32S9-2,4,8-g3

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

Above

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

Below

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

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

\(-4/4097x_{1}^{8} - 65536/16785409\nu^{2}x_{1}^{7}x_{3} + 114688/16785409x_{1}^{6}x_{3}^{2} + 114688/16785409\nu^{2}x_{1}^{5}x_{3}^{3} - 71680/16785409x_{1}^{4}x_{3}^{4} - 28672/16785409\nu^{2}x_{1}^{3}x_{3}^{5} + 7168/16785409x_{1}^{2}x_{3}^{6} + 1024/16785409\nu^{2}x_{1}x_{3}^{7} + x_{2}^{2} - 64/16785409x_{3}^{8}\)

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

Belyi Curve 1 Degree: 8

Belyi Curve 1 Naive Measure: 134287370

Belyi Map 1 Numerator:

\(-4095\nu^{2}x_{1}^{7}x_{2}^{2} + 1725322804709382602618969692800/325232303859151291990153313161217\nu^{2}x_{1}^{7} + 117506041/16388x_{1}^{6}x_{2}^{2} - 5452682363509264492780896748256/325232303859151291990153313161217x_{1}^{6} + 481019760633/67141636\nu^{2}x_{1}^{5}x_{2}^{2} - 7449676342587987131385204563424/325232303859151291990153313161217\nu^{2}x_{1}^{5} - 9870161535311837/2200634261536x_{1}^{4}x_{2}^{2} + 5698082462823664606644467284196/325232303859151291990153313161217x_{1}^{4} - 8050561018631286777/4507999284756496\nu^{2}x_{1}^{3}x_{2}^{2} + 2632368756535455942470435700456/325232303859151291990153313161217\nu^{2}x_{1}^{3} + 33588569699909816188921/73877092278589456448x_{1}^{2}x_{2}^{2} - 733834052352629830779521785826/325232303859151291990153313161217x_{1}^{2} + 17365863547386556216569855/302674447065381003067456\nu^{2}x_{1}x_{2}^{2} - 114217609755680366913866473470/325232303859151291990153313161217\nu^{2}x_{1} - 281474943156225/262144x_{2}^{4} - 201796989634276490199202783231/19840915354029855513077875712x_{2}^{2} + 122433328218157068146891280383/5203716861746420671842453010579472\)

Belyi Map 1 Denominator:

\(-4095\nu^{2}x_{1}^{7}x_{2}^{2} + 1725322804709382602618969692800/325232303859151291990153313161217\nu^{2}x_{1}^{7} + 117506041/16388x_{1}^{6}x_{2}^{2} - 5452682363509264492780896748256/325232303859151291990153313161217x_{1}^{6} + 481019760633/67141636\nu^{2}x_{1}^{5}x_{2}^{2} - 7449676342587987131385204563424/325232303859151291990153313161217\nu^{2}x_{1}^{5} - 9870161535311837/2200634261536x_{1}^{4}x_{2}^{2} + 5698082462823664606644467284196/325232303859151291990153313161217x_{1}^{4} - 8050561018631286777/4507999284756496\nu^{2}x_{1}^{3}x_{2}^{2} + 2632368756535455942470435700456/325232303859151291990153313161217\nu^{2}x_{1}^{3} + 33588569699909816188921/73877092278589456448x_{1}^{2}x_{2}^{2} - 733834052352629830779521785826/325232303859151291990153313161217x_{1}^{2} + 17365863547386556216569855/302674447065381003067456\nu^{2}x_{1}x_{2}^{2} - 114217609755680366913866473470/325232303859151291990153313161217\nu^{2}x_{1} + 16785409/16x_{2}^{4} - 201796989634276490199202783231/19840915354029855513077875712x_{2}^{2} + 122433328218157068146891280383/5203716861746420671842453010579472\)

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

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

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

Belyi Curve 2 Degree: 8

Belyi Curve 2 Naive Measure: 3

Belyi Map 2 Numerator:

\(1/4x_{2}^{4} + x_{2}^{2} + 1\)

Belyi Map 2 Denominator:

\(x_{2}^{2} + 1\)

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

\(65537/8\nu x_{1}^{7}x_{2}^{2} - 1022106390997956033230521112169432271234449539072/22297682844144366589289473329255217787109375\nu x_{1}^{7} + 30065819641/262140x_{1}^{6}x_{2}^{2} - 2867096654570135961175463135436742210504483667968/22297682844144366589289473329255217787109375x_{1}^{6} - 3941070580875278/4294836225\nu x_{1}^{5}x_{2}^{2} + 6279275133363296126051834750462272699505819254784/22297682844144366589289473329255217787109375\nu x_{1}^{5} - 1291705565282492481466/281462092005375x_{1}^{4}x_{2}^{2} + 2109207367898980849455148930915864108001958821888/4459536568828873317857894665851043557421875x_{1}^{4} + 271101043794179948783075552/18445618199572250625\nu x_{1}^{3}x_{2}^{2} - 13173201210296073256654002525621182561104412278784/22297682844144366589289473329255217787109375\nu x_{1}^{3} + 35645887774937788824403356679744/1208833588708967444709375x_{1}^{2}x_{2}^{2} - 11601159123282303851898543259305140980954062389248/22297682844144366589289473329255217787109375x_{1}^{2} - 2716199441484735150965943252660781568/79220909236042181489028890625\nu x_{1}x_{2}^{2} + 6483447520552627799273696217409190194771134513152/22297682844144366589289473329255217787109375\nu x_{1} + 18446744065119617025/17179869184x_{2}^{4} - 104221386164413018539305030961725179952896/5191742286784024363883508347109375x_{2}^{2} + 1748522409136437727608266847459646498926846017536/22297682844144366589289473329255217787109375\)

Belyi Map 3 Denominator:

\(65537/8\nu x_{1}^{7}x_{2}^{2} - 1022106390997956033230521112169432271234449539072/22297682844144366589289473329255217787109375\nu x_{1}^{7} + 30065819641/262140x_{1}^{6}x_{2}^{2} - 2867096654570135961175463135436742210504483667968/22297682844144366589289473329255217787109375x_{1}^{6} - 3941070580875278/4294836225\nu x_{1}^{5}x_{2}^{2} + 6279275133363296126051834750462272699505819254784/22297682844144366589289473329255217787109375\nu x_{1}^{5} - 1291705565282492481466/281462092005375x_{1}^{4}x_{2}^{2} + 2109207367898980849455148930915864108001958821888/4459536568828873317857894665851043557421875x_{1}^{4} + 271101043794179948783075552/18445618199572250625\nu x_{1}^{3}x_{2}^{2} - 13173201210296073256654002525621182561104412278784/22297682844144366589289473329255217787109375\nu x_{1}^{3} + 35645887774937788824403356679744/1208833588708967444709375x_{1}^{2}x_{2}^{2} - 11601159123282303851898543259305140980954062389248/22297682844144366589289473329255217787109375x_{1}^{2} - 2716199441484735150965943252660781568/79220909236042181489028890625\nu x_{1}x_{2}^{2} + 6483447520552627799273696217409190194771134513152/22297682844144366589289473329255217787109375\nu x_{1} + 4294836225/65536x_{2}^{4} - 104221386164413018539305030961725179952896/5191742286784024363883508347109375x_{2}^{2} + 1748522409136437727608266847459646498926846017536/22297682844144366589289473329255217787109375\)

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

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

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

\(-256/65537x_{1}^{8} + 8192/4295098369\nu^{4}x_{1}^{7}x_{3} + 114688/4295098369x_{1}^{6}x_{3}^{2} - 917504/4295098369\nu^{4}x_{1}^{5}x_{3}^{3} - 4587520/4295098369x_{1}^{4}x_{3}^{4} + 14680064/4295098369\nu^{4}x_{1}^{3}x_{3}^{5} + 29360128/4295098369x_{1}^{2}x_{3}^{6} - 33554432/4295098369\nu^{4}x_{1}x_{3}^{7} + x_{2}^{2} - 16777216/4295098369x_{3}^{8}\)

Belyi Curve 5 Base Field: \(\nu^{8}+1\), discriminant [ <2, 24> ]

Belyi Curve 5 Degree: 8

Belyi Curve 5 Naive Measure: 34360852490

Belyi Map 5 Numerator:

\(-65535/8\nu^{4}x_{1}^{7}x_{2}^{2} + 1018858452941178534099658176003906625818800947200/22303807926762253812938859060411589043224577\nu^{4}x_{1}^{7} - 30063722489/262148x_{1}^{6}x_{2}^{2} + 2847609026024960347401148907860250027435266932736/22303807926762253812938859060411589043224577x_{1}^{6} + 3940228767285234/4295098369\nu^{4}x_{1}^{5}x_{2}^{2} - 6189197542290062147901138249570756282596324278272/22303807926762253812938859060411589043224577\nu^{4}x_{1}^{5} + 1290838622354152882106/281487861809153x_{1}^{4}x_{2}^{2} - 10234845955050233060933799947254610993686736011264/22303807926762253812938859060411589043224577x_{1}^{4} - 270497761475612370828001056/18447869999386460161\nu^{4}x_{1}^{3}x_{2}^{2} + 12403524398270065026636465706266428340583658422272/22303807926762253812938859060411589043224577\nu^{4}x_{1}^{3} - 35342592464128532041441795374656/1209018056149790439571457x_{1}^{2}x_{2}^{2} + 10322189002699907972655438608798381425099502780416/22303807926762253812938859060411589043224577x_{1}^{2} + 2600751963855507031506510780104179200/79235416345888816038194577409\nu^{4}x_{1}x_{2}^{2} - 5209450456530727585272433668068785359717641748480/22303807926762253812938859060411589043224577\nu^{4}x_{1} - 18446744065119617025/17179869184x_{2}^{4} + 70028322500697783942746136505243769241344/5192851481060515336695158019653633x_{2}^{2} - 1174902595839151012005551577446375813695167004672/22303807926762253812938859060411589043224577\)

Belyi Map 5 Denominator:

\(-65535/8\nu^{4}x_{1}^{7}x_{2}^{2} + 1018858452941178534099658176003906625818800947200/22303807926762253812938859060411589043224577\nu^{4}x_{1}^{7} - 30063722489/262148x_{1}^{6}x_{2}^{2} + 2847609026024960347401148907860250027435266932736/22303807926762253812938859060411589043224577x_{1}^{6} + 3940228767285234/4295098369\nu^{4}x_{1}^{5}x_{2}^{2} - 6189197542290062147901138249570756282596324278272/22303807926762253812938859060411589043224577\nu^{4}x_{1}^{5} + 1290838622354152882106/281487861809153x_{1}^{4}x_{2}^{2} - 10234845955050233060933799947254610993686736011264/22303807926762253812938859060411589043224577x_{1}^{4} - 270497761475612370828001056/18447869999386460161\nu^{4}x_{1}^{3}x_{2}^{2} + 12403524398270065026636465706266428340583658422272/22303807926762253812938859060411589043224577\nu^{4}x_{1}^{3} - 35342592464128532041441795374656/1209018056149790439571457x_{1}^{2}x_{2}^{2} + 10322189002699907972655438608798381425099502780416/22303807926762253812938859060411589043224577x_{1}^{2} + 2600751963855507031506510780104179200/79235416345888816038194577409\nu^{4}x_{1}x_{2}^{2} - 5209450456530727585272433668068785359717641748480/22303807926762253812938859060411589043224577\nu^{4}x_{1} + 4295098369/65536x_{2}^{4} + 70028322500697783942746136505243769241344/5192851481060515336695158019653633x_{2}^{2} - 1174902595839151012005551577446375813695167004672/22303807926762253812938859060411589043224577\)

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

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 7

Belyi Curve 6 Naive Measure: 17

Belyi Map 6 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 6 Denominator:

\(-29631x_{1}^{6} + 273x_{1}^{5}x_{2}^{2} - 34425x_{1}^{4} - 681x_{1}^{3}x_{2}^{2} + x_{1}^{2}x_{2}^{4} - 5049x_{1}^{2} + 5063x_{1}x_{2}^{2} - 22x_{2}^{4} + 1\)

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

\(-4/4097x_{1}^{8} - 65536/16785409\nu x_{1}^{7}x_{3} + 114688/16785409x_{1}^{6}x_{3}^{2} + 114688/16785409\nu x_{1}^{5}x_{3}^{3} - 71680/16785409x_{1}^{4}x_{3}^{4} - 28672/16785409\nu x_{1}^{3}x_{3}^{5} + 7168/16785409x_{1}^{2}x_{3}^{6} + 1024/16785409\nu x_{1}x_{3}^{7} + x_{2}^{2} - 64/16785409x_{3}^{8}\)

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

Belyi Curve 7 Degree: 8

Belyi Curve 7 Naive Measure: 134287370

Belyi Map 7 Numerator:

\(-4095\nu x_{1}^{7}x_{2}^{2} + 1725322804709382602618969692800/325232303859151291990153313161217\nu x_{1}^{7} + 117506041/16388x_{1}^{6}x_{2}^{2} - 5452682363509264492780896748256/325232303859151291990153313161217x_{1}^{6} + 481019760633/67141636\nu x_{1}^{5}x_{2}^{2} - 7449676342587987131385204563424/325232303859151291990153313161217\nu x_{1}^{5} - 9870161535311837/2200634261536x_{1}^{4}x_{2}^{2} + 5698082462823664606644467284196/325232303859151291990153313161217x_{1}^{4} - 8050561018631286777/4507999284756496\nu x_{1}^{3}x_{2}^{2} + 2632368756535455942470435700456/325232303859151291990153313161217\nu x_{1}^{3} + 33588569699909816188921/73877092278589456448x_{1}^{2}x_{2}^{2} - 733834052352629830779521785826/325232303859151291990153313161217x_{1}^{2} + 17365863547386556216569855/302674447065381003067456\nu x_{1}x_{2}^{2} - 114217609755680366913866473470/325232303859151291990153313161217\nu x_{1} - 281474943156225/262144x_{2}^{4} - 201796989634276490199202783231/19840915354029855513077875712x_{2}^{2} + 122433328218157068146891280383/5203716861746420671842453010579472\)

Belyi Map 7 Denominator:

\(-4095\nu x_{1}^{7}x_{2}^{2} + 1725322804709382602618969692800/325232303859151291990153313161217\nu x_{1}^{7} + 117506041/16388x_{1}^{6}x_{2}^{2} - 5452682363509264492780896748256/325232303859151291990153313161217x_{1}^{6} + 481019760633/67141636\nu x_{1}^{5}x_{2}^{2} - 7449676342587987131385204563424/325232303859151291990153313161217\nu x_{1}^{5} - 9870161535311837/2200634261536x_{1}^{4}x_{2}^{2} + 5698082462823664606644467284196/325232303859151291990153313161217x_{1}^{4} - 8050561018631286777/4507999284756496\nu x_{1}^{3}x_{2}^{2} + 2632368756535455942470435700456/325232303859151291990153313161217\nu x_{1}^{3} + 33588569699909816188921/73877092278589456448x_{1}^{2}x_{2}^{2} - 733834052352629830779521785826/325232303859151291990153313161217x_{1}^{2} + 17365863547386556216569855/302674447065381003067456\nu x_{1}x_{2}^{2} - 114217609755680366913866473470/325232303859151291990153313161217\nu x_{1} + 16785409/16x_{2}^{4} - 201796989634276490199202783231/19840915354029855513077875712x_{2}^{2} + 122433328218157068146891280383/5203716861746420671842453010579472\)

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

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

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

Belyi Curve 8 Degree: 8

Belyi Curve 8 Naive Measure: 60372586

Belyi Map 8 Numerator:

\(4097\nu x_{1}^{7}x_{2}^{2} - 1815471421204627992826658514304/323806196140536893859673787109375\nu x_{1}^{7} - 117374969/16380x_{1}^{6}x_{2}^{2} + 814370125318466329126555488736/46258028020076699122810541015625x_{1}^{6} - 481053315079/67076100\nu x_{1}^{5}x_{2}^{2} + 1108455271280632174709043234016/46258028020076699122810541015625\nu x_{1}^{5} + 9833053011566557/2197413036000x_{1}^{4}x_{2}^{2} - 169135206892599095378120851508/9251605604015339824562108203125x_{1}^{4} + 8090530536258269191/4499203191210000\nu x_{1}^{3}x_{2}^{2} - 389954102104701530639742173224/46258028020076699122810541015625\nu x_{1}^{3} - 32518654025741367607289/73696948272019800000x_{1}^{2}x_{2}^{2} + 15507652403311225937783375998/6608289717153814160401505859375x_{1}^{2} - 21361039591743726640754689/301789003173921081000000\nu x_{1}x_{2}^{2} + 118136630430118994671648415746/323806196140536893859673787109375\nu x_{1} + 281474943156225/262144x_{2}^{4} - 47424363654667081448578867201/19773215487955309227120000000x_{2}^{2} - 126517225606488318357058867201/5180899138248590301754780593750000\)

Belyi Map 8 Denominator:

\(4097\nu x_{1}^{7}x_{2}^{2} - 1815471421204627992826658514304/323806196140536893859673787109375\nu x_{1}^{7} - 117374969/16380x_{1}^{6}x_{2}^{2} + 814370125318466329126555488736/46258028020076699122810541015625x_{1}^{6} - 481053315079/67076100\nu x_{1}^{5}x_{2}^{2} + 1108455271280632174709043234016/46258028020076699122810541015625\nu x_{1}^{5} + 9833053011566557/2197413036000x_{1}^{4}x_{2}^{2} - 169135206892599095378120851508/9251605604015339824562108203125x_{1}^{4} + 8090530536258269191/4499203191210000\nu x_{1}^{3}x_{2}^{2} - 389954102104701530639742173224/46258028020076699122810541015625\nu x_{1}^{3} - 32518654025741367607289/73696948272019800000x_{1}^{2}x_{2}^{2} + 15507652403311225937783375998/6608289717153814160401505859375x_{1}^{2} - 21361039591743726640754689/301789003173921081000000\nu x_{1}x_{2}^{2} + 118136630430118994671648415746/323806196140536893859673787109375\nu x_{1} + 16769025/16x_{2}^{4} - 47424363654667081448578867201/19773215487955309227120000000x_{2}^{2} - 126517225606488318357058867201/5180899138248590301754780593750000\)

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

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

Belyi Curve 9 Base Field: Rationals

Belyi Curve 9 Degree: 8

Belyi Curve 9 Naive Measure: 6

Belyi Map 9 Numerator:

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

Belyi Map 9 Denominator:

\(64x_{1}^{8}\)