32S9-2,8,4-g3

graph data
Name 32S9-2,8,4-g3
Type Hyperbolic
Degree 32
Genus 3
Galois orbit size for 32S9-2,8,4-g3-path1 1
Galois orbit size for 32S9-2,8,4-g3-path10 1
Galois orbit size for 32S9-2,8,4-g3-path11 1
Galois orbit size for 32S9-2,8,4-g3-path12 1
Galois orbit size for 32S9-2,8,4-g3-path13 1
Galois orbit size for 32S9-2,8,4-g3-path14 1
Galois orbit size for 32S9-2,8,4-g3-path15 1
Galois orbit size for 32S9-2,8,4-g3-path16 1
Galois orbit size for 32S9-2,8,4-g3-path17 1
Galois orbit size for 32S9-2,8,4-g3-path2 1
Galois orbit size for 32S9-2,8,4-g3-path3 1
Galois orbit size for 32S9-2,8,4-g3-path4 1
Galois orbit size for 32S9-2,8,4-g3-path5 1
Galois orbit size for 32S9-2,8,4-g3-path6 1
Galois orbit size for 32S9-2,8,4-g3-path7 1
Galois orbit size for 32S9-2,8,4-g3-path8 1
Galois orbit size for 32S9-2,8,4-g3-path9 1
Passport size 1
Pointed size 1

Above

64S21-4,8,4-g13 64S38-2,16,4-g7 64S41-2,16,4-g7 64S40-4,16,8-g19 64S43-4,16,8-g19 64S13-4,8,8-g17 64S12-2,8,8-g9 64S8-2,8,4-g5 64S41-4,16,4-g15 64S6-4,8,8-g17 64S40-2,16,8-g11 64S39-4,16,4-g15 64S42-2,16,8-g11 64S6-2,8,8-g9 64S8-4,8,4-g13

Below

16T12-2,8,4-g2 16T10-2,4,4-g1 16T13-2,8,2-g0

Belyi Curve 1: 32S9-2,8,4-g3-path10

\(-\nu^{2}x_{1}^{8} + x_{2}^{2} + \nu^{2}x_{3}^{8}\)

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

Belyi Curve 1 Degree: 8

Belyi Curve 1 Naive Measure: 3

Belyi Map 1 Numerator:

\(-x_{2}^{4} - 4\nu^{2}x_{2}^{2} + 4\)

Belyi Map 1 Denominator:

\(-x_{2}^{4}\)

Belyi Curve 2: 32S9-2,8,4-g3-path11

\(4/4095x_{1}^{8} + 65536/16769025\nu x_{1}^{7}x_{3} - 16384/2395575x_{1}^{6}x_{3}^{2} - 16384/2395575\nu x_{1}^{5}x_{3}^{3} + 2048/479115x_{1}^{4}x_{3}^{4} + 4096/2395575\nu x_{1}^{3}x_{3}^{5} - 1024/2395575x_{1}^{2}x_{3}^{6} - 1024/16769025\nu x_{1}x_{3}^{7} + x_{2}^{2} + 64/16769025x_{3}^{8}\)

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

Belyi Curve 2 Degree: 8

Belyi Curve 2 Naive Measure: 60372586

Belyi Map 2 Numerator:

\(67125248/16769025\nu x_{1}^{7}x_{2}^{2} - 29744683765016625034471973098356736/5429914198235566686555216227881787109375\nu x_{1}^{7} - 480767873024/68669157375x_{1}^{6}x_{2}^{2} + 13342640133217752336409485127450624/775702028319366669507888032554541015625x_{1}^{6} - 1970394378563584/281200199450625\nu x_{1}^{5}x_{2}^{2} + 18160931164661877550432964346118144/775702028319366669507888032554541015625\nu x_{1}^{5} + 5034523141922077184/1151514816750309375x_{1}^{4}x_{2}^{2} - 2771111229728343578675132031107072/155140405663873333901577606510908203125x_{1}^{4} + 8284703269128467651584/4715453174592516890625\nu x_{1}^{3}x_{2}^{2} - 6389008008883429878001535766102016/775702028319366669507888032554541015625\nu x_{1}^{3} - 8324775430589790107465984/19309780749956356667109375x_{1}^{2}x_{2}^{2} + 254077376975851125764642832351232/110814575474195238501126861793505859375x_{1}^{2} - 5468426135486394020033200384/79073552171071280551812890625\nu x_{1}x_{2}^{2} + 1935550552967069608700287643582464/5429914198235566686555216227881787109375\nu x_{1} + 16785409/16x_{2}^{4} - 758789818474673303177261875216/323806196140536893859673787109375x_{2}^{2} - 129553639021044037997628280013824/5429914198235566686555216227881787109375\)

Belyi Map 2 Denominator:

\(16769025/16x_{2}^{4}\)

Belyi Curve 3: 32S9-2,8,4-g3-path15

\(256/65535x_{1}^{8} + 8192/4294836225\nu x_{1}^{7}x_{3} + 114688/4294836225x_{1}^{6}x_{3}^{2} - 917504/4294836225\nu x_{1}^{5}x_{3}^{3} - 917504/858967245x_{1}^{4}x_{3}^{4} + 14680064/4294836225\nu x_{1}^{3}x_{3}^{5} + 29360128/4294836225x_{1}^{2}x_{3}^{6} - 33554432/4294836225\nu x_{1}x_{3}^{7} + x_{2}^{2} - 16777216/4294836225x_{3}^{8}\)

Belyi Curve 3 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 3 Degree: 8

Belyi Curve 3 Naive Measure: 30922886356

Belyi Map 3 Numerator:

\(2147516416/4294836225\nu x_{1}^{7}x_{2}^{2} - 267939057761768186375181726428543653310483539970490368/95764896012592254757360127065656661622341681787109375\nu x_{1}^{7} + 1970393555992576/281462092005375x_{1}^{6}x_{2}^{2} - 751592185415633721406380608175929350030487366655803392/95764896012592254757360127065656661622341681787109375x_{1}^{6} - 1033128006352968876032/18445618199572250625\nu x_{1}^{5}x_{2}^{2} + 1646074300560387899667732168825182014539253482726096896/95764896012592254757360127065656661622341681787109375\nu x_{1}^{5} - 338612863705413709061423104/1208833588708967444709375x_{1}^{4}x_{2}^{2} + 552916056250510435799570561346008280728065493405007872/19152979202518450951472025413131332324468336357421875x_{1}^{4} + 71067512024381508493790557503488/79220909236042181489028890625\nu x_{1}^{3}x_{2}^{2} - 3453275658071853827792306838076439281298155052409552896/95764896012592254757360127065656661622341681787109375\nu x_{1}^{3} + 9344355604873291713584393533454811136/5191742286784024363883508347109375x_{1}^{2}x_{2}^{2} - 3041174257213716260952091724167286877311221730967027712/95764896012592254757360127065656661622341681787109375x_{1}^{2} - 712035386388574411414816228025507923361792/340240830764391036687105719527812890625\nu x_{1}x_{2}^{2} + 1699596866827748061812803821216514754418084285815717888/95764896012592254757360127065656661622341681787109375\nu x_{1} + 4295098369/65536x_{2}^{4} - 27321011054683886331967578036430485573571969024/22297682844144366589289473329255217787109375x_{2}^{2} + 458364658420662331666141504460461571814679122420957184/95764896012592254757360127065656661622341681787109375\)

Belyi Map 3 Denominator:

\(4294836225/65536x_{2}^{4}\)

Belyi Curve 4: 32S9-2,8,4-g3-path16

\(x_{1}^{8} + x_{2}^{2} - x_{3}^{8}\)

Belyi Curve 4 Base Field: Rationals

Belyi Curve 4 Degree: 8

Belyi Curve 4 Naive Measure: 3

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

\(x_{2}^{4}\)

Belyi Curve 5: 32S9-2,8,4-g3-path17

\(256/65535x_{1}^{8} + 8192/4294836225\nu x_{1}^{7}x_{3} + 114688/4294836225x_{1}^{6}x_{3}^{2} - 917504/4294836225\nu x_{1}^{5}x_{3}^{3} - 917504/858967245x_{1}^{4}x_{3}^{4} + 14680064/4294836225\nu x_{1}^{3}x_{3}^{5} + 29360128/4294836225x_{1}^{2}x_{3}^{6} - 33554432/4294836225\nu x_{1}x_{3}^{7} + x_{2}^{2} - 16777216/4294836225x_{3}^{8}\)

Belyi Curve 5 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 5 Degree: 8

Belyi Curve 5 Naive Measure: 30922886356

Belyi Map 5 Numerator:

\(2147516416/4294836225\nu x_{1}^{7}x_{2}^{2} - 267939057761768186375181726428543653310483539970490368/95764896012592254757360127065656661622341681787109375\nu x_{1}^{7} + 1970393555992576/281462092005375x_{1}^{6}x_{2}^{2} - 751592185415633721406380608175929350030487366655803392/95764896012592254757360127065656661622341681787109375x_{1}^{6} - 1033128006352968876032/18445618199572250625\nu x_{1}^{5}x_{2}^{2} + 1646074300560387899667732168825182014539253482726096896/95764896012592254757360127065656661622341681787109375\nu x_{1}^{5} - 338612863705413709061423104/1208833588708967444709375x_{1}^{4}x_{2}^{2} + 552916056250510435799570561346008280728065493405007872/19152979202518450951472025413131332324468336357421875x_{1}^{4} + 71067512024381508493790557503488/79220909236042181489028890625\nu x_{1}^{3}x_{2}^{2} - 3453275658071853827792306838076439281298155052409552896/95764896012592254757360127065656661622341681787109375\nu x_{1}^{3} + 9344355604873291713584393533454811136/5191742286784024363883508347109375x_{1}^{2}x_{2}^{2} - 3041174257213716260952091724167286877311221730967027712/95764896012592254757360127065656661622341681787109375x_{1}^{2} - 712035386388574411414816228025507923361792/340240830764391036687105719527812890625\nu x_{1}x_{2}^{2} + 1699596866827748061812803821216514754418084285815717888/95764896012592254757360127065656661622341681787109375\nu x_{1} + 4295098369/65536x_{2}^{4} - 27321011054683886331967578036430485573571969024/22297682844144366589289473329255217787109375x_{2}^{2} + 458364658420662331666141504460461571814679122420957184/95764896012592254757360127065656661622341681787109375\)

Belyi Map 5 Denominator:

\(4294836225/65536x_{2}^{4}\)

Belyi Curve 6: 32S9-2,8,4-g3-path2

\(x_{1}^{4}x_{2} + x_{2}^{2} + 4x_{3}^{8}\)

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 8

Belyi Curve 6 Naive Measure: 6

Belyi Map 6 Numerator:

\(x_{1}^{16} + 32x_{1}^{8} + 256\)

Belyi Map 6 Denominator:

\(x_{1}^{16} - 32x_{1}^{8} + 256\)

Belyi Curve 7: 32S9-2,8,4-g3-path3

\(-\nu x_{1}^{8} + x_{2}^{2} + \nu x_{3}^{8}\)

Belyi Curve 7 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 7 Degree: 8

Belyi Curve 7 Naive Measure: 3

Belyi Map 7 Numerator:

\(-x_{2}^{4} - 4\nu x_{2}^{2} + 4\)

Belyi Map 7 Denominator:

\(-x_{2}^{4}\)

Belyi Curve 8: 32S9-2,8,4-g3-path8

\(-x_{1}^{8} + x_{2}^{2} + x_{3}^{8}\)

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

Belyi Curve 8 Degree: 8

Belyi Curve 8 Naive Measure: 3

Belyi Map 8 Numerator:

\(x_{2}^{4} + 4x_{2}^{2} + 4\)

Belyi Map 8 Denominator:

\(x_{2}^{4}\)

Belyi Curve 9: 32S9-2,8,4-g3-path9

\(-x_{1}^{7}x_{3} - 7x_{1}^{5}x_{3}^{3} - 7x_{1}^{3}x_{3}^{5} - x_{1}x_{3}^{7} + x_{2}^{2}\)

Belyi Curve 9 Base Field: Rationals

Belyi Curve 9 Degree: 7

Belyi Curve 9 Naive Measure: 17

Belyi Map 9 Numerator:

\(-29631x_{1}^{6} + 273x_{1}^{5}x_{2}^{2} - 34425x_{1}^{4} - 681x_{1}^{3}x_{2}^{2} + x_{1}^{2}x_{2}^{4} - 5049x_{1}^{2} + 5063x_{1}x_{2}^{2} + 42x_{2}^{4} + 1\)

Belyi Map 9 Denominator:

\(64x_{2}^{4}\)