21\!\cdots\!00}{30\!\cdots\!67}a^{22}-\frac{33\!\cdots\!34}{15\!\cdots\!35}a^{21}+\frac{22\!\cdots\!18}{51\!\cdots\!45}a^{20}+\frac{72\!\cdots\!99}{15\!\cdots\!35}a^{19}-\frac{52\!\cdots\!72}{25\!\cdots\!35}a^{18}+\frac{86\!\cdots\!98}{15\!\cdots\!35}a^{17}+\frac{13\!\cdots\!47}{51\!\cdots\!45}a^{16}-\frac{80\!\cdots\!06}{15\!\cdots\!35}a^{15}+\frac{34\!\cdots\!92}{15\!\cdots\!35}a^{14}+\frac{81\!\cdots\!17}{15\!\cdots\!35}a^{13}-\frac{67\!\cdots\!31}{51\!\cdots\!45}a^{12}+\frac{11\!\cdots\!59}{17\!\cdots\!15}a^{11}+\frac{48\!\cdots\!96}{17\!\cdots\!15}a^{10}-\frac{41\!\cdots\!33}{17\!\cdots\!15}a^{9}-\frac{86\!\cdots\!14}{30\!\cdots\!67}a^{8}+\frac{80\!\cdots\!46}{30\!\cdots\!67}a^{7}+\frac{12\!\cdots\!51}{51\!\cdots\!45}a^{6}-\frac{31\!\cdots\!61}{30\!\cdots\!67}a^{5}-\frac{60\!\cdots\!47}{15\!\cdots\!35}a^{4}+\frac{51\!\cdots\!03}{15\!\cdots\!35}a^{3}-\frac{17\!\cdots\!84}{19\!\cdots\!65}a^{2}+\frac{88\!\cdots\!96}{30\!\cdots\!67}a-\frac{18\!\cdots\!35}{30\!\cdots\!67}$, $\frac{57\!\cdots\!68}{51\!\cdots\!45}a^{31}-\frac{10\!\cdots\!98}{41\!\cdots\!05}a^{30}-\frac{60\!\cdots\!87}{15\!\cdots\!35}a^{29}+\frac{49\!\cdots\!42}{15\!\cdots\!35}a^{28}+\frac{10\!\cdots\!32}{15\!\cdots\!35}a^{27}+\frac{72\!\cdots\!61}{51\!\cdots\!45}a^{26}-\frac{15\!\cdots\!78}{24\!\cdots\!35}a^{25}-\frac{38\!\cdots\!11}{15\!\cdots\!35}a^{24}+\frac{52\!\cdots\!07}{15\!\cdots\!35}a^{23}+\frac{59\!\cdots\!22}{51\!\cdots\!45}a^{22}-\frac{39\!\cdots\!40}{30\!\cdots\!67}a^{21}+\frac{17\!\cdots\!77}{30\!\cdots\!67}a^{20}+\frac{65\!\cdots\!07}{15\!\cdots\!35}a^{19}-\frac{20\!\cdots\!90}{34\!\cdots\!63}a^{18}-\frac{42\!\cdots\!78}{51\!\cdots\!45}a^{17}+\frac{10\!\cdots\!89}{15\!\cdots\!35}a^{16}-\frac{93\!\cdots\!41}{51\!\cdots\!45}a^{15}+\frac{15\!\cdots\!59}{15\!\cdots\!35}a^{14}+\frac{10\!\cdots\!87}{51\!\cdots\!45}a^{13}-\frac{79\!\cdots\!29}{15\!\cdots\!35}a^{12}-\frac{49\!\cdots\!79}{10\!\cdots\!89}a^{11}+\frac{18\!\cdots\!24}{15\!\cdots\!35}a^{10}+\frac{82\!\cdots\!28}{15\!\cdots\!35}a^{9}-\frac{73\!\cdots\!44}{51\!\cdots\!45}a^{8}-\frac{10\!\cdots\!48}{15\!\cdots\!35}a^{7}+\frac{32\!\cdots\!92}{30\!\cdots\!67}a^{6}+\frac{17\!\cdots\!98}{15\!\cdots\!35}a^{5}-\frac{54\!\cdots\!51}{51\!\cdots\!45}a^{4}-\frac{15\!\cdots\!64}{15\!\cdots\!35}a^{3}+\frac{32\!\cdots\!07}{21\!\cdots\!85}a^{2}-\frac{21\!\cdots\!19}{30\!\cdots\!67}a+\frac{19\!\cdots\!24}{15\!\cdots\!35}$, $\frac{49\!\cdots\!68}{30\!\cdots\!67}a^{31}-\frac{11\!\cdots\!11}{20\!\cdots\!15}a^{30}-\frac{77\!\cdots\!68}{15\!\cdots\!35}a^{29}+\frac{19\!\cdots\!07}{15\!\cdots\!35}a^{28}+\frac{15\!\cdots\!54}{17\!\cdots\!15}a^{27}-\frac{42\!\cdots\!66}{30\!\cdots\!67}a^{26}-\frac{11\!\cdots\!89}{13\!\cdots\!91}a^{25}+\frac{14\!\cdots\!63}{15\!\cdots\!35}a^{24}+\frac{26\!\cdots\!73}{51\!\cdots\!45}a^{23}-\frac{82\!\cdots\!99}{15\!\cdots\!35}a^{22}-\frac{97\!\cdots\!11}{51\!\cdots\!45}a^{21}+\frac{10\!\cdots\!12}{30\!\cdots\!67}a^{20}+\frac{66\!\cdots\!16}{15\!\cdots\!35}a^{19}-\frac{26\!\cdots\!41}{15\!\cdots\!35}a^{18}+\frac{44\!\cdots\!98}{17\!\cdots\!15}a^{17}+\frac{11\!\cdots\!56}{51\!\cdots\!45}a^{16}-\frac{41\!\cdots\!40}{10\!\cdots\!89}a^{15}+\frac{28\!\cdots\!22}{15\!\cdots\!35}a^{14}+\frac{11\!\cdots\!92}{17\!\cdots\!15}a^{13}-\frac{18\!\cdots\!53}{17\!\cdots\!15}a^{12}+\frac{66\!\cdots\!86}{15\!\cdots\!35}a^{11}+\frac{74\!\cdots\!31}{30\!\cdots\!67}a^{10}-\frac{88\!\cdots\!12}{51\!\cdots\!45}a^{9}-\frac{64\!\cdots\!04}{25\!\cdots\!35}a^{8}+\frac{63\!\cdots\!35}{34\!\cdots\!63}a^{7}+\frac{35\!\cdots\!46}{15\!\cdots\!35}a^{6}-\frac{81\!\cdots\!24}{15\!\cdots\!35}a^{5}-\frac{56\!\cdots\!21}{17\!\cdots\!15}a^{4}+\frac{36\!\cdots\!12}{15\!\cdots\!35}a^{3}-\frac{10\!\cdots\!79}{19\!\cdots\!65}a^{2}+\frac{34\!\cdots\!38}{15\!\cdots\!35}a-\frac{72\!\cdots\!78}{30\!\cdots\!67}$ (assuming GRH)
Regulator:		\( 298974970196971.9 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 298974970196971.9 \cdot 1800}{6\cdot\sqrt{7231362775399344187879888625220455562201364498198121976692736}}\cr\approx \mathstrut & 0.196799461191503 \end{aligned}\] (assuming GRH)

Galois group

$C_2\times C_4^2$ (as 32T36):

An abelian group of order 32

The 32 conjugacy class representatives for $C_2\times C_4^2$

Character table for $C_2\times C_4^2$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-78}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{16})^+\), 4.0.18432.2, \(\Q(\sqrt{-6}, \sqrt{13})\), \(\Q(\sqrt{-6}, \sqrt{26})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-39})\), 4.4.346112.1, 4.0.3115008.1, \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{26})\), 4.0.2197.1, 4.4.1265472.2, 4.0.140608.2, 4.4.19773.1, 4.0.4499456.2, 4.4.40495104.2, 4.0.4499456.1, 4.4.40495104.1, 8.0.339738624.2, 8.0.9475854336.2, 8.0.9703274840064.7, 8.8.119793516544.1, 8.0.9703274840064.5, 8.0.9703274840064.4, 8.0.9703274840064.1, 8.0.1601419382784.3, 8.0.1601419382784.1, 8.0.1639853447970816.40, 8.0.1639853447970816.63, 8.0.19770609664.2, 8.8.1601419382784.1, 8.0.20245104295936.1, 8.8.1639853447970816.2, 8.0.390971529.1, 8.0.1601419382784.5, 8.0.1639853447970816.65, 8.0.1639853447970816.1, 16.0.94153542621819044779524096.1, 16.0.2564544039556287515590656.1, 16.0.2689119330821773737947987705856.2, 16.0.409864247953326282266116096.2, 16.16.2689119330821773737947987705856.1, 16.0.2689119330821773737947987705856.4, 16.0.2689119330821773737947987705856.6

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	R	R	${\href{/padicField/5.4.0.1}{4} }^{8}$	${\href{/padicField/7.4.0.1}{4} }^{8}$	${\href{/padicField/11.4.0.1}{4} }^{8}$	R	${\href{/padicField/17.2.0.1}{2} }^{16}$	${\href{/padicField/19.4.0.1}{4} }^{8}$	${\href{/padicField/23.2.0.1}{2} }^{16}$	${\href{/padicField/29.4.0.1}{4} }^{8}$	${\href{/padicField/31.4.0.1}{4} }^{8}$	${\href{/padicField/37.4.0.1}{4} }^{8}$	${\href{/padicField/41.4.0.1}{4} }^{8}$	${\href{/padicField/43.4.0.1}{4} }^{8}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.4.0.1}{4} }^{8}$	${\href{/padicField/59.4.0.1}{4} }^{8}$

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
\(2\)		Deg $16$	$4$	$4$	$44$
		Deg $16$	$4$	$4$	$44$
\(3\)	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
\(13\)	13.16.12.1	$x^{16} + 12 x^{14} + 48 x^{13} + 114 x^{12} + 432 x^{11} + 888 x^{10} - 5280 x^{9} + 4933 x^{8} + 13680 x^{7} + 64788 x^{6} - 10416 x^{5} + 182568 x^{4} + 90432 x^{3} + 720840 x^{2} + 400992 x + 573316$	$4$	$4$	$12$	$C_4^2$	$[\ ]_{4}^{4}$
	13.16.12.1	$x^{16} + 12 x^{14} + 48 x^{13} + 114 x^{12} + 432 x^{11} + 888 x^{10} - 5280 x^{9} + 4933 x^{8} + 13680 x^{7} + 64788 x^{6} - 10416 x^{5} + 182568 x^{4} + 90432 x^{3} + 720840 x^{2} + 400992 x + 573316$	$4$	$4$	$12$	$C_4^2$	$[\ ]_{4}^{4}$