32S3-4,8,8-g9

graph data
Name 32S3-4,8,8-g9
Type Hyperbolic
Degree 32
Genus 9
Galois orbit size for 32S3-4,8,8-g9-path1 1
Galois orbit size for 32S3-4,8,8-g9-path10 1
Galois orbit size for 32S3-4,8,8-g9-path11 1
Galois orbit size for 32S3-4,8,8-g9-path12 1
Galois orbit size for 32S3-4,8,8-g9-path13 1
Galois orbit size for 32S3-4,8,8-g9-path14 1
Galois orbit size for 32S3-4,8,8-g9-path15 1
Galois orbit size for 32S3-4,8,8-g9-path16 1
Galois orbit size for 32S3-4,8,8-g9-path17 1
Galois orbit size for 32S3-4,8,8-g9-path18 1
Galois orbit size for 32S3-4,8,8-g9-path19 1
Galois orbit size for 32S3-4,8,8-g9-path2 1
Galois orbit size for 32S3-4,8,8-g9-path20 1
Galois orbit size for 32S3-4,8,8-g9-path21 1
Galois orbit size for 32S3-4,8,8-g9-path22 1
Galois orbit size for 32S3-4,8,8-g9-path23 1
Galois orbit size for 32S3-4,8,8-g9-path24 1
Galois orbit size for 32S3-4,8,8-g9-path25 1
Galois orbit size for 32S3-4,8,8-g9-path26 1
Galois orbit size for 32S3-4,8,8-g9-path27 1
Galois orbit size for 32S3-4,8,8-g9-path28 1
Galois orbit size for 32S3-4,8,8-g9-path29 1
Galois orbit size for 32S3-4,8,8-g9-path3 1
Galois orbit size for 32S3-4,8,8-g9-path4 1
Galois orbit size for 32S3-4,8,8-g9-path5 1
Galois orbit size for 32S3-4,8,8-g9-path6 1
Galois orbit size for 32S3-4,8,8-g9-path7 1
Galois orbit size for 32S3-4,8,8-g9-path8 1
Galois orbit size for 32S3-4,8,8-g9-path9 1
Passport size 1
Pointed size 1

Above

64S27-4,16,16-g21 64S17-4,8,8-g17 64S26-8,16,16-g25 64S26-4,16,16-g21 64S3-8,8,8-g21 64S2-8,8,8-g21 64S27-8,16,16-g25

Below

16T5-4,8,8-g5 16T4-4,4,4-g3 16T5-2,8,8-g3

Belyi Curve 1: 32S3-4,8,8-g9-path10

\(x_{1}^{6} - 143848/960015\nu^{2}x_{1}^{5} + 16565776/14400225x_{1}^{4} - 1413010688/216003375\nu^{2}x_{1}^{3} - 153718528/14529375x_{1}^{2} + 16320255/2097152\nu^{2}x_{1}x_{2}^{2}x_{3}^{2} - 16320255/2097152\nu^{2}x_{1}x_{2}^{2} + 37933285376/6942965625\nu^{2}x_{1}x_{3}^{2} + 20285188096/3240050625\nu^{2}x_{1} - 20672323/1048576x_{2}^{2}x_{3}^{2} + 12480195/1048576x_{2}^{2} + 240149069824/48600759375x_{3}^{2} + 17236815872/3240050625\)
\(536870912/3672057375\nu^{2}x_{1}^{5} + 15778548593721344/59928912734630625x_{1}^{4} - 45745207043424256/299644563673153125\nu^{2}x_{1}^{3} + 10546236824609619968/13484005365291890625x_{1}^{2} + 64/255\nu^{2}x_{1}x_{2}^{2}x_{3}^{2} - 64/255\nu^{2}x_{1}x_{2}^{2} - 422282833810161664/699861870170859375\nu^{2}x_{1}x_{3}^{4} - 19057122560706609152/11897651792904609375\nu^{2}x_{1}x_{3}^{2} - 8965642663013384192/13484005365291890625\nu^{2}x_{1} + x_{2}^{2}x_{3}^{4} - 10082/3825x_{2}^{2}x_{3}^{2} + 5999/4335x_{2}^{2} - 353215911399784448/699861870170859375x_{3}^{4} - 19111517599955943424/11897651792904609375x_{3}^{2} - 11830292528395976704/13484005365291890625\)
\(x_{1}^{2}x_{3}^{2} - x_{1}^{2} + 8/15\nu^{2}x_{1}x_{3}^{2} + 8/17\nu^{2}x_{1} + 16/15x_{3}^{2} + 16/17\)

Belyi Curve 1 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 1 Degree: 20

Belyi Curve 1 Naive Measure: 80379953794214208189

Belyi Map 1 Numerator:

\(1/1666995368452984760434688(-1108462489395508348250625\nu^{3} + 1108462489395508348250625\nu )x_{2}^{3}x_{3}^{28} + 1/208374421056623095054336(1108462489395508348250625\nu^{3} - 1108462489395508348250625\nu )x_{2}^{3}x_{3}^{26} + 1/1666995368452984760434688(-30173959875690187459126875\nu^{3} + 30173959875690187459126875\nu )x_{2}^{3}x_{3}^{24} + 1/104187210528311547527168(3448123799192256072931875\nu^{3} - 3448123799192256072931875\nu )x_{2}^{3}x_{3}^{22} + 1/1666995368452984760434688(-52756781197640210641715625\nu^{3} + 52756781197640210641715625\nu )x_{2}^{3}x_{3}^{20} + 1/208374421056623095054336(1046430740788156831629375\nu^{3} - 1046430740788156831629375\nu )x_{2}^{3}x_{3}^{18} + 1/481761661482912595765624832(13774269130319577118163833125\nu^{3} - 13774269130319577118163833125\nu )x_{2}^{3}x_{3}^{16} + 1/15055051921341018617675776(-658379803375444798481911875\nu^{3} + 658379803375444798481911875\nu )x_{2}^{3}x_{3}^{14} + 1/481761661482912595765624832(16022227077315041132400913125\nu^{3} - 16022227077315041132400913125\nu )x_{2}^{3}x_{3}^{12} + 1/60220207685364074470703104(-550405356308840126531390625\nu^{3} + 550405356308840126531390625\nu )x_{2}^{3}x_{3}^{10} + 1/481761661482912595765624832(-4895836846716740598260015625\nu^{3} + 4895836846716740598260015625\nu )x_{2}^{3}x_{3}^{8} + 1/30110103842682037235351552(432017485386805872938671875\nu^{3} - 432017485386805872938671875\nu )x_{2}^{3}x_{3}^{6} + 1/481761661482912595765624832(-4038658016578678230535546875\nu^{3} + 4038658016578678230535546875\nu )x_{2}^{3}x_{3}^{4} + 1/60220207685364074470703104(151172525990717451650390625\nu^{3} - 151172525990717451650390625\nu )x_{2}^{3}x_{3}^{2} + 1/481761661482912595765624832(-151172525990717451650390625\nu^{3} + 151172525990717451650390625\nu )x_{2}^{3} + 1/2398554685440000(2393741243606531\nu^{3} - 2393741243606531\nu )x_{2}x_{3}^{28} + 1/2665060761600(-6345022467427\nu^{3} + 6345022467427\nu )x_{2}x_{3}^{26} + 1/10660243046400(-37543083049507\nu^{3} + 37543083049507\nu )x_{2}x_{3}^{24} + 1/5922357248(73559741353\nu^{3} - 73559741353\nu )x_{2}x_{3}^{22} + 1/47378857984(-321756643369\nu^{3} + 321756643369\nu )x_{2}x_{3}^{20} + 1/11844714496(-66773843325\nu^{3} + 66773843325\nu )x_{2}x_{3}^{18} + 1/13692489957376(132125074022025\nu^{3} - 132125074022025\nu )x_{2}x_{3}^{16} + 1/855780622336(-7494237095625\nu^{3} + 7494237095625\nu )x_{2}x_{3}^{14} + 1/3957129597681664(11715326449115625\nu^{3} - 11715326449115625\nu )x_{2}x_{3}^{12} + 1/989282399420416(4504561382671875\nu^{3} - 4504561382671875\nu )x_{2}x_{3}^{10} + 1/1143610453730000896(-5223810581765015625\nu^{3} + 5223810581765015625\nu )x_{2}x_{3}^{8} + 1/142951306716250112(106453888316015625\nu^{3} - 106453888316015625\nu )x_{2}x_{3}^{6} + 1/330503421127970258944(158085336369786328125\nu^{3} - 158085336369786328125\nu )x_{2}x_{3}^{4} + 1/82625855281992564736(-13101701585888671875\nu^{3} + 13101701585888671875\nu )x_{2}x_{3}^{2} + 1/95515488705983404834816(744139743876826171875\nu^{3} - 744139743876826171875\nu )x_{2} + 83521/405000x_{3}^{28} - 83521/317700x_{3}^{26} - 324547/211800x_{3}^{24} + 1445/706x_{3}^{22} + 9333/2824x_{3}^{20} - 6975/1412x_{3}^{18} - 2371725/816136x_{3}^{16} + 556875/102017x_{3}^{14} + 11491875/13874312x_{3}^{12} - 353109375/117931652x_{3}^{10} + 16641703125/68164494856x_{3}^{8} + 12814453125/17041123714x_{3}^{6} - 3221553515625/19699539013384x_{3}^{4} - 576650390625/9849769506692x_{3}^{2} + 92840712890625/5693166774867976\)

Belyi Map 1 Denominator:

\(x_{3}^{20} - 450/353x_{3}^{18} - 440325/102017x_{3}^{16} + 607500/102017x_{3}^{14} - 31033125/29482913x_{3}^{12} - 22781250/29482913x_{3}^{10} + 1833890625/8520561857x_{3}^{8}\)

Belyi Curve 2: 32S3-4,8,8-g9-path11

\(x_{1}^{3}x_{3}^{2} + x_{1}x_{3}^{2} - x_{2}2\)
\(-10x_{1}^{2}x_{3}^{2} - 2x_{1}x_{2} + x_{2}x_{3}^{4} - 2x_{3}^{2}\)
\(x_{1}^{4} + 6x_{1}^{2} - 1/2x_{2}x_{3}^{2} + 1\)

Belyi Curve 2 Base Field: Rationals

Belyi Curve 2 Degree: 17

Belyi Curve 2 Naive Measure: 28

Belyi Map 2 Numerator:

\(384x_{1}^{2} - 25/3x_{1}x_{3}^{12} + 32x_{1}x_{3}^{4} + 5/48x_{2}^{3}x_{3}^{6} - 5/384x_{2}^{2}x_{3}^{20} + 35/48x_{2}^{2}x_{3}^{12} - 13/12x_{2}^{2}x_{3}^{4} - 5/64x_{2}x_{3}^{18} + 31/3x_{2}x_{3}^{10} - 72x_{2}x_{3}^{2} + 5/24x_{3}^{16} - 2/3x_{3}^{8} + 64\)

Belyi Map 2 Denominator:

\(384x_{1}^{2} - 25/3x_{1}x_{3}^{12} + 32x_{1}x_{3}^{4} + 5/48x_{2}^{3}x_{3}^{6} - 5/384x_{2}^{2}x_{3}^{20} + 35/48x_{2}^{2}x_{3}^{12} - 5/6x_{2}^{2}x_{3}^{4} - 5/64x_{2}x_{3}^{18} + 31/3x_{2}x_{3}^{10} - 72x_{2}x_{3}^{2} + 5/24x_{3}^{16} - 2/3x_{3}^{8} + 64\)

Belyi Curve 3: 32S3-4,8,8-g9-path12

\(x_{1}^{3} + 1/4x_{1}^{2}x_{2} + x_{1}^{2}x_{3}^{4} - 7/4x_{1}^{2}x_{3}^{2} + 1/2x_{1}^{2} - 1/4x_{1}x_{2}x_{3}^{4} + x_{1}x_{2}x_{3}^{2} - 3/4x_{1}x_{2} + x_{1}x_{3}^{4} + 2x_{1}x_{3}^{2} + x_{1} - 1/4x_{2}x_{3}^{4} + x_{2}x_{3}^{2} + 5/4x_{2} - 1/4x_{3}^{2} + 1/2\)
\(-16x_{1}^{3} - 9x_{1}^{2}x_{2}x_{3}^{2} + 10x_{1}^{2}x_{2} - x_{1}x_{2}^{2} - 16x_{1}x_{3}^{4} - 32x_{1}x_{3}^{2} - 16x_{1} + x_{2}^{2}x_{3}^{4} - 4x_{2}^{2}x_{3}^{2} + 4x_{2}^{2} - 15x_{2}x_{3}^{2} - 6x_{2}2\)
\(x_{1}^{5} + x_{1}^{4} + 8x_{1}^{3} + 4x_{1}^{2}x_{3}^{2} - x_{1}x_{2}x_{3}^{2} + 2x_{1}x_{2} + 4x_{1}x_{3}^{2} + 7x_{1} - x_{2}x_{3}^{2} + 2x_{2} - 1\)
\(x_{1}^{4}x_{2} - 4x_{1}^{4} + 8x_{1}^{2}x_{2} - 16x_{1}^{2}x_{3}^{2} + 4x_{1}x_{2}x_{3}^{2} - 8x_{1}x_{2} - x_{2}^{2}x_{3}^{2} + 2x_{2}^{2} + 4x_{2}x_{3}^{2} + 7x_{2} + 4\)
\(x_{1}^{3}x_{3}^{2} - 2x_{1}^{3} - x_{1}x_{3}^{2} - 2x_{1} - x_{2}2\)

Belyi Curve 3 Base Field: Rationals

Belyi Curve 3 Degree: 18

Belyi Curve 3 Naive Measure: 272

Belyi Map 3 Numerator:

\(787451901073744410578467438863/14645612474695151003058795712x_{1}x_{3}^{40} - 83405627893897484767553490208409/58582449898780604012235182848x_{1}x_{3}^{38} + 3917617785290547626153886304659887/234329799595122416048940731392x_{1}x_{3}^{36} - 104040614102228075784710687767819209/937319198380489664195762925568x_{1}x_{3}^{34} + 102556351976391593100451781873020787/234329799595122416048940731392x_{1}x_{3}^{32} - 107683525793496338332712258221017973/117164899797561208024470365696x_{1}x_{3}^{30} + 18956093161333691100511537476833431/117164899797561208024470365696x_{1}x_{3}^{28} + 2191034122101408818334634363796589175/468659599190244832097881462784x_{1}x_{3}^{26} - 3008142349959754906677779334698016379/234329799595122416048940731392x_{1}x_{3}^{24} + 1586599818877112129659158491494517353/117164899797561208024470365696x_{1}x_{3}^{22} + 253590999136754050666987650546513643/58582449898780604012235182848x_{1}x_{3}^{20} - 3435079766652973758034355030833083391/117164899797561208024470365696x_{1}x_{3}^{18} + 87437287347652113636587944672719143/2547063039077417565749355776x_{1}x_{3}^{16} - 238829117583606195685400037930447933/14645612474695151003058795712x_{1}x_{3}^{14} - 194705235985130125088571056562525/457675389834223468845587366x_{1}x_{3}^{12} + 25416086073376347006268392958378657/7322806237347575501529397856x_{1}x_{3}^{10} - 4000431040905964404971058881001371/3661403118673787750764698928x_{1}x_{3}^{8} - 14172697019648584977590117713743/457675389834223468845587366x_{1}x_{3}^{6} + 48330714813748809102602110963187/457675389834223468845587366x_{1}x_{3}^{4} - 5029566233589778954499613134476/228837694917111734422793683x_{1}x_{3}^{2} - 118905237062135405997348583268313/14997107174087834627132206809088x_{2}^{4}x_{3}^{22} + 973356994490413821664879163401393/7498553587043917313566103404544x_{2}^{4}x_{3}^{20} - 7043015379275910061239284028698053/7498553587043917313566103404544x_{2}^{4}x_{3}^{18} + 14760864554894387029800659787698145/3749276793521958656783051702272x_{2}^{4}x_{3}^{16} - 78503520206015037404878939284506699/7498553587043917313566103404544x_{2}^{4}x_{3}^{14} + 33963011557511439718877464562829157/1874638396760979328391525851136x_{2}^{4}x_{3}^{12} - 74972615675481129815800440070039235/3749276793521958656783051702272x_{2}^{4}x_{3}^{10} + 12041153401122527506327232798458445/937319198380489664195762925568x_{2}^{4}x_{3}^{8} - 1505180188480298764434378386010665/468659599190244832097881462784x_{2}^{4}x_{3}^{6} - 283924731541094236107681944965679/234329799595122416048940731392x_{2}^{4}x_{3}^{4} + 229629774114418160957247757023795/234329799595122416048940731392x_{2}^{4}x_{3}^{2} - 1258764584566947409031440045717/7322806237347575501529397856x_{2}^{4} + 118905237062135405997348583268313/14997107174087834627132206809088x_{2}^{3}x_{3}^{36} - 2453388116746415303867281584619/10188252156309670262997423104x_{2}^{3}x_{3}^{34} + 44945767528461279538417220339031/13132318015838734349502808064x_{2}^{3}x_{3}^{32} - 28316583143193154097053743420056233/937319198380489664195762925568x_{2}^{3}x_{3}^{30} + 1387589771118078207444858160652452509/7498553587043917313566103404544x_{2}^{3}x_{3}^{28} - 67784786915221325347795622910043721/81506017250477362103979384832x_{2}^{3}x_{3}^{26} + 10594907779337944966227312397966836581/3749276793521958656783051702272x_{2}^{3}x_{3}^{24} - 6906969232845091095732884701140892191/937319198380489664195762925568x_{2}^{3}x_{3}^{22} + 6940683235844220700670789438841842303/468659599190244832097881462784x_{2}^{3}x_{3}^{20} - 42776627512470580610638138560683205341/1874638396760979328391525851136x_{2}^{3}x_{3}^{18} + 24851636431550550884216909658723002707/937319198380489664195762925568x_{2}^{3}x_{3}^{16} - 914632267916040366431426183021614675/40753008625238681051989692416x_{2}^{3}x_{3}^{14} + 6012475005858623230004353192035744139/468659599190244832097881462784x_{2}^{3}x_{3}^{12} - 1830976295938322012935320917409872075/468659599190244832097881462784x_{2}^{3}x_{3}^{10} - 39231972010647639687824091002852537/117164899797561208024470365696x_{2}^{3}x_{3}^{8} + 1547397176131026548768567893831665/1830701559336893875382349464x_{2}^{3}x_{3}^{6} - 4765063741810938900341581583255783/14645612474695151003058795712x_{2}^{3}x_{3}^{4} + 63876510528360718207091556753803/1273531519538708782874677888x_{2}^{3}x_{3}^{2} - 3775836078311008003625474549785/1830701559336893875382349464x_{2}^{3} - 787451901073744410578467438863/234329799595122416048940731392x_{2}^{2}x_{3}^{40} + 102304473519667350621436708741121/937319198380489664195762925568x_{2}^{2}x_{3}^{38} - 24275966084557583650555059997013/14645612474695151003058795712x_{2}^{2}x_{3}^{36} + 227815257077032932676946026478732241/14997107174087834627132206809088x_{2}^{2}x_{3}^{34} - 685666012632553002128259171521478617/7498553587043917313566103404544x_{2}^{2}x_{3}^{32} + 2760312328118181994328363345442676207/7498553587043917313566103404544x_{2}^{2}x_{3}^{30} - 3503753459544223012058128905014563203/3749276793521958656783051702272x_{2}^{2}x_{3}^{28} + 7564884071578623578220896460033589549/7498553587043917313566103404544x_{2}^{2}x_{3}^{26} + 2323384093281047789224598190358854031/937319198380489664195762925568x_{2}^{2}x_{3}^{24} - 53382250986665035726141396974003855441/3749276793521958656783051702272x_{2}^{2}x_{3}^{22} + 16167808027122308511887925905377452095/468659599190244832097881462784x_{2}^{2}x_{3}^{20} - 98095126049412365902182112650957668755/1874638396760979328391525851136x_{2}^{2}x_{3}^{18} + 24466907969650970805592759517204688119/468659599190244832097881462784x_{2}^{2}x_{3}^{16} - 1331957620939133073182787443598359813/40753008625238681051989692416x_{2}^{2}x_{3}^{14} + 2447706481267276568543857325925390341/234329799595122416048940731392x_{2}^{2}x_{3}^{12} + 30547859669685824497077687454517123/234329799595122416048940731392x_{2}^{2}x_{3}^{10} - 14295847125703675934612868761315183/14645612474695151003058795712x_{2}^{2}x_{3}^{8} + 2344576693027136007387849134286991/58582449898780604012235182848x_{2}^{2}x_{3}^{6} + 682048291092534012706141657809489/14645612474695151003058795712x_{2}^{2}x_{3}^{4} + 343185063538141664832773538304963/14645612474695151003058795712x_{2}^{2}x_{3}^{2} - 3773090025972002662812401025589/457675389834223468845587366x_{2}^{2} + 2362355703221233231735402316589/29291224949390302006117591424x_{2}x_{3}^{38} - 1933226754060123870700957097820735/937319198380489664195762925568x_{2}x_{3}^{36} + 11212868169913506844995402842903229/468659599190244832097881462784x_{2}x_{3}^{34} - 303999640681898845707135415119942735/1874638396760979328391525851136x_{2}x_{3}^{32} + 647254992317059618203106868564799523/937319198380489664195762925568x_{2}x_{3}^{30} - 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Belyi Map 3 Denominator:

\(35478922500764731599182947739515/3661403118673787750764698928x_{1}x_{3}^{40} - 3762080255346195846397260533453197/14645612474695151003058795712x_{1}x_{3}^{38} + 176913531789088230029006059524151387/58582449898780604012235182848x_{1}x_{3}^{36} - 4704409224701828845805720947435653757/234329799595122416048940731392x_{1}x_{3}^{34} + 4644446511076230574523947523867812139/58582449898780604012235182848x_{1}x_{3}^{32} - 4887881898646681795878790175831052369/29291224949390302006117591424x_{1}x_{3}^{30} + 892324216994254801498568398267050267/29291224949390302006117591424x_{1}x_{3}^{28} + 99184095262043787960467160476386253763/117164899797561208024470365696x_{1}x_{3}^{26} - 136510031864228422435776252621207711263/58582449898780604012235182848x_{1}x_{3}^{24} + 72122029524779081993191928435961692965/29291224949390302006117591424x_{1}x_{3}^{22} + 11473038516009764762527803190286820535/14645612474695151003058795712x_{1}x_{3}^{20} - 156007448058078860481913036967715453611/29291224949390302006117591424x_{1}x_{3}^{18} + 3969113313557385177285037458994189291/636765759769354391437338944x_{1}x_{3}^{16} - 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9622386377629389622941935466473249061/29291224949390302006117591424x_{2}x_{3}^{28} + 12313539887451507573056705093298422773/29291224949390302006117591424x_{2}x_{3}^{26} + 78000656672198059903525610395553441709/234329799595122416048940731392x_{2}x_{3}^{24} - 37849719957391245038228373618407854127/14645612474695151003058795712x_{2}x_{3}^{22} + 313341374491758107359945330770273976997/58582449898780604012235182848x_{2}x_{3}^{20} - 177740433521301249041127709764733111143/29291224949390302006117591424x_{2}x_{3}^{18} + 225791389398857107256742501692283884243/58582449898780604012235182848x_{2}x_{3}^{16} - 4137115783217923495338232576383993099/3661403118673787750764698928x_{2}x_{3}^{14} + 23248966032659902025229158478684527/228837694917111734422793683x_{2}x_{3}^{12} - 565297763251896482533406487317992801/3661403118673787750764698928x_{2}x_{3}^{10} + 479465083298468056980636613706793601/3661403118673787750764698928x_{2}x_{3}^{8} - 3734692789146644853969338548435549/228837694917111734422793683x_{2}x_{3}^{6} - 3276203106087718362331791971872937/228837694917111734422793683x_{2}x_{3}^{4} + 2189799016455208994499581444123023/228837694917111734422793683x_{2}x_{3}^{2} - 453074489334133446953204991997528/228837694917111734422793683x_{2} + 319310302506882584392646529655635/58582449898780604012235182848x_{3}^{36} - 935017559391589919064563675062269/7322806237347575501529397856x_{3}^{34} + 19710676250628724910402476133912349/14645612474695151003058795712x_{3}^{32} - 963646880979749346394708426791726361/117164899797561208024470365696x_{3}^{30} + 1825454532012690013451539822964988867/58582449898780604012235182848x_{3}^{28} - 2058647227721358750607150333584252369/29291224949390302006117591424x_{3}^{26} + 953383996143424651523572529683027071/14645612474695151003058795712x_{3}^{24} + 6463134134872644323402005860588563439/58582449898780604012235182848x_{3}^{22} - 3271550067590146520701036639676076547/7322806237347575501529397856x_{3}^{20} + 8339503545484460612418808297811202667/14645612474695151003058795712x_{3}^{18} - 20341651897000326034314798988878044/228837694917111734422793683x_{3}^{16} - 9673471274668396655800875613701369939/14645612474695151003058795712x_{3}^{14} + 3212707869998259287920129775014296231/3661403118673787750764698928x_{3}^{12} - 423377270162593980974618155217310891/915350779668446937691174732x_{3}^{10} + 49600463780846797534660393621545285/915350779668446937691174732x_{3}^{8} + 35017650198695208665463092177236839/915350779668446937691174732x_{3}^{6} - 2265368785267548560978274195288712/228837694917111734422793683x_{3}^{4}\)

Belyi Curve 4: 32S3-4,8,8-g9-path6

\(x_{1}^{3} - 1/4x_{1}^{2}x_{2} + x_{1}^{2}x_{3}^{4} + 7/4x_{1}^{2}x_{3}^{2} + 1/2x_{1}^{2} + 1/4x_{1}x_{2}x_{3}^{4} + x_{1}x_{2}x_{3}^{2} + 3/4x_{1}x_{2} + x_{1}x_{3}^{4} - 2x_{1}x_{3}^{2} + x_{1} + 1/4x_{2}x_{3}^{4} + x_{2}x_{3}^{2} - 5/4x_{2} + 1/4x_{3}^{2} + 1/2\)
\(-16x_{1}^{3} - 9x_{1}^{2}x_{2}x_{3}^{2} - 10x_{1}^{2}x_{2} - x_{1}x_{2}^{2} - 16x_{1}x_{3}^{4} + 32x_{1}x_{3}^{2} - 16x_{1} + x_{2}^{2}x_{3}^{4} + 4x_{2}^{2}x_{3}^{2} + 4x_{2}^{2} - 15x_{2}x_{3}^{2} + 6x_{2}2\)
\(x_{1}^{5} + x_{1}^{4} + 8x_{1}^{3} - 4x_{1}^{2}x_{3}^{2} - x_{1}x_{2}x_{3}^{2} - 2x_{1}x_{2} - 4x_{1}x_{3}^{2} + 7x_{1} - x_{2}x_{3}^{2} - 2x_{2} - 1\)
\(x_{1}^{4}x_{2} + 4x_{1}^{4} + 8x_{1}^{2}x_{2} - 16x_{1}^{2}x_{3}^{2} - 4x_{1}x_{2}x_{3}^{2} - 8x_{1}x_{2} - x_{2}^{2}x_{3}^{2} - 2x_{2}^{2} - 4x_{2}x_{3}^{2} + 7x_{2} - 4\)
\(x_{1}^{3}x_{3}^{2} + 2x_{1}^{3} - x_{1}x_{3}^{2} + 2x_{1} - x_{2}2\)

Belyi Curve 4 Base Field: Rationals

Belyi Curve 4 Degree: 18

Belyi Curve 4 Naive Measure: 272

Belyi Map 4 Numerator:

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Belyi Map 4 Denominator:

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417059476093625255203039826150475483109/937319198380489664195762925568x_{2}^{2}x_{3}^{24} - 9619320951225686231151184246339030306883/3749276793521958656783051702272x_{2}^{2}x_{3}^{22} - 2916562444548736952008087624920448192405/468659599190244832097881462784x_{2}^{2}x_{3}^{20} - 17703620140994964436347160998804764286121/1874638396760979328391525851136x_{2}^{2}x_{3}^{18} - 4415192661082568857974865837555733583045/468659599190244832097881462784x_{2}^{2}x_{3}^{16} - 240139837490046972100682641192966980895/40753008625238681051989692416x_{2}^{2}x_{3}^{14} - 439973589224277358193275197718933620367/234329799595122416048940731392x_{2}^{2}x_{3}^{12} + 6317794880329586274894899455687254329/234329799595122416048940731392x_{2}^{2}x_{3}^{10} + 2578776171847733125749180564961912461/14645612474695151003058795712x_{2}^{2}x_{3}^{8} + 404496595918894183443201631230070237/58582449898780604012235182848x_{2}^{2}x_{3}^{6} - 121931867897137062005236551784182067/14645612474695151003058795712x_{2}^{2}x_{3}^{4} + 62404655375329870021781709086745209/14645612474695151003058795712x_{2}^{2}x_{3}^{2} + 675836813273668830873119704621239/457675389834223468845587366x_{2}^{2} - 423384714305955545958459970557591/29291224949390302006117591424x_{2}x_{3}^{38} - 346879812728908550786920388300334957/937319198380489664195762925568x_{2}x_{3}^{36} - 2014249473194261576251473084302025063/468659599190244832097881462784x_{2}x_{3}^{34} - 54675397677410616526160887772302911901/1874638396760979328391525851136x_{2}x_{3}^{32} - 116562471846517167841975479658587741305/937319198380489664195762925568x_{2}x_{3}^{30} - 38277319858366606965639225843942503325/117164899797561208024470365696x_{2}x_{3}^{28} - 48983301046939192243542102881585069343/117164899797561208024470365696x_{2}x_{3}^{26} + 310269677025612219200698387822421041019/937319198380489664195762925568x_{2}x_{3}^{24} + 37640960065258801954560453538924578721/14645612474695151003058795712x_{2}x_{3}^{22} + 1246460070100308801990489309425450857379/234329799595122416048940731392x_{2}x_{3}^{20} + 707044883697013896600610687160031233617/117164899797561208024470365696x_{2}x_{3}^{18} + 898183900351452249931798041134000887717/234329799595122416048940731392x_{2}x_{3}^{16} + 8228402640071467884184463367266561333/7322806237347575501529397856x_{2}x_{3}^{14} + 23118550105044272626142464548736020/228837694917111734422793683x_{2}x_{3}^{12} + 2248770554409646801223648926403243539/14645612474695151003058795712x_{2}x_{3}^{10} + 1907279044389603458803359129530127143/14645612474695151003058795712x_{2}x_{3}^{8} + 14855166861733963187088264502916575/915350779668446937691174732x_{2}x_{3}^{6} - 13032785070032467711221829159935819/915350779668446937691174732x_{2}x_{3}^{4} - 8710879996619561864046726724324617/915350779668446937691174732x_{2}x_{3}^{2} - 450559706217338557475955185430290/228837694917111734422793683x_{2} - 1270154142917866637875379911672773/234329799595122416048940731392x_{3}^{36} - 3719343878503050559591574340056517/29291224949390302006117591424x_{3}^{34} - 78406326602120824108121113479100299/58582449898780604012235182848x_{3}^{32} - 3833280524311906004714705086015155247/468659599190244832097881462784x_{3}^{30} - 7261511381209391258914822868566587149/234329799595122416048940731392x_{3}^{28} - 8189207820741898127809979617800219327/117164899797561208024470365696x_{3}^{26} - 3792593605224406603776717721859581849/58582449898780604012235182848x_{3}^{24} + 25709575664955298524074614621665033865/234329799595122416048940731392x_{3}^{22} + 13014103696831528027821378388138019159/29291224949390302006117591424x_{3}^{20} + 33174457494906622098357034582545586157/58582449898780604012235182848x_{3}^{18} + 80919647609899262304047338697512433/915350779668446937691174732x_{3}^{16} - 38480982825097950004391710939566816901/58582449898780604012235182848x_{3}^{14} - 12780065123677442988422418226511270021/14645612474695151003058795712x_{3}^{12} - 1684160597380254998722148456050264921/3661403118673787750764698928x_{3}^{10} - 197291949283593781171030317461472671/3661403118673787750764698928x_{3}^{8} + 139300416317964186731354756907495393/3661403118673787750764698928x_{3}^{6} + 2252793038982014776698149780103058/228837694917111734422793683x_{3}^{4} + 16\)

Belyi Curve 5: 32S3-4,8,8-g9-path7

\(2x_{1}^{3} + x_{1}x_{2}x_{3}^{4} - 2x_{3}^{2}\)
\(-2x_{1}^{3}x_{3}^{2} + 2x_{1}^{2}x_{2} + 2x_{1}^{2} + x_{2}^{2}x_{3}^{4} - x_{2}x_{3}^{4}\)
\(x_{1}^{4} + x_{1}x_{2}x_{3}^{2} - x_{2} - 1\)

Belyi Curve 5 Base Field: Rationals

Belyi Curve 5 Degree: 21

Belyi Curve 5 Naive Measure: 17

Belyi Map 5 Numerator:

\(-1/4x_{1}x_{3}^{18} - 5/2x_{1}x_{3}^{10} + 2x_{1}x_{3}^{2} + 1/32x_{2}^{4}x_{3}^{24} + 1/4x_{2}^{4}x_{3}^{16} - 1/32x_{2}^{3}x_{3}^{24} - 3/8x_{2}^{3}x_{3}^{16} + 1/2x_{2}^{2}x_{3}^{16} + 5x_{2}^{2}x_{3}^{8} + 5x_{2}^{2} + 3/8x_{2}x_{3}^{16} + 3x_{2}x_{3}^{8} + 10x_{2} - 1/4x_{3}^{16} - 3x_{3}^{8} + 4\)

Belyi Map 5 Denominator:

\(-1/2x_{1}x_{3}^{18} - 5x_{1}x_{3}^{10} + 4x_{1}x_{3}^{2} + 1/16x_{2}^{4}x_{3}^{24} + 1/2x_{2}^{4}x_{3}^{16} - 1/16x_{2}^{3}x_{3}^{24} - 3/4x_{2}^{3}x_{3}^{16} + x_{2}^{2}x_{3}^{16} + 10x_{2}^{2}x_{3}^{8} + 8x_{2}^{2} + 3/4x_{2}x_{3}^{16} + 6x_{2}x_{3}^{8} + 20x_{2} - 1/2x_{3}^{16} - 6x_{3}^{8} + 12\)

Belyi Curve 6: 32S3-4,8,8-g9-path9

\(-8589934592/281462092005375\nu x_{1}^{3}x_{2} - 3448016925097984/18445618199572250625x_{1}^{2}x_{2} + 838417128229055758336/239885264685437119378125x_{1}^{2}x_{3}^{2} + 2513255020116312064/4703632640890923909375\nu x_{1}x_{2} - 2881439070388662304768/239885264685437119378125\nu x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{6} - 1024/65535\nu x_{2}^{2}x_{3}^{2} - 126583155195904/277098404709375\nu x_{2}x_{3}^{4} - 30036579731302776832/4703632640890923909375x_{2} - 579036811290279411712/44423197163969836921875x_{3}^{2}\)
\(-33554432/858967245x_{1}^{2}x_{3}^{2} + x_{1}x_{2}x_{3}^{4} - 1024/65535\nu x_{1}x_{2} + 134217728/858967245\nu x_{1}x_{3}^{2} + 1546/255\nu x_{2}x_{3}^{4} + 1562624/16842495x_{2} + 268435456/1431612075x_{3}^{2}\)
\(x_{1}^{3}x_{3}^{2} - 6\nu x_{1}^{2}x_{3}^{2} - 16x_{1}x_{3}^{2} - 4294836225/8388608x_{2}x_{3}^{4} + 65535/8192\nu x_{2} + 16\nu x_{3}^{2}\)
\(x_{1}^{4} - 16/257\nu x_{1}^{3} - 96/257x_{1}^{2} + 256/257\nu x_{1} + 65535/1024\nu x_{2}x_{3}^{2} + 256/257\)

Belyi Curve 6 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 6 Degree: 20

Belyi Curve 6 Naive Measure: 533619437993393453952291

Belyi Map 6 Numerator:

\(-319229886726144/95298165909845\nu x_{1}^{3} - 155732211323109376/24301032307010475x_{1}^{2} + 206842649490594874221386675/48354564243456\nu x_{1}x_{3}^{16} + 158366007427330836593047/1204141980672x_{1}x_{3}^{12} - 77671320271627861/76857510\nu x_{1}x_{3}^{8} + 9716899650732032/647236113943725x_{1}x_{3}^{4} + 54867038975295488/8100344102336825\nu x_{1} - 748155095279074544632061172796795133116683761179078125/24452635277439283076474197728431177728\nu x_{2}^{3}x_{3}^{22} - 56757597840747395171075292062747456911483522209375/23879526638124299879369333719171072x_{2}^{3}x_{3}^{18} + 861137114161178139432437302467880137106910625/11659925116271630800473307480064\nu x_{2}^{3}x_{3}^{14} + 13064908828358926204301134945306929057375/11386645621359014453587214336x_{2}^{3}x_{3}^{10} - 5357033306186489530763362698885325/601068708897752029855744\nu x_{2}^{3}x_{3}^{6} - 601396916968120783660082925075/21718302958219555766272x_{2}^{3}x_{3}^{2} - 2740130601308450740519362185296150775527634335/546558989825232693772186288128x_{2}^{2}x_{3}^{20} + 13130229179927805291822140520894407616715/44479084458433650209325056\nu x_{2}^{2}x_{3}^{16} + 1695673905168933207373239724005802205/260619635498634669195264x_{2}^{2}x_{3}^{12} - 2698129303541582307435425319115/42418560465272569856\nu x_{2}^{2}x_{3}^{8} - 115661183175627798200989087/497092505452412928x_{2}^{2}x_{3}^{4} - 9652497235/77158912\nu x_{2}^{2} + 22445203605195785512233589955142851677/69227090679324834004992\nu x_{2}x_{3}^{18} + 60427677880153177284139628626265/3976740043619303424x_{2}x_{3}^{14} - 234977350211958477337993073753/990301475705978880\nu x_{2}x_{3}^{10} - 1560649479991815979729/1264171614208x_{2}x_{3}^{6} - 16380669159800832/217437491873905\nu x_{2}x_{3}^{2} + 2684866068406259863475564774377/394573244226600960x_{3}^{16} - 40384339012524969958431767/192662716907520\nu x_{3}^{12} - 3036065512903273239437/1881471844800x_{3}^{8} - 17932622332166144/647236113943725\nu x_{3}^{4} + 215727978120740864/72903096921031425\)

Belyi Map 6 Denominator:

\(-319229886726144/95298165909845\nu x_{1}^{3} - 155732211323109376/24301032307010475x_{1}^{2} + 206842649490594874221386675/48354564243456\nu x_{1}x_{3}^{16} + 158366007427330836593047/1204141980672x_{1}x_{3}^{12} - 77671320271627861/76857510\nu x_{1}x_{3}^{8} + 9716899650732032/647236113943725x_{1}x_{3}^{4} + 54867038975295488/8100344102336825\nu x_{1} - 748155095279074544632061172796795133116683761179078125/24452635277439283076474197728431177728\nu x_{2}^{3}x_{3}^{22} - 56757597840747395171075292062747456911483522209375/23879526638124299879369333719171072x_{2}^{3}x_{3}^{18} + 861137114161178139432437302467880137106910625/11659925116271630800473307480064\nu x_{2}^{3}x_{3}^{14} + 13064908828358926204301134945306929057375/11386645621359014453587214336x_{2}^{3}x_{3}^{10} - 5357033306186489530763362698885325/601068708897752029855744\nu x_{2}^{3}x_{3}^{6} - 601396916968120783660082925075/21718302958219555766272x_{2}^{3}x_{3}^{2} - 2740130601308450740519362185296150775527634335/546558989825232693772186288128x_{2}^{2}x_{3}^{20} + 13130229179927805291822140520894407616715/44479084458433650209325056\nu x_{2}^{2}x_{3}^{16} + 1695673905168933207373239724005802205/260619635498634669195264x_{2}^{2}x_{3}^{12} - 2698129303541582307435425319115/42418560465272569856\nu x_{2}^{2}x_{3}^{8} - 115663251267346085747635615/497092505452412928x_{2}^{2}x_{3}^{4} - 9652497235/77158912\nu x_{2}^{2} + 22445203605195785512233589955142851677/69227090679324834004992\nu x_{2}x_{3}^{18} + 60427677880153177284139628626265/3976740043619303424x_{2}x_{3}^{14} - 234977350211958477337993073753/990301475705978880\nu x_{2}x_{3}^{10} - 1560649479991815979729/1264171614208x_{2}x_{3}^{6} - 16380669159800832/217437491873905\nu x_{2}x_{3}^{2} + 2684866068406259863475564774377/394573244226600960x_{3}^{16} - 40384339012524969958431767/192662716907520\nu x_{3}^{12} - 3036065512903273239437/1881471844800x_{3}^{8} - 17932622332166144/647236113943725\nu x_{3}^{4} + 215727978120740864/72903096921031425\)