Number field 32.0.6142666889587199870339155304469168186187744140625.1

• Label: 32.0.614...625.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $6.143\times 10^{48}$
• Root discriminant: \(33.47\)
• Ramified primes: $5,67$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_8.A_4$ (as 32T402)

Normalized defining polynomial

x^32 - 4*x^31 + 38*x^30 - 130*x^29 + 699*x^28 - 2191*x^27 + 8745*x^26 - 25571*x^25 + 83386*x^24 - 222738*x^23 + 612551*x^22 - 1445800*x^21 + 3366820*x^20 - 6839883*x^19 + 13440516*x^18 - 23161945*x^17 + 38310126*x^16 - 55684805*x^15 + 77474162*x^14 - 94829103*x^13 + 110710129*x^12 - 113727761*x^11 + 110357640*x^10 - 94103904*x^9 + 74479117*x^8 - 51698016*x^7 + 32429638*x^6 - 17496051*x^5 + 7973462*x^4 - 3036857*x^3 + 868340*x^2 - 166271*x + 16531

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(6142666889587199870339155304469168186187744140625\) \(\medspace = 5^{28}\cdot 67^{16}\)
Root discriminant:		\(33.47\)
Galois root discriminant:		$5^{7/8}67^{2/3}\approx 67.45000024642073$
Ramified primes:		\(5\), \(67\)
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{11111}a^{30}+\frac{3099}{11111}a^{29}-\frac{1480}{11111}a^{28}-\frac{568}{11111}a^{27}-\frac{1279}{11111}a^{26}+\frac{1337}{11111}a^{25}+\frac{4661}{11111}a^{24}+\frac{1889}{11111}a^{23}+\frac{4889}{11111}a^{22}+\frac{2663}{11111}a^{21}-\frac{2921}{11111}a^{20}+\frac{3509}{11111}a^{19}+\frac{4063}{11111}a^{18}+\frac{1573}{11111}a^{17}-\frac{654}{11111}a^{16}+\frac{2569}{11111}a^{15}-\frac{2627}{11111}a^{14}-\frac{5052}{11111}a^{13}-\frac{500}{11111}a^{12}+\frac{505}{11111}a^{11}-\frac{5033}{11111}a^{10}+\frac{3346}{11111}a^{9}-\frac{3744}{11111}a^{8}+\frac{3427}{11111}a^{7}-\frac{1000}{11111}a^{6}+\frac{3973}{11111}a^{5}+\frac{2816}{11111}a^{4}+\frac{321}{11111}a^{3}+\frac{2226}{11111}a^{2}-\frac{2484}{11111}a+\frac{6}{41}$, $\frac{1}{10\!\cdots\!29}a^{31}+\frac{30\!\cdots\!25}{10\!\cdots\!29}a^{30}+\frac{71\!\cdots\!54}{10\!\cdots\!29}a^{29}+\frac{38\!\cdots\!28}{10\!\cdots\!29}a^{28}-\frac{49\!\cdots\!70}{10\!\cdots\!29}a^{27}-\frac{11\!\cdots\!61}{10\!\cdots\!29}a^{26}-\frac{81\!\cdots\!58}{10\!\cdots\!29}a^{25}-\frac{24\!\cdots\!83}{10\!\cdots\!29}a^{24}+\frac{39\!\cdots\!12}{10\!\cdots\!29}a^{23}-\frac{44\!\cdots\!17}{10\!\cdots\!29}a^{22}+\frac{68\!\cdots\!52}{10\!\cdots\!29}a^{21}-\frac{31\!\cdots\!84}{10\!\cdots\!29}a^{20}-\frac{42\!\cdots\!89}{10\!\cdots\!29}a^{19}-\frac{64\!\cdots\!93}{10\!\cdots\!29}a^{18}-\frac{18\!\cdots\!67}{10\!\cdots\!29}a^{17}-\frac{58\!\cdots\!60}{10\!\cdots\!29}a^{16}+\frac{24\!\cdots\!36}{10\!\cdots\!29}a^{15}-\frac{42\!\cdots\!40}{10\!\cdots\!29}a^{14}-\frac{27\!\cdots\!71}{10\!\cdots\!29}a^{13}-\frac{51\!\cdots\!29}{10\!\cdots\!29}a^{12}+\frac{35\!\cdots\!01}{10\!\cdots\!29}a^{11}+\frac{18\!\cdots\!32}{10\!\cdots\!29}a^{10}+\frac{12\!\cdots\!02}{10\!\cdots\!29}a^{9}+\frac{18\!\cdots\!69}{10\!\cdots\!29}a^{8}-\frac{90\!\cdots\!10}{25\!\cdots\!69}a^{7}-\frac{46\!\cdots\!13}{10\!\cdots\!29}a^{6}-\frac{50\!\cdots\!38}{10\!\cdots\!29}a^{5}+\frac{14\!\cdots\!19}{10\!\cdots\!29}a^{4}+\frac{47\!\cdots\!50}{10\!\cdots\!29}a^{3}+\frac{39\!\cdots\!84}{10\!\cdots\!29}a^{2}+\frac{46\!\cdots\!04}{10\!\cdots\!29}a+\frac{14\!\cdots\!58}{38\!\cdots\!99}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( \frac{90291092229261887443385456616074916398978755267412463416561909877613940161435669771329}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{31} - \frac{326293861374252267718960501123735594187278074872001671062473218909249456142379348799725}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{30} + \frac{3305154096143990392604649798762942607867306264720207923477624652218036895218836003202534}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{29} - \frac{10461690802345099154530269581539479045321507098775569027949299742209818807011522328606725}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{28} + \frac{59076540449768798365339443556406147821189119998109577865019001278374041805237391000487993}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{27} - \frac{175020865780655680054521017559555208821739954355861526584956218623243415861830902558917403}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{26} + \frac{722054428148362810852832897517949444384106217547533358114802914764800217807440121482780103}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{25} - \frac{2030102867700341092154519098787788542561405180585632607518354735676306694452163049517591225}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{24} + \frac{6745538009756878158670077952511831026407870567468592795902440264488931742230875893875663363}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{23} - \frac{17507456506922919028605150368419511115381553393756873992029625268648309339120729287570929712}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{22} + \frac{48550964685839112673543639217140380073995718013091683010443195043120534890322897081827171836}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{21} - \frac{2726887888613371311358439032071704143516050344997100249618906216885775007281681106714092074}{1926471398139242548692299969926083464153523628756370639078055861330368926975524629054759} a^{20} + \frac{260842161685592046478217690638601576196459598024569296311552414516854444441961965979294902467}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{19} - \frac{12607148598487985715691601977749226198458987208740649592266176275690648007890158250641956570}{1926471398139242548692299969926083464153523628756370639078055861330368926975524629054759} a^{18} + \frac{1014038302805579516883748690454047666626744080361581545680800514963866383412354709181118128334}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{17} - \frac{1699840212878636981318756132668485384640593490633787573152854743388825786644931185549283217772}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{16} + \frac{2802796859392509866194196150951587189128843535895512389770489831999055839979632407181712131291}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{15} - \frac{3945514531436039917140596518512799971134658899455870720061861706664154040159572196753446606073}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{14} + \frac{5471411309429574305726100198981996873782371108304937591515796293751668712357740845213176992967}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{13} - \frac{6448418347151372248884875151093679389592854692923362773680691665323768383649750950175658841641}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{12} + \frac{7504193913797034248152721199072468804507080997237263521587185760349461083497808181229392761477}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{11} - \frac{7367404887694245182183355247759212899870499318961047997562356619769340060480154655320472262050}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{10} + \frac{7114652527498930046212196958911145559339685665261738767429766698825685263160161798308668654987}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{9} - \frac{5743225305234312339309698853333507080650223695207073057414408368042493066144871539076000396105}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{8} + \frac{4500273156231488719635591244227838501192296891953780224759379956536807694684089138747815724966}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{7} - \frac{2923324478271884015395228983314938090778539961640820594981063836806908283906316601681829979214}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{6} + \frac{1793317501876108688802041042857819091330637363687747341190229034398474359440555154595714596640}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{5} - \frac{882720900799482757398234079484700737952316904690387446099147908964017412130318991753413929884}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{4} + \frac{375941787770800027486369121930360849013952961878983102335331883844167432202116229709525004752}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{3} - \frac{127182923862640809174451022277550069287811252883267530116690283917743722115071570374504254120}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{2} + \frac{28383098606015499999371450964778404350196797663676067794692006283517482652058428111959088483}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a - \frac{13695001339095496795758407553150380970912238336373525155267460422275494561694364526388153}{291458772412210127292930991760034767639462984424395557941698488245553970501832139451089} \)  (order $10$)
Fundamental units:		$\frac{57\!\cdots\!67}{78\!\cdots\!19}a^{31}-\frac{20\!\cdots\!39}{78\!\cdots\!19}a^{30}+\frac{21\!\cdots\!77}{78\!\cdots\!19}a^{29}-\frac{67\!\cdots\!54}{78\!\cdots\!19}a^{28}+\frac{37\!\cdots\!62}{78\!\cdots\!19}a^{27}-\frac{11\!\cdots\!46}{78\!\cdots\!19}a^{26}+\frac{46\!\cdots\!11}{78\!\cdots\!19}a^{25}-\frac{13\!\cdots\!97}{78\!\cdots\!19}a^{24}+\frac{43\!\cdots\!29}{78\!\cdots\!19}a^{23}-\frac{11\!\cdots\!83}{78\!\cdots\!19}a^{22}+\frac{31\!\cdots\!22}{78\!\cdots\!19}a^{21}-\frac{17\!\cdots\!60}{19\!\cdots\!59}a^{20}+\frac{16\!\cdots\!22}{78\!\cdots\!19}a^{19}-\frac{81\!\cdots\!32}{19\!\cdots\!59}a^{18}+\frac{65\!\cdots\!38}{78\!\cdots\!19}a^{17}-\frac{11\!\cdots\!23}{78\!\cdots\!19}a^{16}+\frac{18\!\cdots\!31}{78\!\cdots\!19}a^{15}-\frac{25\!\cdots\!68}{78\!\cdots\!19}a^{14}+\frac{35\!\cdots\!01}{78\!\cdots\!19}a^{13}-\frac{42\!\cdots\!44}{78\!\cdots\!19}a^{12}+\frac{48\!\cdots\!96}{78\!\cdots\!19}a^{11}-\frac{48\!\cdots\!47}{78\!\cdots\!19}a^{10}+\frac{46\!\cdots\!52}{78\!\cdots\!19}a^{9}-\frac{37\!\cdots\!39}{78\!\cdots\!19}a^{8}+\frac{29\!\cdots\!68}{78\!\cdots\!19}a^{7}-\frac{19\!\cdots\!08}{78\!\cdots\!19}a^{6}+\frac{11\!\cdots\!97}{78\!\cdots\!19}a^{5}-\frac{58\!\cdots\!57}{78\!\cdots\!19}a^{4}+\frac{25\!\cdots\!23}{78\!\cdots\!19}a^{3}-\frac{85\!\cdots\!07}{78\!\cdots\!19}a^{2}+\frac{19\!\cdots\!25}{78\!\cdots\!19}a-\frac{93\!\cdots\!55}{29\!\cdots\!89}$, $\frac{29\!\cdots\!70}{10\!\cdots\!29}a^{31}-\frac{11\!\cdots\!69}{10\!\cdots\!29}a^{30}+\frac{10\!\cdots\!17}{10\!\cdots\!29}a^{29}-\frac{35\!\cdots\!77}{10\!\cdots\!29}a^{28}+\frac{19\!\cdots\!94}{10\!\cdots\!29}a^{27}-\frac{60\!\cdots\!30}{10\!\cdots\!29}a^{26}+\frac{24\!\cdots\!04}{10\!\cdots\!29}a^{25}-\frac{69\!\cdots\!96}{10\!\cdots\!29}a^{24}+\frac{22\!\cdots\!28}{10\!\cdots\!29}a^{23}-\frac{60\!\cdots\!57}{10\!\cdots\!29}a^{22}+\frac{16\!\cdots\!40}{10\!\cdots\!29}a^{21}-\frac{38\!\cdots\!63}{10\!\cdots\!29}a^{20}+\frac{90\!\cdots\!28}{10\!\cdots\!29}a^{19}-\frac{18\!\cdots\!26}{10\!\cdots\!29}a^{18}+\frac{35\!\cdots\!91}{10\!\cdots\!29}a^{17}-\frac{60\!\cdots\!87}{10\!\cdots\!29}a^{16}+\frac{99\!\cdots\!81}{10\!\cdots\!29}a^{15}-\frac{34\!\cdots\!42}{25\!\cdots\!69}a^{14}+\frac{19\!\cdots\!82}{10\!\cdots\!29}a^{13}-\frac{23\!\cdots\!70}{10\!\cdots\!29}a^{12}+\frac{27\!\cdots\!12}{10\!\cdots\!29}a^{11}-\frac{27\!\cdots\!40}{10\!\cdots\!29}a^{10}+\frac{26\!\cdots\!61}{10\!\cdots\!29}a^{9}-\frac{22\!\cdots\!33}{10\!\cdots\!29}a^{8}+\frac{17\!\cdots\!20}{10\!\cdots\!29}a^{7}-\frac{11\!\cdots\!90}{10\!\cdots\!29}a^{6}+\frac{72\!\cdots\!22}{10\!\cdots\!29}a^{5}-\frac{37\!\cdots\!43}{10\!\cdots\!29}a^{4}+\frac{16\!\cdots\!01}{10\!\cdots\!29}a^{3}-\frac{57\!\cdots\!71}{10\!\cdots\!29}a^{2}+\frac{13\!\cdots\!33}{10\!\cdots\!29}a-\frac{62\!\cdots\!88}{38\!\cdots\!99}$, $\frac{94\!\cdots\!81}{10\!\cdots\!29}a^{31}-\frac{33\!\cdots\!21}{10\!\cdots\!29}a^{30}+\frac{34\!\cdots\!71}{10\!\cdots\!29}a^{29}-\frac{10\!\cdots\!25}{10\!\cdots\!29}a^{28}+\frac{61\!\cdots\!92}{10\!\cdots\!29}a^{27}-\frac{18\!\cdots\!98}{10\!\cdots\!29}a^{26}+\frac{75\!\cdots\!86}{10\!\cdots\!29}a^{25}-\frac{21\!\cdots\!94}{10\!\cdots\!29}a^{24}+\frac{70\!\cdots\!70}{10\!\cdots\!29}a^{23}-\frac{18\!\cdots\!25}{10\!\cdots\!29}a^{22}+\frac{50\!\cdots\!91}{10\!\cdots\!29}a^{21}-\frac{11\!\cdots\!91}{10\!\cdots\!29}a^{20}+\frac{27\!\cdots\!54}{10\!\cdots\!29}a^{19}-\frac{53\!\cdots\!39}{10\!\cdots\!29}a^{18}+\frac{10\!\cdots\!62}{10\!\cdots\!29}a^{17}-\frac{17\!\cdots\!03}{10\!\cdots\!29}a^{16}+\frac{28\!\cdots\!96}{10\!\cdots\!29}a^{15}-\frac{40\!\cdots\!43}{10\!\cdots\!29}a^{14}+\frac{56\!\cdots\!98}{10\!\cdots\!29}a^{13}-\frac{66\!\cdots\!67}{10\!\cdots\!29}a^{12}+\frac{77\!\cdots\!26}{10\!\cdots\!29}a^{11}-\frac{75\!\cdots\!68}{10\!\cdots\!29}a^{10}+\frac{73\!\cdots\!96}{10\!\cdots\!29}a^{9}-\frac{58\!\cdots\!40}{10\!\cdots\!29}a^{8}+\frac{46\!\cdots\!18}{10\!\cdots\!29}a^{7}-\frac{29\!\cdots\!85}{10\!\cdots\!29}a^{6}+\frac{18\!\cdots\!26}{10\!\cdots\!29}a^{5}-\frac{89\!\cdots\!68}{10\!\cdots\!29}a^{4}+\frac{37\!\cdots\!13}{10\!\cdots\!29}a^{3}-\frac{12\!\cdots\!13}{10\!\cdots\!29}a^{2}+\frac{28\!\cdots\!91}{10\!\cdots\!29}a-\frac{13\!\cdots\!46}{38\!\cdots\!99}$, $\frac{10\!\cdots\!64}{10\!\cdots\!29}a^{31}-\frac{36\!\cdots\!42}{10\!\cdots\!29}a^{30}+\frac{36\!\cdots\!43}{10\!\cdots\!29}a^{29}-\frac{11\!\cdots\!10}{10\!\cdots\!29}a^{28}+\frac{65\!\cdots\!47}{10\!\cdots\!29}a^{27}-\frac{19\!\cdots\!15}{10\!\cdots\!29}a^{26}+\frac{80\!\cdots\!40}{10\!\cdots\!29}a^{25}-\frac{22\!\cdots\!58}{10\!\cdots\!29}a^{24}+\frac{74\!\cdots\!91}{10\!\cdots\!29}a^{23}-\frac{19\!\cdots\!82}{10\!\cdots\!29}a^{22}+\frac{53\!\cdots\!96}{10\!\cdots\!29}a^{21}-\frac{12\!\cdots\!11}{10\!\cdots\!29}a^{20}+\frac{28\!\cdots\!58}{10\!\cdots\!29}a^{19}-\frac{57\!\cdots\!71}{10\!\cdots\!29}a^{18}+\frac{11\!\cdots\!95}{10\!\cdots\!29}a^{17}-\frac{18\!\cdots\!43}{10\!\cdots\!29}a^{16}+\frac{30\!\cdots\!81}{10\!\cdots\!29}a^{15}-\frac{43\!\cdots\!76}{10\!\cdots\!29}a^{14}+\frac{60\!\cdots\!91}{10\!\cdots\!29}a^{13}-\frac{70\!\cdots\!03}{10\!\cdots\!29}a^{12}+\frac{82\!\cdots\!59}{10\!\cdots\!29}a^{11}-\frac{80\!\cdots\!47}{10\!\cdots\!29}a^{10}+\frac{77\!\cdots\!96}{10\!\cdots\!29}a^{9}-\frac{62\!\cdots\!47}{10\!\cdots\!29}a^{8}+\frac{49\!\cdots\!89}{10\!\cdots\!29}a^{7}-\frac{31\!\cdots\!26}{10\!\cdots\!29}a^{6}+\frac{19\!\cdots\!91}{10\!\cdots\!29}a^{5}-\frac{95\!\cdots\!97}{10\!\cdots\!29}a^{4}+\frac{40\!\cdots\!12}{10\!\cdots\!29}a^{3}-\frac{13\!\cdots\!52}{10\!\cdots\!29}a^{2}+\frac{31\!\cdots\!61}{10\!\cdots\!29}a-\frac{37\!\cdots\!94}{94\!\cdots\!39}$, $\frac{58\!\cdots\!86}{10\!\cdots\!29}a^{31}-\frac{20\!\cdots\!55}{10\!\cdots\!29}a^{30}+\frac{21\!\cdots\!86}{10\!\cdots\!29}a^{29}-\frac{67\!\cdots\!73}{10\!\cdots\!29}a^{28}+\frac{38\!\cdots\!04}{10\!\cdots\!29}a^{27}-\frac{11\!\cdots\!34}{10\!\cdots\!29}a^{26}+\frac{46\!\cdots\!43}{10\!\cdots\!29}a^{25}-\frac{13\!\cdots\!44}{10\!\cdots\!29}a^{24}+\frac{43\!\cdots\!50}{10\!\cdots\!29}a^{23}-\frac{11\!\cdots\!81}{10\!\cdots\!29}a^{22}+\frac{31\!\cdots\!92}{10\!\cdots\!29}a^{21}-\frac{71\!\cdots\!05}{10\!\cdots\!29}a^{20}+\frac{16\!\cdots\!34}{10\!\cdots\!29}a^{19}-\frac{33\!\cdots\!92}{10\!\cdots\!29}a^{18}+\frac{65\!\cdots\!33}{10\!\cdots\!29}a^{17}-\frac{10\!\cdots\!10}{10\!\cdots\!29}a^{16}+\frac{18\!\cdots\!28}{10\!\cdots\!29}a^{15}-\frac{25\!\cdots\!02}{10\!\cdots\!29}a^{14}+\frac{35\!\cdots\!91}{10\!\cdots\!29}a^{13}-\frac{41\!\cdots\!57}{10\!\cdots\!29}a^{12}+\frac{48\!\cdots\!61}{10\!\cdots\!29}a^{11}-\frac{47\!\cdots\!24}{10\!\cdots\!29}a^{10}+\frac{45\!\cdots\!69}{10\!\cdots\!29}a^{9}-\frac{36\!\cdots\!93}{10\!\cdots\!29}a^{8}+\frac{28\!\cdots\!06}{10\!\cdots\!29}a^{7}-\frac{18\!\cdots\!44}{10\!\cdots\!29}a^{6}+\frac{11\!\cdots\!29}{10\!\cdots\!29}a^{5}-\frac{56\!\cdots\!44}{10\!\cdots\!29}a^{4}+\frac{24\!\cdots\!96}{10\!\cdots\!29}a^{3}-\frac{81\!\cdots\!81}{10\!\cdots\!29}a^{2}+\frac{18\!\cdots\!66}{10\!\cdots\!29}a-\frac{90\!\cdots\!53}{38\!\cdots\!99}$, $\frac{50\!\cdots\!58}{10\!\cdots\!29}a^{31}-\frac{18\!\cdots\!36}{10\!\cdots\!29}a^{30}+\frac{18\!\cdots\!05}{10\!\cdots\!29}a^{29}-\frac{58\!\cdots\!66}{10\!\cdots\!29}a^{28}+\frac{32\!\cdots\!13}{10\!\cdots\!29}a^{27}-\frac{23\!\cdots\!91}{25\!\cdots\!69}a^{26}+\frac{40\!\cdots\!18}{10\!\cdots\!29}a^{25}-\frac{11\!\cdots\!14}{10\!\cdots\!29}a^{24}+\frac{37\!\cdots\!78}{10\!\cdots\!29}a^{23}-\frac{96\!\cdots\!19}{10\!\cdots\!29}a^{22}+\frac{26\!\cdots\!65}{10\!\cdots\!29}a^{21}-\frac{61\!\cdots\!06}{10\!\cdots\!29}a^{20}+\frac{14\!\cdots\!37}{10\!\cdots\!29}a^{19}-\frac{28\!\cdots\!53}{10\!\cdots\!29}a^{18}+\frac{55\!\cdots\!58}{10\!\cdots\!29}a^{17}-\frac{93\!\cdots\!50}{10\!\cdots\!29}a^{16}+\frac{15\!\cdots\!49}{10\!\cdots\!29}a^{15}-\frac{21\!\cdots\!09}{10\!\cdots\!29}a^{14}+\frac{29\!\cdots\!41}{10\!\cdots\!29}a^{13}-\frac{34\!\cdots\!06}{10\!\cdots\!29}a^{12}+\frac{40\!\cdots\!32}{10\!\cdots\!29}a^{11}-\frac{39\!\cdots\!05}{10\!\cdots\!29}a^{10}+\frac{38\!\cdots\!49}{10\!\cdots\!29}a^{9}-\frac{30\!\cdots\!55}{10\!\cdots\!29}a^{8}+\frac{23\!\cdots\!42}{10\!\cdots\!29}a^{7}-\frac{15\!\cdots\!39}{10\!\cdots\!29}a^{6}+\frac{93\!\cdots\!45}{10\!\cdots\!29}a^{5}-\frac{44\!\cdots\!14}{10\!\cdots\!29}a^{4}+\frac{19\!\cdots\!82}{10\!\cdots\!29}a^{3}-\frac{62\!\cdots\!67}{10\!\cdots\!29}a^{2}+\frac{13\!\cdots\!97}{10\!\cdots\!29}a-\frac{63\!\cdots\!48}{38\!\cdots\!99}$, $\frac{91\!\cdots\!14}{10\!\cdots\!29}a^{31}-\frac{33\!\cdots\!90}{10\!\cdots\!29}a^{30}+\frac{33\!\cdots\!98}{10\!\cdots\!29}a^{29}-\frac{10\!\cdots\!29}{10\!\cdots\!29}a^{28}+\frac{60\!\cdots\!11}{10\!\cdots\!29}a^{27}-\frac{17\!\cdots\!11}{10\!\cdots\!29}a^{26}+\frac{73\!\cdots\!61}{10\!\cdots\!29}a^{25}-\frac{20\!\cdots\!11}{10\!\cdots\!29}a^{24}+\frac{68\!\cdots\!12}{10\!\cdots\!29}a^{23}-\frac{17\!\cdots\!20}{10\!\cdots\!29}a^{22}+\frac{49\!\cdots\!88}{10\!\cdots\!29}a^{21}-\frac{11\!\cdots\!46}{10\!\cdots\!29}a^{20}+\frac{26\!\cdots\!68}{10\!\cdots\!29}a^{19}-\frac{53\!\cdots\!17}{10\!\cdots\!29}a^{18}+\frac{10\!\cdots\!70}{10\!\cdots\!29}a^{17}-\frac{17\!\cdots\!75}{10\!\cdots\!29}a^{16}+\frac{28\!\cdots\!74}{10\!\cdots\!29}a^{15}-\frac{40\!\cdots\!14}{10\!\cdots\!29}a^{14}+\frac{56\!\cdots\!58}{10\!\cdots\!29}a^{13}-\frac{66\!\cdots\!37}{10\!\cdots\!29}a^{12}+\frac{76\!\cdots\!16}{10\!\cdots\!29}a^{11}-\frac{75\!\cdots\!81}{10\!\cdots\!29}a^{10}+\frac{72\!\cdots\!25}{10\!\cdots\!29}a^{9}-\frac{58\!\cdots\!80}{10\!\cdots\!29}a^{8}+\frac{45\!\cdots\!48}{10\!\cdots\!29}a^{7}-\frac{29\!\cdots\!36}{10\!\cdots\!29}a^{6}+\frac{18\!\cdots\!18}{10\!\cdots\!29}a^{5}-\frac{89\!\cdots\!43}{10\!\cdots\!29}a^{4}+\frac{37\!\cdots\!33}{10\!\cdots\!29}a^{3}-\frac{12\!\cdots\!79}{10\!\cdots\!29}a^{2}+\frac{27\!\cdots\!81}{10\!\cdots\!29}a-\frac{12\!\cdots\!64}{38\!\cdots\!99}$, $\frac{11\!\cdots\!10}{10\!\cdots\!29}a^{31}-\frac{35\!\cdots\!65}{10\!\cdots\!29}a^{30}+\frac{38\!\cdots\!33}{10\!\cdots\!29}a^{29}-\frac{11\!\cdots\!86}{10\!\cdots\!29}a^{28}+\frac{67\!\cdots\!09}{10\!\cdots\!29}a^{27}-\frac{18\!\cdots\!95}{10\!\cdots\!29}a^{26}+\frac{80\!\cdots\!84}{10\!\cdots\!29}a^{25}-\frac{21\!\cdots\!20}{10\!\cdots\!29}a^{24}+\frac{73\!\cdots\!86}{10\!\cdots\!29}a^{23}-\frac{18\!\cdots\!47}{10\!\cdots\!29}a^{22}+\frac{51\!\cdots\!65}{10\!\cdots\!29}a^{21}-\frac{11\!\cdots\!58}{10\!\cdots\!29}a^{20}+\frac{26\!\cdots\!96}{10\!\cdots\!29}a^{19}-\frac{51\!\cdots\!49}{10\!\cdots\!29}a^{18}+\frac{10\!\cdots\!48}{10\!\cdots\!29}a^{17}-\frac{16\!\cdots\!33}{10\!\cdots\!29}a^{16}+\frac{27\!\cdots\!68}{10\!\cdots\!29}a^{15}-\frac{36\!\cdots\!01}{10\!\cdots\!29}a^{14}+\frac{51\!\cdots\!00}{10\!\cdots\!29}a^{13}-\frac{57\!\cdots\!82}{10\!\cdots\!29}a^{12}+\frac{68\!\cdots\!34}{10\!\cdots\!29}a^{11}-\frac{63\!\cdots\!30}{10\!\cdots\!29}a^{10}+\frac{62\!\cdots\!07}{10\!\cdots\!29}a^{9}-\frac{47\!\cdots\!72}{10\!\cdots\!29}a^{8}+\frac{38\!\cdots\!18}{10\!\cdots\!29}a^{7}-\frac{23\!\cdots\!08}{10\!\cdots\!29}a^{6}+\frac{15\!\cdots\!40}{10\!\cdots\!29}a^{5}-\frac{72\!\cdots\!73}{10\!\cdots\!29}a^{4}+\frac{32\!\cdots\!45}{10\!\cdots\!29}a^{3}-\frac{12\!\cdots\!62}{10\!\cdots\!29}a^{2}+\frac{28\!\cdots\!86}{10\!\cdots\!29}a-\frac{16\!\cdots\!59}{38\!\cdots\!99}$, $\frac{15\!\cdots\!67}{10\!\cdots\!29}a^{31}-\frac{57\!\cdots\!51}{10\!\cdots\!29}a^{30}+\frac{57\!\cdots\!49}{10\!\cdots\!29}a^{29}-\frac{18\!\cdots\!08}{10\!\cdots\!29}a^{28}+\frac{10\!\cdots\!72}{10\!\cdots\!29}a^{27}-\frac{30\!\cdots\!32}{10\!\cdots\!29}a^{26}+\frac{12\!\cdots\!92}{10\!\cdots\!29}a^{25}-\frac{35\!\cdots\!76}{10\!\cdots\!29}a^{24}+\frac{11\!\cdots\!51}{10\!\cdots\!29}a^{23}-\frac{30\!\cdots\!90}{10\!\cdots\!29}a^{22}+\frac{85\!\cdots\!21}{10\!\cdots\!29}a^{21}-\frac{19\!\cdots\!40}{10\!\cdots\!29}a^{20}+\frac{45\!\cdots\!61}{10\!\cdots\!29}a^{19}-\frac{90\!\cdots\!65}{10\!\cdots\!29}a^{18}+\frac{17\!\cdots\!44}{10\!\cdots\!29}a^{17}-\frac{29\!\cdots\!63}{10\!\cdots\!29}a^{16}+\frac{48\!\cdots\!65}{10\!\cdots\!29}a^{15}-\frac{68\!\cdots\!46}{10\!\cdots\!29}a^{14}+\frac{94\!\cdots\!20}{10\!\cdots\!29}a^{13}-\frac{11\!\cdots\!21}{10\!\cdots\!29}a^{12}+\frac{12\!\cdots\!37}{10\!\cdots\!29}a^{11}-\frac{12\!\cdots\!38}{10\!\cdots\!29}a^{10}+\frac{12\!\cdots\!05}{10\!\cdots\!29}a^{9}-\frac{96\!\cdots\!52}{10\!\cdots\!29}a^{8}+\frac{74\!\cdots\!65}{10\!\cdots\!29}a^{7}-\frac{47\!\cdots\!22}{10\!\cdots\!29}a^{6}+\frac{28\!\cdots\!34}{10\!\cdots\!29}a^{5}-\frac{13\!\cdots\!27}{10\!\cdots\!29}a^{4}+\frac{53\!\cdots\!44}{10\!\cdots\!29}a^{3}-\frac{17\!\cdots\!53}{10\!\cdots\!29}a^{2}+\frac{26\!\cdots\!89}{10\!\cdots\!29}a+\frac{36\!\cdots\!88}{38\!\cdots\!99}$, $\frac{35\!\cdots\!96}{10\!\cdots\!29}a^{31}-\frac{12\!\cdots\!36}{10\!\cdots\!29}a^{30}+\frac{12\!\cdots\!74}{10\!\cdots\!29}a^{29}-\frac{40\!\cdots\!78}{10\!\cdots\!29}a^{28}+\frac{23\!\cdots\!25}{10\!\cdots\!29}a^{27}-\frac{68\!\cdots\!14}{10\!\cdots\!29}a^{26}+\frac{28\!\cdots\!82}{10\!\cdots\!29}a^{25}-\frac{79\!\cdots\!30}{10\!\cdots\!29}a^{24}+\frac{26\!\cdots\!17}{10\!\cdots\!29}a^{23}-\frac{68\!\cdots\!05}{10\!\cdots\!29}a^{22}+\frac{18\!\cdots\!59}{10\!\cdots\!29}a^{21}-\frac{43\!\cdots\!71}{10\!\cdots\!29}a^{20}+\frac{10\!\cdots\!72}{10\!\cdots\!29}a^{19}-\frac{20\!\cdots\!11}{10\!\cdots\!29}a^{18}+\frac{39\!\cdots\!60}{10\!\cdots\!29}a^{17}-\frac{66\!\cdots\!31}{10\!\cdots\!29}a^{16}+\frac{10\!\cdots\!88}{10\!\cdots\!29}a^{15}-\frac{15\!\cdots\!01}{10\!\cdots\!29}a^{14}+\frac{51\!\cdots\!89}{25\!\cdots\!69}a^{13}-\frac{25\!\cdots\!46}{10\!\cdots\!29}a^{12}+\frac{29\!\cdots\!33}{10\!\cdots\!29}a^{11}-\frac{28\!\cdots\!24}{10\!\cdots\!29}a^{10}+\frac{27\!\cdots\!11}{10\!\cdots\!29}a^{9}-\frac{22\!\cdots\!54}{10\!\cdots\!29}a^{8}+\frac{17\!\cdots\!87}{10\!\cdots\!29}a^{7}-\frac{11\!\cdots\!93}{10\!\cdots\!29}a^{6}+\frac{69\!\cdots\!46}{10\!\cdots\!29}a^{5}-\frac{83\!\cdots\!75}{25\!\cdots\!69}a^{4}+\frac{35\!\cdots\!51}{25\!\cdots\!69}a^{3}-\frac{49\!\cdots\!65}{10\!\cdots\!29}a^{2}+\frac{11\!\cdots\!24}{10\!\cdots\!29}a-\frac{56\!\cdots\!99}{38\!\cdots\!99}$, $\frac{11\!\cdots\!90}{10\!\cdots\!29}a^{31}-\frac{42\!\cdots\!33}{10\!\cdots\!29}a^{30}+\frac{42\!\cdots\!09}{10\!\cdots\!29}a^{29}-\frac{13\!\cdots\!39}{10\!\cdots\!29}a^{28}+\frac{76\!\cdots\!29}{10\!\cdots\!29}a^{27}-\frac{22\!\cdots\!44}{10\!\cdots\!29}a^{26}+\frac{93\!\cdots\!70}{10\!\cdots\!29}a^{25}-\frac{26\!\cdots\!18}{10\!\cdots\!29}a^{24}+\frac{87\!\cdots\!10}{10\!\cdots\!29}a^{23}-\frac{22\!\cdots\!39}{10\!\cdots\!29}a^{22}+\frac{62\!\cdots\!80}{10\!\cdots\!29}a^{21}-\frac{14\!\cdots\!21}{10\!\cdots\!29}a^{20}+\frac{33\!\cdots\!54}{10\!\cdots\!29}a^{19}-\frac{67\!\cdots\!63}{10\!\cdots\!29}a^{18}+\frac{13\!\cdots\!67}{10\!\cdots\!29}a^{17}-\frac{22\!\cdots\!38}{10\!\cdots\!29}a^{16}+\frac{36\!\cdots\!15}{10\!\cdots\!29}a^{15}-\frac{51\!\cdots\!45}{10\!\cdots\!29}a^{14}+\frac{70\!\cdots\!59}{10\!\cdots\!29}a^{13}-\frac{83\!\cdots\!94}{10\!\cdots\!29}a^{12}+\frac{96\!\cdots\!26}{10\!\cdots\!29}a^{11}-\frac{95\!\cdots\!91}{10\!\cdots\!29}a^{10}+\frac{91\!\cdots\!34}{10\!\cdots\!29}a^{9}-\frac{18\!\cdots\!14}{25\!\cdots\!69}a^{8}+\frac{57\!\cdots\!86}{10\!\cdots\!29}a^{7}-\frac{37\!\cdots\!44}{10\!\cdots\!29}a^{6}+\frac{22\!\cdots\!46}{10\!\cdots\!29}a^{5}-\frac{11\!\cdots\!78}{10\!\cdots\!29}a^{4}+\frac{47\!\cdots\!66}{10\!\cdots\!29}a^{3}-\frac{15\!\cdots\!92}{10\!\cdots\!29}a^{2}+\frac{35\!\cdots\!39}{10\!\cdots\!29}a-\frac{15\!\cdots\!94}{38\!\cdots\!99}$, $\frac{14\!\cdots\!22}{10\!\cdots\!29}a^{31}-\frac{50\!\cdots\!14}{10\!\cdots\!29}a^{30}+\frac{51\!\cdots\!62}{10\!\cdots\!29}a^{29}-\frac{16\!\cdots\!34}{10\!\cdots\!29}a^{28}+\frac{92\!\cdots\!30}{10\!\cdots\!29}a^{27}-\frac{27\!\cdots\!50}{10\!\cdots\!29}a^{26}+\frac{27\!\cdots\!24}{25\!\cdots\!69}a^{25}-\frac{31\!\cdots\!04}{10\!\cdots\!29}a^{24}+\frac{10\!\cdots\!90}{10\!\cdots\!29}a^{23}-\frac{27\!\cdots\!99}{10\!\cdots\!29}a^{22}+\frac{75\!\cdots\!83}{10\!\cdots\!29}a^{21}-\frac{17\!\cdots\!00}{10\!\cdots\!29}a^{20}+\frac{40\!\cdots\!20}{10\!\cdots\!29}a^{19}-\frac{80\!\cdots\!12}{10\!\cdots\!29}a^{18}+\frac{15\!\cdots\!58}{10\!\cdots\!29}a^{17}-\frac{26\!\cdots\!31}{10\!\cdots\!29}a^{16}+\frac{43\!\cdots\!60}{10\!\cdots\!29}a^{15}-\frac{61\!\cdots\!13}{10\!\cdots\!29}a^{14}+\frac{85\!\cdots\!12}{10\!\cdots\!29}a^{13}-\frac{10\!\cdots\!80}{10\!\cdots\!29}a^{12}+\frac{11\!\cdots\!62}{10\!\cdots\!29}a^{11}-\frac{11\!\cdots\!16}{10\!\cdots\!29}a^{10}+\frac{27\!\cdots\!38}{25\!\cdots\!69}a^{9}-\frac{89\!\cdots\!89}{10\!\cdots\!29}a^{8}+\frac{70\!\cdots\!51}{10\!\cdots\!29}a^{7}-\frac{45\!\cdots\!35}{10\!\cdots\!29}a^{6}+\frac{28\!\cdots\!32}{10\!\cdots\!29}a^{5}-\frac{13\!\cdots\!09}{10\!\cdots\!29}a^{4}+\frac{59\!\cdots\!24}{10\!\cdots\!29}a^{3}-\frac{19\!\cdots\!82}{10\!\cdots\!29}a^{2}+\frac{45\!\cdots\!26}{10\!\cdots\!29}a-\frac{22\!\cdots\!40}{38\!\cdots\!99}$, $\frac{49\!\cdots\!17}{10\!\cdots\!29}a^{31}-\frac{17\!\cdots\!02}{10\!\cdots\!29}a^{30}+\frac{18\!\cdots\!95}{10\!\cdots\!29}a^{29}-\frac{57\!\cdots\!06}{10\!\cdots\!29}a^{28}+\frac{32\!\cdots\!48}{10\!\cdots\!29}a^{27}-\frac{95\!\cdots\!41}{10\!\cdots\!29}a^{26}+\frac{39\!\cdots\!21}{10\!\cdots\!29}a^{25}-\frac{11\!\cdots\!35}{10\!\cdots\!29}a^{24}+\frac{36\!\cdots\!23}{10\!\cdots\!29}a^{23}-\frac{95\!\cdots\!39}{10\!\cdots\!29}a^{22}+\frac{26\!\cdots\!32}{10\!\cdots\!29}a^{21}-\frac{61\!\cdots\!38}{10\!\cdots\!29}a^{20}+\frac{14\!\cdots\!83}{10\!\cdots\!29}a^{19}-\frac{28\!\cdots\!19}{10\!\cdots\!29}a^{18}+\frac{55\!\cdots\!56}{10\!\cdots\!29}a^{17}-\frac{92\!\cdots\!01}{10\!\cdots\!29}a^{16}+\frac{15\!\cdots\!32}{10\!\cdots\!29}a^{15}-\frac{21\!\cdots\!71}{10\!\cdots\!29}a^{14}+\frac{29\!\cdots\!09}{10\!\cdots\!29}a^{13}-\frac{35\!\cdots\!88}{10\!\cdots\!29}a^{12}+\frac{40\!\cdots\!55}{10\!\cdots\!29}a^{11}-\frac{40\!\cdots\!11}{10\!\cdots\!29}a^{10}+\frac{38\!\cdots\!91}{10\!\cdots\!29}a^{9}-\frac{31\!\cdots\!08}{10\!\cdots\!29}a^{8}+\frac{24\!\cdots\!22}{10\!\cdots\!29}a^{7}-\frac{15\!\cdots\!43}{10\!\cdots\!29}a^{6}+\frac{97\!\cdots\!77}{10\!\cdots\!29}a^{5}-\frac{48\!\cdots\!96}{10\!\cdots\!29}a^{4}+\frac{20\!\cdots\!94}{10\!\cdots\!29}a^{3}-\frac{69\!\cdots\!88}{10\!\cdots\!29}a^{2}+\frac{15\!\cdots\!04}{10\!\cdots\!29}a-\frac{18\!\cdots\!52}{94\!\cdots\!39}$, $\frac{39\!\cdots\!60}{10\!\cdots\!29}a^{31}-\frac{14\!\cdots\!43}{10\!\cdots\!29}a^{30}+\frac{14\!\cdots\!83}{10\!\cdots\!29}a^{29}-\frac{45\!\cdots\!83}{10\!\cdots\!29}a^{28}+\frac{25\!\cdots\!38}{10\!\cdots\!29}a^{27}-\frac{76\!\cdots\!77}{10\!\cdots\!29}a^{26}+\frac{31\!\cdots\!69}{10\!\cdots\!29}a^{25}-\frac{88\!\cdots\!53}{10\!\cdots\!29}a^{24}+\frac{29\!\cdots\!65}{10\!\cdots\!29}a^{23}-\frac{76\!\cdots\!88}{10\!\cdots\!29}a^{22}+\frac{21\!\cdots\!40}{10\!\cdots\!29}a^{21}-\frac{48\!\cdots\!04}{10\!\cdots\!29}a^{20}+\frac{11\!\cdots\!58}{10\!\cdots\!29}a^{19}-\frac{22\!\cdots\!42}{10\!\cdots\!29}a^{18}+\frac{10\!\cdots\!93}{25\!\cdots\!69}a^{17}-\frac{73\!\cdots\!87}{10\!\cdots\!29}a^{16}+\frac{12\!\cdots\!01}{10\!\cdots\!29}a^{15}-\frac{17\!\cdots\!73}{10\!\cdots\!29}a^{14}+\frac{23\!\cdots\!48}{10\!\cdots\!29}a^{13}-\frac{27\!\cdots\!36}{10\!\cdots\!29}a^{12}+\frac{32\!\cdots\!82}{10\!\cdots\!29}a^{11}-\frac{31\!\cdots\!64}{10\!\cdots\!29}a^{10}+\frac{30\!\cdots\!13}{10\!\cdots\!29}a^{9}-\frac{24\!\cdots\!63}{10\!\cdots\!29}a^{8}+\frac{19\!\cdots\!03}{10\!\cdots\!29}a^{7}-\frac{12\!\cdots\!70}{10\!\cdots\!29}a^{6}+\frac{75\!\cdots\!28}{10\!\cdots\!29}a^{5}-\frac{37\!\cdots\!04}{10\!\cdots\!29}a^{4}+\frac{15\!\cdots\!28}{10\!\cdots\!29}a^{3}-\frac{53\!\cdots\!65}{10\!\cdots\!29}a^{2}+\frac{11\!\cdots\!73}{10\!\cdots\!29}a-\frac{51\!\cdots\!97}{38\!\cdots\!99}$, $\frac{19\!\cdots\!71}{10\!\cdots\!29}a^{31}-\frac{78\!\cdots\!36}{10\!\cdots\!29}a^{30}+\frac{74\!\cdots\!79}{10\!\cdots\!29}a^{29}-\frac{25\!\cdots\!76}{10\!\cdots\!29}a^{28}+\frac{13\!\cdots\!45}{10\!\cdots\!29}a^{27}-\frac{43\!\cdots\!47}{10\!\cdots\!29}a^{26}+\frac{17\!\cdots\!71}{10\!\cdots\!29}a^{25}-\frac{50\!\cdots\!03}{10\!\cdots\!29}a^{24}+\frac{16\!\cdots\!49}{10\!\cdots\!29}a^{23}-\frac{43\!\cdots\!21}{10\!\cdots\!29}a^{22}+\frac{12\!\cdots\!32}{10\!\cdots\!29}a^{21}-\frac{28\!\cdots\!01}{10\!\cdots\!29}a^{20}+\frac{66\!\cdots\!27}{10\!\cdots\!29}a^{19}-\frac{13\!\cdots\!84}{10\!\cdots\!29}a^{18}+\frac{26\!\cdots\!80}{10\!\cdots\!29}a^{17}-\frac{45\!\cdots\!02}{10\!\cdots\!29}a^{16}+\frac{74\!\cdots\!69}{10\!\cdots\!29}a^{15}-\frac{10\!\cdots\!64}{10\!\cdots\!29}a^{14}+\frac{14\!\cdots\!95}{10\!\cdots\!29}a^{13}-\frac{18\!\cdots\!80}{10\!\cdots\!29}a^{12}+\frac{21\!\cdots\!45}{10\!\cdots\!29}a^{11}-\frac{21\!\cdots\!90}{10\!\cdots\!29}a^{10}+\frac{20\!\cdots\!83}{10\!\cdots\!29}a^{9}-\frac{17\!\cdots\!64}{10\!\cdots\!29}a^{8}+\frac{14\!\cdots\!94}{10\!\cdots\!29}a^{7}-\frac{99\!\cdots\!85}{10\!\cdots\!29}a^{6}+\frac{63\!\cdots\!57}{10\!\cdots\!29}a^{5}-\frac{34\!\cdots\!69}{10\!\cdots\!29}a^{4}+\frac{16\!\cdots\!38}{10\!\cdots\!29}a^{3}-\frac{61\!\cdots\!50}{10\!\cdots\!29}a^{2}+\frac{16\!\cdots\!73}{10\!\cdots\!29}a-\frac{83\!\cdots\!10}{38\!\cdots\!99}$ (assuming GRH)
Regulator:		\( 879491729100.8733 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 879491729100.8733 \cdot 1}{10\cdot\sqrt{6142666889587199870339155304469168186187744140625}}\cr\approx \mathstrut & 0.209378059774320 \end{aligned}\] (assuming GRH)

Galois group

$C_8.A_4$ (as 32T402):

A solvable group of order 96

The 28 conjugacy class representatives for $C_8.A_4$

Character table for $C_8.A_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.112225.1, 8.0.12594450625.1, 16.0.99137616590976806640625.1

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	$24{,}\,{\href{/padicField/2.8.0.1}{8} }$	${\href{/padicField/3.8.0.1}{8} }^{4}$	R	$24{,}\,{\href{/padicField/7.8.0.1}{8} }$	${\href{/padicField/11.6.0.1}{6} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	$24{,}\,{\href{/padicField/13.8.0.1}{8} }$	$24{,}\,{\href{/padicField/17.8.0.1}{8} }$	${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$	$24{,}\,{\href{/padicField/23.8.0.1}{8} }$	${\href{/padicField/29.12.0.1}{12} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$	${\href{/padicField/31.3.0.1}{3} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$	$24{,}\,{\href{/padicField/37.8.0.1}{8} }$	${\href{/padicField/41.6.0.1}{6} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$	${\href{/padicField/43.8.0.1}{8} }^{4}$	$24{,}\,{\href{/padicField/47.8.0.1}{8} }$	${\href{/padicField/53.8.0.1}{8} }^{4}$	${\href{/padicField/59.2.0.1}{2} }^{16}$

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
\(5\)	5.16.14.1	$x^{16} - 20 x^{8} - 100$	$8$	$2$	$14$	$C_8: C_2$	$[\ ]_{8}^{2}$
	5.16.14.1	$x^{16} - 20 x^{8} - 100$	$8$	$2$	$14$	$C_8: C_2$	$[\ ]_{8}^{2}$
\(67\)	67.8.0.1	$x^{8} + 3 x^{4} + 46 x^{3} + 17 x^{2} + 64 x + 2$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$
		Deg $24$	$3$	$8$	$16$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.5.2t1.a.a	$1$	$ 5 $	\(\Q(\sqrt{5}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.335.6t1.b.a	$1$	$ 5 \cdot 67 $	6.6.2518890125.1	$C_6$ (as 6T1)	$0$	$1$
	1.335.6t1.b.b	$1$	$ 5 \cdot 67 $	6.6.2518890125.1	$C_6$ (as 6T1)	$0$	$1$
	1.67.3t1.a.a	$1$	$ 67 $	3.3.4489.1	$C_3$ (as 3T1)	$0$	$1$
	1.67.3t1.a.b	$1$	$ 67 $	3.3.4489.1	$C_3$ (as 3T1)	$0$	$1$
*	1.5.4t1.a.a	$1$	$ 5 $	\(\Q(\zeta_{5})\)	$C_4$ (as 4T1)	$0$	$-1$
*	1.5.4t1.a.b	$1$	$ 5 $	\(\Q(\zeta_{5})\)	$C_4$ (as 4T1)	$0$	$-1$
	1.335.12t1.a.a	$1$	$ 5 \cdot 67 $	12.0.793100932727814453125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.335.12t1.a.b	$1$	$ 5 \cdot 67 $	12.0.793100932727814453125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.335.12t1.a.c	$1$	$ 5 \cdot 67 $	12.0.793100932727814453125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.335.12t1.a.d	$1$	$ 5 \cdot 67 $	12.0.793100932727814453125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	2.112225.48.a.a	$2$	$ 5^{2} \cdot 67^{2}$	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.112225.48.a.b	$2$	$ 5^{2} \cdot 67^{2}$	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.112225.48.a.c	$2$	$ 5^{2} \cdot 67^{2}$	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.112225.48.a.d	$2$	$ 5^{2} \cdot 67^{2}$	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.1675.32t402.a.a	$2$	$ 5^{2} \cdot 67 $	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.1675.32t402.a.b	$2$	$ 5^{2} \cdot 67 $	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.1675.32t402.a.c	$2$	$ 5^{2} \cdot 67 $	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.1675.32t402.a.d	$2$	$ 5^{2} \cdot 67 $	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.1675.32t402.a.e	$2$	$ 5^{2} \cdot 67 $	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.1675.32t402.a.f	$2$	$ 5^{2} \cdot 67 $	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.1675.32t402.a.g	$2$	$ 5^{2} \cdot 67 $	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.1675.32t402.a.h	$2$	$ 5^{2} \cdot 67 $	32.0.6142666889587199870339155304469168186187744140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	3.112225.4t4.a.a	$3$	$ 5^{2} \cdot 67^{2}$	4.0.112225.1	$A_4$ (as 4T4)	$1$	$-1$
*	3.22445.6t6.a.a	$3$	$ 5 \cdot 67^{2}$	6.2.100755605.1	$A_4\times C_2$ (as 6T6)	$1$	$-1$
*	3.561125.12t29.a.a	$3$	$ 5^{3} \cdot 67^{2}$	12.8.793100932727814453125.1	$C_4\times A_4$ (as 12T29)	$0$	$1$
*	3.561125.12t29.a.b	$3$	$ 5^{3} \cdot 67^{2}$	12.8.793100932727814453125.1	$C_4\times A_4$ (as 12T29)	$0$	$1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.