32S12-8,4,8-g9

graph data
Name 32S12-8,4,8-g9
Type Hyperbolic
Degree 32
Genus 9
Galois orbit size for 32S12-8,4,8-g9-path1 1
Galois orbit size for 32S12-8,4,8-g9-path10 1
Galois orbit size for 32S12-8,4,8-g9-path11 1
Galois orbit size for 32S12-8,4,8-g9-path12 1
Galois orbit size for 32S12-8,4,8-g9-path13 1
Galois orbit size for 32S12-8,4,8-g9-path14 1
Galois orbit size for 32S12-8,4,8-g9-path15 1
Galois orbit size for 32S12-8,4,8-g9-path16 1
Galois orbit size for 32S12-8,4,8-g9-path17 1
Galois orbit size for 32S12-8,4,8-g9-path18 1
Galois orbit size for 32S12-8,4,8-g9-path19 1
Galois orbit size for 32S12-8,4,8-g9-path2 1
Galois orbit size for 32S12-8,4,8-g9-path20 1
Galois orbit size for 32S12-8,4,8-g9-path21 1
Galois orbit size for 32S12-8,4,8-g9-path22 1
Galois orbit size for 32S12-8,4,8-g9-path23 1
Galois orbit size for 32S12-8,4,8-g9-path24 1
Galois orbit size for 32S12-8,4,8-g9-path25 1
Galois orbit size for 32S12-8,4,8-g9-path26 1
Galois orbit size for 32S12-8,4,8-g9-path27 1
Galois orbit size for 32S12-8,4,8-g9-path28 1
Galois orbit size for 32S12-8,4,8-g9-path29 1
Galois orbit size for 32S12-8,4,8-g9-path3 1
Galois orbit size for 32S12-8,4,8-g9-path30 1
Galois orbit size for 32S12-8,4,8-g9-path31 1
Galois orbit size for 32S12-8,4,8-g9-path32 1
Galois orbit size for 32S12-8,4,8-g9-path33 1
Galois orbit size for 32S12-8,4,8-g9-path34 1
Galois orbit size for 32S12-8,4,8-g9-path35 1
Galois orbit size for 32S12-8,4,8-g9-path36 1
Galois orbit size for 32S12-8,4,8-g9-path37 1
Galois orbit size for 32S12-8,4,8-g9-path38 1
Galois orbit size for 32S12-8,4,8-g9-path39 1
Galois orbit size for 32S12-8,4,8-g9-path4 1
Galois orbit size for 32S12-8,4,8-g9-path40 1
Galois orbit size for 32S12-8,4,8-g9-path41 1
Galois orbit size for 32S12-8,4,8-g9-path42 1
Galois orbit size for 32S12-8,4,8-g9-path43 1
Galois orbit size for 32S12-8,4,8-g9-path44 1
Galois orbit size for 32S12-8,4,8-g9-path45 1
Galois orbit size for 32S12-8,4,8-g9-path46 1
Galois orbit size for 32S12-8,4,8-g9-path5 1
Galois orbit size for 32S12-8,4,8-g9-path6 1
Galois orbit size for 32S12-8,4,8-g9-path7 1
Galois orbit size for 32S12-8,4,8-g9-path8 1
Galois orbit size for 32S12-8,4,8-g9-path9 1
Passport size 2
Pointed size 2

Above

64S44-16,8,16-g25 64S45-16,4,16-g21 64S17-8,4,8-g17 64S44-16,4,16-g21 64S15-8,8,8-g21 64S16-8,8,8-g21 64S45-16,8,16-g25

Below

16T6-8,2,8-g3 16T5-8,4,8-g5 16T8-4,4,4-g3 16T6-8,4,8-g5 16T5-8,2,8-g3

Belyi Curve 1: 32S12-8,4,8-g9-path1

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

Belyi Curve 1 Base Field: Rationals

Belyi Curve 1 Degree: 20

Belyi Curve 1 Naive Measure: 7

Belyi Map 1 Numerator:

\(1\)

Belyi Map 1 Denominator:

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

Belyi Curve 2: 32S12-8,4,8-g9-path15

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

Belyi Curve 2 Base Field: Rationals

Belyi Curve 2 Degree: 14

Belyi Curve 2 Naive Measure: 20

Belyi Map 2 Numerator:

\(x_{2}^{4} + 8x_{2}^{2} + 16\)

Belyi Map 2 Denominator:

\(16x_{2}^{2}\)

Belyi Curve 3: 32S12-8,4,8-g9-path17

\(536870912/281462092005375\nu x_{1}^{3}x_{2} + 215501057818624/18445618199572250625x_{1}^{2}x_{2} - 52401070514315984896/239885264685437119378125x_{1}^{2}x_{3}^{2} - 157078438757269504/4703632640890923909375\nu x_{1}x_{2} + 180089941899291394048/239885264685437119378125\nu x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{6} + 256/65535x_{2}^{2}x_{3}^{2} + 31645788798976/277098404709375x_{2}x_{3}^{4} + 1877286233206423552/4703632640890923909375x_{2} + 36189800705642463232/44423197163969836921875x_{3}^{2}\)
\(-8388608/858967245\nu x_{1}^{2}x_{3}^{2} + x_{1}x_{2}x_{3}^{4} + 256/65535x_{1}x_{2} - 33554432/858967245x_{1}x_{3}^{2} + 1546/255\nu x_{2}x_{3}^{4} + 390656/16842495\nu x_{2} + 67108864/1431612075\nu x_{3}^{2}\)
\(x_{1}^{3}x_{3}^{2} - 6\nu x_{1}^{2}x_{3}^{2} - 16x_{1}x_{3}^{2} + 4294836225/2097152\nu x_{2}x_{3}^{4} + 65535/8192\nu x_{2} + 16\nu x_{3}^{2}\)
\(x_{1}^{4} - 16/257\nu x_{1}^{3} - 96/257x_{1}^{2} + 256/257\nu x_{1} + 65535/256x_{2}x_{3}^{2} + 256/257\)

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

Belyi Curve 3 Degree: 20

Belyi Curve 3 Naive Measure: 533619437993393453952291

Belyi Map 3 Numerator:

\(-931590512640/1121154893057\nu x_{1}^{3} - 1782213722112/1121154893057x_{1}^{2} + 87908126033502821544089336875/323627933237248\nu x_{1}x_{3}^{16} + 3959150185683270914826175/1896257421312\nu x_{1}x_{3}^{12} + 19806186669265104555/4938170368\nu x_{1}x_{3}^{8} - 148268122112/9953650349\nu x_{1}x_{3}^{4} + 1883710291968/1121154893057\nu x_{1} + 48648785070521822264699777761111603530912361570669555078125/391242164439028529223587163654898843648x_{2}^{3}x_{3}^{22} + 3690662799594599370999170866380153385669216031664609375/1528289704839955192279637358026948608x_{2}^{3}x_{3}^{18} + 55995440848330608516594235592973905915376863390625/2984940829765537484921166714896384x_{2}^{3}x_{3}^{14} + 849545696564039176434681299818583061955809375/11659925116271630800473307480064x_{2}^{3}x_{3}^{10} + 348341090734776481737887659495018258125/2461977431645192314289127424x_{2}^{3}x_{3}^{6} + 39105834525852053957496892203001875/355832675667469201674600448x_{2}^{3}x_{3}^{2} + 59392330783360669800757175366294068059561474211125/11659925116271630800473307480064x_{2}^{2}x_{3}^{20} + 853793152424805539100734687371158855276892875/11386645621359014453587214336x_{2}^{2}x_{3}^{16} + 36753731894536627269814971017825762793375/88958168916867300418650112x_{2}^{2}x_{3}^{12} + 175445857962791389540988531375452875/173746423665756446130176x_{2}^{2}x_{3}^{8} + 2507045797107720291153565449525/2714787869777444470784x_{2}^{2}x_{3}^{4} + 9791240618925/2528343228416x_{2}^{2} + 28617634596624626528097827192807135888175/347492847331512892260352x_{2}x_{3}^{18} + 1309769918052320117633726450474293875/1357393934888722235392x_{2}x_{3}^{14} + 19973074768016470573729411269005/5302320058159071232x_{2}x_{3}^{10} + 101481232436467834081878225/20712187727183872x_{2}x_{3}^{6} - 191211119190180/2558088139693x_{2}x_{3}^{2} + 13424330342031299317377823871885/31068281590775808x_{3}^{16} + 201921695062624849792158835/60680237481984x_{3}^{12} + 3036065512903273239437/474064355328x_{3}^{8} - 273630101504/9953650349x_{3}^{4} + 822936928256/1121154893057\)

Belyi Map 3 Denominator:

\(-931590512640/1121154893057\nu x_{1}^{3} - 1782213722112/1121154893057x_{1}^{2} + 87908126033502821544089336875/323627933237248\nu x_{1}x_{3}^{16} + 3959150185683270914826175/1896257421312\nu x_{1}x_{3}^{12} + 19806186669265104555/4938170368\nu x_{1}x_{3}^{8} - 148268122112/9953650349\nu x_{1}x_{3}^{4} + 1883710291968/1121154893057\nu x_{1} + 48648785070521822264699777761111603530912361570669555078125/391242164439028529223587163654898843648x_{2}^{3}x_{3}^{22} + 3690662799594599370999170866380153385669216031664609375/1528289704839955192279637358026948608x_{2}^{3}x_{3}^{18} + 55995440848330608516594235592973905915376863390625/2984940829765537484921166714896384x_{2}^{3}x_{3}^{14} + 849545696564039176434681299818583061955809375/11659925116271630800473307480064x_{2}^{3}x_{3}^{10} + 348341090734776481737887659495018258125/2461977431645192314289127424x_{2}^{3}x_{3}^{6} + 39105834525852053957496892203001875/355832675667469201674600448x_{2}^{3}x_{3}^{2} + 59392330783360669800757175366294068059561474211125/11659925116271630800473307480064x_{2}^{2}x_{3}^{20} + 853793152424805539100734687371158855276892875/11386645621359014453587214336x_{2}^{2}x_{3}^{16} + 36753731894536627269814971017825762793375/88958168916867300418650112x_{2}^{2}x_{3}^{12} + 175445857962791389540988531375452875/173746423665756446130176x_{2}^{2}x_{3}^{8} + 2507045797107720291153565449525/2714787869777444470784x_{2}^{2}x_{3}^{4} - 2490019036425/9876340736x_{2}^{2} + 28617634596624626528097827192807135888175/347492847331512892260352x_{2}x_{3}^{18} + 1309769918052320117633726450474293875/1357393934888722235392x_{2}x_{3}^{14} + 19973074768016470573729411269005/5302320058159071232x_{2}x_{3}^{10} + 101481232436467834081878225/20712187727183872x_{2}x_{3}^{6} - 191211119190180/2558088139693x_{2}x_{3}^{2} + 13424330342031299317377823871885/31068281590775808x_{3}^{16} + 201921695062624849792158835/60680237481984x_{3}^{12} + 3036065512903273239437/474064355328x_{3}^{8} - 273630101504/9953650349x_{3}^{4} + 822936928256/1121154893057\)

Belyi Curve 4: 32S12-8,4,8-g9-path18

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

Belyi Curve 4 Base Field: Rationals

Belyi Curve 4 Degree: 20

Belyi Curve 4 Naive Measure: 7

Belyi Map 4 Numerator:

\(1\)

Belyi Map 4 Denominator:

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

Belyi Curve 5: 32S12-8,4,8-g9-path25

\(x_{2}x_{3}^{4} + 2x_{2}x_{3}^{2} + 2x_{2} + 8x_{3}^{4} - 16\)
\(x_{1}^{4} + x_{2}x_{3}^{2} + x_{2} + 8x_{3}^{2} - 12\)

Belyi Curve 5 Base Field: Rationals

Belyi Curve 5 Degree: 20

Belyi Curve 5 Naive Measure: 52

Belyi Map 5 Numerator:

\(x_{2}^{4} + 32x_{2}^{3}x_{3}^{2} + 64x_{2}^{3} + 1152x_{2}^{2}x_{3}^{2} + 768x_{2}^{2} + 13312x_{2}x_{3}^{2} - 4096x_{2} + 49152x_{3}^{2} - 69632\)

Belyi Map 5 Denominator:

\(x_{2}^{4} + 32x_{2}^{3}x_{3}^{2} + 32x_{2}^{3} + 256x_{2}^{2}x_{3}^{2} - 640x_{2}^{2} - 6144x_{2}x_{3}^{2} - 6144x_{2} - 49152x_{3}^{2} + 69632\)

Belyi Curve 6: 32S12-8,4,8-g9-path26

\(x_{2}x_{3}^{4} + 2x_{2}x_{3}^{2} + 2x_{2} + 8x_{3}^{4} - 16\)
\(x_{1}^{4} + x_{2}x_{3}^{2} + x_{2} + 8x_{3}^{2} - 12\)

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 20

Belyi Curve 6 Naive Measure: 52

Belyi Map 6 Numerator:

\(x_{2}^{4} + 32x_{2}^{3}x_{3}^{2} + 64x_{2}^{3} + 1152x_{2}^{2}x_{3}^{2} + 768x_{2}^{2} + 13312x_{2}x_{3}^{2} - 4096x_{2} + 49152x_{3}^{2} - 69632\)

Belyi Map 6 Denominator:

\(x_{2}^{4} + 32x_{2}^{3}x_{3}^{2} + 32x_{2}^{3} + 256x_{2}^{2}x_{3}^{2} - 640x_{2}^{2} - 6144x_{2}x_{3}^{2} - 6144x_{2} - 49152x_{3}^{2} + 69632\)

Belyi Curve 7: 32S12-8,4,8-g9-path27

\(536870912/281462092005375\nu x_{1}^{3}x_{2} + 215501057818624/18445618199572250625x_{1}^{2}x_{2} - 52401070514315984896/239885264685437119378125x_{1}^{2}x_{3}^{2} - 157078438757269504/4703632640890923909375\nu x_{1}x_{2} + 180089941899291394048/239885264685437119378125\nu x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{6} + 256/65535x_{2}^{2}x_{3}^{2} + 31645788798976/277098404709375x_{2}x_{3}^{4} + 1877286233206423552/4703632640890923909375x_{2} + 36189800705642463232/44423197163969836921875x_{3}^{2}\)
\(-8388608/858967245\nu x_{1}^{2}x_{3}^{2} + x_{1}x_{2}x_{3}^{4} + 256/65535x_{1}x_{2} - 33554432/858967245x_{1}x_{3}^{2} + 1546/255\nu x_{2}x_{3}^{4} + 390656/16842495\nu x_{2} + 67108864/1431612075\nu x_{3}^{2}\)
\(x_{1}^{3}x_{3}^{2} - 6\nu x_{1}^{2}x_{3}^{2} - 16x_{1}x_{3}^{2} + 4294836225/2097152\nu x_{2}x_{3}^{4} + 65535/8192\nu x_{2} + 16\nu x_{3}^{2}\)
\(x_{1}^{4} - 16/257\nu x_{1}^{3} - 96/257x_{1}^{2} + 256/257\nu x_{1} + 65535/256x_{2}x_{3}^{2} + 256/257\)

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

Belyi Curve 7 Degree: 20

Belyi Curve 7 Naive Measure: 533619437993393453952291

Belyi Map 7 Numerator:

\(4362470401/65536x_{2}^{2}x_{3}^{4}\)

Belyi Map 7 Denominator:

\(-319229886726144/95298165909845\nu x_{1}^{3} - 155732211323109376/24301032307010475x_{1}^{2} + 206842649490594874221386675/188885016576\nu x_{1}x_{3}^{16} + 158366007427330836593047/18814718448\nu x_{1}x_{3}^{12} + 621370562173022888/38428755\nu x_{1}x_{3}^{8} - 38867598602928128/647236113943725\nu x_{1}x_{3}^{4} + 54867038975295488/8100344102336825\nu x_{1} + 748155095279074544632061172796795133116683761179078125/1492470414882768742460583357448192x_{2}^{3}x_{3}^{22} + 56757597840747395171075292062747456911483522209375/5829962558135815400236653740032x_{2}^{3}x_{3}^{18} + 861137114161178139432437302467880137106910625/11386645621359014453587214336x_{2}^{3}x_{3}^{14} + 13064908828358926204301134945306929057375/44479084458433650209325056x_{2}^{3}x_{3}^{10} + 5357033306186489530763362698885325/9391698576527375466496x_{2}^{3}x_{3}^{6} + 601396916968120783660082925075/1357393934888722235392x_{2}^{3}x_{3}^{2} + 2740130601308450740519362185296150775527634335/133437253375300950627975168x_{2}^{2}x_{3}^{20} + 13130229179927805291822140520894407616715/43436605916439111532544x_{2}^{2}x_{3}^{16} + 1695673905168933207373239724005802205/1018045451166541676544x_{2}^{2}x_{3}^{12} + 2698129303541582307435425319115/662790007269883904x_{2}^{2}x_{3}^{8} + 115663251267346085747635615/31068281590775808x_{2}^{2}x_{3}^{4} - 9652497235/19289728x_{2}^{2} + 22445203605195785512233589955142851677/67604580741528158208x_{2}x_{3}^{18} + 60427677880153177284139628626265/15534140795387904x_{2}x_{3}^{14} + 234977350211958477337993073753/15473460557905920x_{2}x_{3}^{10} + 1560649479991815979729/79010725888x_{2}x_{3}^{6} - 65522676639203328/217437491873905x_{2}x_{3}^{2} + 2684866068406259863475564774377/1541301735260160x_{3}^{16} + 40384339012524969958431767/3010354951680x_{3}^{12} + 3036065512903273239437/117591990300x_{3}^{8} - 71730489328664576/647236113943725x_{3}^{4} + 215727978120740864/72903096921031425\)

Belyi Curve 8: 32S12-8,4,8-g9-path28

\(536870912/281462092005375\nu x_{1}^{3}x_{2} + 215501057818624/18445618199572250625x_{1}^{2}x_{2} - 52401070514315984896/239885264685437119378125x_{1}^{2}x_{3}^{2} - 157078438757269504/4703632640890923909375\nu x_{1}x_{2} + 180089941899291394048/239885264685437119378125\nu x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{6} + 256/65535x_{2}^{2}x_{3}^{2} + 31645788798976/277098404709375x_{2}x_{3}^{4} + 1877286233206423552/4703632640890923909375x_{2} + 36189800705642463232/44423197163969836921875x_{3}^{2}\)
\(-8388608/858967245\nu x_{1}^{2}x_{3}^{2} + x_{1}x_{2}x_{3}^{4} + 256/65535x_{1}x_{2} - 33554432/858967245x_{1}x_{3}^{2} + 1546/255\nu x_{2}x_{3}^{4} + 390656/16842495\nu x_{2} + 67108864/1431612075\nu x_{3}^{2}\)
\(x_{1}^{3}x_{3}^{2} - 6\nu x_{1}^{2}x_{3}^{2} - 16x_{1}x_{3}^{2} + 4294836225/2097152\nu x_{2}x_{3}^{4} + 65535/8192\nu x_{2} + 16\nu x_{3}^{2}\)
\(x_{1}^{4} - 16/257\nu x_{1}^{3} - 96/257x_{1}^{2} + 256/257\nu x_{1} + 65535/256x_{2}x_{3}^{2} + 256/257\)

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

Belyi Curve 8 Degree: 20

Belyi Curve 8 Naive Measure: 533619437993393453952291

Belyi Map 8 Numerator:

\(4362470401/65536x_{2}^{2}x_{3}^{4}\)

Belyi Map 8 Denominator:

\(-319229886726144/95298165909845\nu x_{1}^{3} - 155732211323109376/24301032307010475x_{1}^{2} + 206842649490594874221386675/188885016576\nu x_{1}x_{3}^{16} + 158366007427330836593047/18814718448\nu x_{1}x_{3}^{12} + 621370562173022888/38428755\nu x_{1}x_{3}^{8} - 38867598602928128/647236113943725\nu x_{1}x_{3}^{4} + 54867038975295488/8100344102336825\nu x_{1} + 748155095279074544632061172796795133116683761179078125/1492470414882768742460583357448192x_{2}^{3}x_{3}^{22} + 56757597840747395171075292062747456911483522209375/5829962558135815400236653740032x_{2}^{3}x_{3}^{18} + 861137114161178139432437302467880137106910625/11386645621359014453587214336x_{2}^{3}x_{3}^{14} + 13064908828358926204301134945306929057375/44479084458433650209325056x_{2}^{3}x_{3}^{10} + 5357033306186489530763362698885325/9391698576527375466496x_{2}^{3}x_{3}^{6} + 601396916968120783660082925075/1357393934888722235392x_{2}^{3}x_{3}^{2} + 2740130601308450740519362185296150775527634335/133437253375300950627975168x_{2}^{2}x_{3}^{20} + 13130229179927805291822140520894407616715/43436605916439111532544x_{2}^{2}x_{3}^{16} + 1695673905168933207373239724005802205/1018045451166541676544x_{2}^{2}x_{3}^{12} + 2698129303541582307435425319115/662790007269883904x_{2}^{2}x_{3}^{8} + 115663251267346085747635615/31068281590775808x_{2}^{2}x_{3}^{4} - 9652497235/19289728x_{2}^{2} + 22445203605195785512233589955142851677/67604580741528158208x_{2}x_{3}^{18} + 60427677880153177284139628626265/15534140795387904x_{2}x_{3}^{14} + 234977350211958477337993073753/15473460557905920x_{2}x_{3}^{10} + 1560649479991815979729/79010725888x_{2}x_{3}^{6} - 65522676639203328/217437491873905x_{2}x_{3}^{2} + 2684866068406259863475564774377/1541301735260160x_{3}^{16} + 40384339012524969958431767/3010354951680x_{3}^{12} + 3036065512903273239437/117591990300x_{3}^{8} - 71730489328664576/647236113943725x_{3}^{4} + 215727978120740864/72903096921031425\)

Belyi Curve 9: 32S12-8,4,8-g9-path29

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

Belyi Curve 9 Base Field: Rationals

Belyi Curve 9 Degree: 20

Belyi Curve 9 Naive Measure: 7

Belyi Map 9 Numerator:

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

Belyi Map 9 Denominator:

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

Belyi Curve 10: 32S12-8,4,8-g9-path3

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

Belyi Curve 10 Base Field: Rationals

Belyi Curve 10 Degree: 14

Belyi Curve 10 Naive Measure: 20

Belyi Map 10 Numerator:

\(x_{2}^{4} + 8x_{2}^{2} + 16\)

Belyi Map 10 Denominator:

\(16x_{2}^{2}\)

Belyi Curve 11: 32S12-8,4,8-g9-path30

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

Belyi Curve 11 Base Field: Rationals

Belyi Curve 11 Degree: 20

Belyi Curve 11 Naive Measure: 7

Belyi Map 11 Numerator:

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

Belyi Map 11 Denominator:

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

Belyi Curve 12: 32S12-8,4,8-g9-path31

\((16\nu - 16)x_{1}^{2}x_{3}^{2} + x_{2}x_{3}^{4} + 2\nu x_{2}2\)
\(x_{1}^{4} + (4\nu + 4)x_{1}^{2} - 1/2\nu x_{2}x_{3}^{2} + 16\nu \)

Belyi Curve 12 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 12 Degree: 20

Belyi Curve 12 Naive Measure: 42

Belyi Map 12 Numerator:

\((-128\nu - 128)x_{1}^{2} + 1/2\nu x_{2}^{2} + 16\nu x_{2}x_{3}^{2} - 512\nu + 512\)

Belyi Map 12 Denominator:

\((128\nu + 128)x_{1}^{2} + 1/2\nu x_{2}^{2} - 16\nu x_{2}x_{3}^{2} + 512\nu + 512\)

Belyi Curve 13: 32S12-8,4,8-g9-path32

\((16\nu + 16)x_{1}^{2}x_{3}^{2} + x_{2}x_{3}^{4} + 2\nu x_{2}2\)
\(x_{1}^{4} + (-4\nu + 4)x_{1}^{2} - 1/2\nu x_{2}x_{3}^{2} - 16\nu \)

Belyi Curve 13 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 13 Degree: 20

Belyi Curve 13 Naive Measure: 42

Belyi Map 13 Numerator:

\((128\nu - 128)x_{1}^{2} + 1/2\nu x_{2}^{2} + 16\nu x_{2}x_{3}^{2} + 512\nu + 512\)

Belyi Map 13 Denominator:

\((-128\nu + 128)x_{1}^{2} + 1/2\nu x_{2}^{2} - 16\nu x_{2}x_{3}^{2} - 512\nu + 512\)

Belyi Curve 14: 32S12-8,4,8-g9-path39

\(x_{1}^{3}x_{3}^{2} + x_{1}x_{3}^{2} - x_{2}2\)
\(-5x_{1}^{2}x_{3}^{2} - x_{1}x_{2} + x_{2}x_{3}^{4} - x_{3}^{2}\)
\(x_{1}^{4} + 6x_{1}^{2} - x_{2}x_{3}^{2} + 1\)

Belyi Curve 14 Base Field: Rationals

Belyi Curve 14 Degree: 17

Belyi Curve 14 Naive Measure: 20

Belyi Map 14 Numerator:

\(x_{2}^{2}x_{3}^{4}\)

Belyi Map 14 Denominator:

\(384x_{1}^{2} - 200/3x_{1}x_{3}^{12} + 64x_{1}x_{3}^{4} + 5/6x_{2}^{3}x_{3}^{6} - 5/6x_{2}^{2}x_{3}^{20} + 35/3x_{2}^{2}x_{3}^{12} - 10/3x_{2}^{2}x_{3}^{4} - 5/2x_{2}x_{3}^{18} + 248/3x_{2}x_{3}^{10} - 144x_{2}x_{3}^{2} + 10/3x_{3}^{16} - 8/3x_{3}^{8} + 64\)

Belyi Curve 15: 32S12-8,4,8-g9-path40

\(x_{1}^{3}x_{3}^{2} + x_{1}x_{3}^{2} - x_{2}2\)
\(-5x_{1}^{2}x_{3}^{2} - x_{1}x_{2} + x_{2}x_{3}^{4} - x_{3}^{2}\)
\(x_{1}^{4} + 6x_{1}^{2} - x_{2}x_{3}^{2} + 1\)

Belyi Curve 15 Base Field: Rationals

Belyi Curve 15 Degree: 17

Belyi Curve 15 Naive Measure: 20

Belyi Map 15 Numerator:

\(x_{2}^{2}x_{3}^{4}\)

Belyi Map 15 Denominator:

\(384x_{1}^{2} - 200/3x_{1}x_{3}^{12} + 64x_{1}x_{3}^{4} + 5/6x_{2}^{3}x_{3}^{6} - 5/6x_{2}^{2}x_{3}^{20} + 35/3x_{2}^{2}x_{3}^{12} - 10/3x_{2}^{2}x_{3}^{4} - 5/2x_{2}x_{3}^{18} + 248/3x_{2}x_{3}^{10} - 144x_{2}x_{3}^{2} + 10/3x_{3}^{16} - 8/3x_{3}^{8} + 64\)

Belyi Curve 16: 32S12-8,4,8-g9-path43

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

Belyi Curve 16 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 16 Degree: 20

Belyi Curve 16 Naive Measure: 533619437993393453952291

Belyi Map 16 Numerator:

\(4362470401/1048576x_{2}^{2}x_{3}^{4}\)

Belyi Map 16 Denominator:

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

Belyi Curve 17: 32S12-8,4,8-g9-path44

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

Belyi Curve 17 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 17 Degree: 20

Belyi Curve 17 Naive Measure: 533619437993393453952291

Belyi Map 17 Numerator:

\(4362470401/1048576x_{2}^{2}x_{3}^{4}\)

Belyi Map 17 Denominator:

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

Belyi Curve 18: 32S12-8,4,8-g9-path45

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

Belyi Curve 18 Base Field: Rationals

Belyi Curve 18 Degree: 20

Belyi Curve 18 Naive Measure: 7

Belyi Map 18 Numerator:

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

Belyi Map 18 Denominator:

\(x_{2}^{2} - 2x_{2}x_{3}^{2}\)

Belyi Curve 19: 32S12-8,4,8-g9-path46

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

Belyi Curve 19 Base Field: Rationals

Belyi Curve 19 Degree: 20

Belyi Curve 19 Naive Measure: 7

Belyi Map 19 Numerator:

\(x_{2}^{2} + 2x_{2}x_{3}^{2}\)

Belyi Map 19 Denominator:

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

Belyi Curve 20: 32S12-8,4,8-g9-path5

\(536870912/281462092005375\nu x_{1}^{3}x_{2} + 215501057818624/18445618199572250625x_{1}^{2}x_{2} - 52401070514315984896/239885264685437119378125x_{1}^{2}x_{3}^{2} - 157078438757269504/4703632640890923909375\nu x_{1}x_{2} + 180089941899291394048/239885264685437119378125\nu x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{6} + 256/65535x_{2}^{2}x_{3}^{2} + 31645788798976/277098404709375x_{2}x_{3}^{4} + 1877286233206423552/4703632640890923909375x_{2} + 36189800705642463232/44423197163969836921875x_{3}^{2}\)
\(-8388608/858967245\nu x_{1}^{2}x_{3}^{2} + x_{1}x_{2}x_{3}^{4} + 256/65535x_{1}x_{2} - 33554432/858967245x_{1}x_{3}^{2} + 1546/255\nu x_{2}x_{3}^{4} + 390656/16842495\nu x_{2} + 67108864/1431612075\nu x_{3}^{2}\)
\(x_{1}^{3}x_{3}^{2} - 6\nu x_{1}^{2}x_{3}^{2} - 16x_{1}x_{3}^{2} + 4294836225/2097152\nu x_{2}x_{3}^{4} + 65535/8192\nu x_{2} + 16\nu x_{3}^{2}\)
\(x_{1}^{4} - 16/257\nu x_{1}^{3} - 96/257x_{1}^{2} + 256/257\nu x_{1} + 65535/256x_{2}x_{3}^{2} + 256/257\)

Belyi Curve 20 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 20 Degree: 20

Belyi Curve 20 Naive Measure: 533619437993393453952291

Belyi Map 20 Numerator:

\(-931590512640/1121154893057\nu x_{1}^{3} - 1782213722112/1121154893057x_{1}^{2} + 87908126033502821544089336875/323627933237248\nu x_{1}x_{3}^{16} + 3959150185683270914826175/1896257421312\nu x_{1}x_{3}^{12} + 19806186669265104555/4938170368\nu x_{1}x_{3}^{8} - 148268122112/9953650349\nu x_{1}x_{3}^{4} + 1883710291968/1121154893057\nu x_{1} + 48648785070521822264699777761111603530912361570669555078125/391242164439028529223587163654898843648x_{2}^{3}x_{3}^{22} + 3690662799594599370999170866380153385669216031664609375/1528289704839955192279637358026948608x_{2}^{3}x_{3}^{18} + 55995440848330608516594235592973905915376863390625/2984940829765537484921166714896384x_{2}^{3}x_{3}^{14} + 849545696564039176434681299818583061955809375/11659925116271630800473307480064x_{2}^{3}x_{3}^{10} + 348341090734776481737887659495018258125/2461977431645192314289127424x_{2}^{3}x_{3}^{6} + 39105834525852053957496892203001875/355832675667469201674600448x_{2}^{3}x_{3}^{2} + 59392330783360669800757175366294068059561474211125/11659925116271630800473307480064x_{2}^{2}x_{3}^{20} + 853793152424805539100734687371158855276892875/11386645621359014453587214336x_{2}^{2}x_{3}^{16} + 36753731894536627269814971017825762793375/88958168916867300418650112x_{2}^{2}x_{3}^{12} + 175445857962791389540988531375452875/173746423665756446130176x_{2}^{2}x_{3}^{8} + 2507045797107720291153565449525/2714787869777444470784x_{2}^{2}x_{3}^{4} + 9791240618925/2528343228416x_{2}^{2} + 28617634596624626528097827192807135888175/347492847331512892260352x_{2}x_{3}^{18} + 1309769918052320117633726450474293875/1357393934888722235392x_{2}x_{3}^{14} + 19973074768016470573729411269005/5302320058159071232x_{2}x_{3}^{10} + 101481232436467834081878225/20712187727183872x_{2}x_{3}^{6} - 191211119190180/2558088139693x_{2}x_{3}^{2} + 13424330342031299317377823871885/31068281590775808x_{3}^{16} + 201921695062624849792158835/60680237481984x_{3}^{12} + 3036065512903273239437/474064355328x_{3}^{8} - 273630101504/9953650349x_{3}^{4} + 822936928256/1121154893057\)

Belyi Map 20 Denominator:

\(-931590512640/1121154893057\nu x_{1}^{3} - 1782213722112/1121154893057x_{1}^{2} + 87908126033502821544089336875/323627933237248\nu x_{1}x_{3}^{16} + 3959150185683270914826175/1896257421312\nu x_{1}x_{3}^{12} + 19806186669265104555/4938170368\nu x_{1}x_{3}^{8} - 148268122112/9953650349\nu x_{1}x_{3}^{4} + 1883710291968/1121154893057\nu x_{1} + 48648785070521822264699777761111603530912361570669555078125/391242164439028529223587163654898843648x_{2}^{3}x_{3}^{22} + 3690662799594599370999170866380153385669216031664609375/1528289704839955192279637358026948608x_{2}^{3}x_{3}^{18} + 55995440848330608516594235592973905915376863390625/2984940829765537484921166714896384x_{2}^{3}x_{3}^{14} + 849545696564039176434681299818583061955809375/11659925116271630800473307480064x_{2}^{3}x_{3}^{10} + 348341090734776481737887659495018258125/2461977431645192314289127424x_{2}^{3}x_{3}^{6} + 39105834525852053957496892203001875/355832675667469201674600448x_{2}^{3}x_{3}^{2} + 59392330783360669800757175366294068059561474211125/11659925116271630800473307480064x_{2}^{2}x_{3}^{20} + 853793152424805539100734687371158855276892875/11386645621359014453587214336x_{2}^{2}x_{3}^{16} + 36753731894536627269814971017825762793375/88958168916867300418650112x_{2}^{2}x_{3}^{12} + 175445857962791389540988531375452875/173746423665756446130176x_{2}^{2}x_{3}^{8} + 2507045797107720291153565449525/2714787869777444470784x_{2}^{2}x_{3}^{4} - 2490019036425/9876340736x_{2}^{2} + 28617634596624626528097827192807135888175/347492847331512892260352x_{2}x_{3}^{18} + 1309769918052320117633726450474293875/1357393934888722235392x_{2}x_{3}^{14} + 19973074768016470573729411269005/5302320058159071232x_{2}x_{3}^{10} + 101481232436467834081878225/20712187727183872x_{2}x_{3}^{6} - 191211119190180/2558088139693x_{2}x_{3}^{2} + 13424330342031299317377823871885/31068281590775808x_{3}^{16} + 201921695062624849792158835/60680237481984x_{3}^{12} + 3036065512903273239437/474064355328x_{3}^{8} - 273630101504/9953650349x_{3}^{4} + 822936928256/1121154893057\)

Belyi Curve 21: 32S12-8,4,8-g9-path6

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

Belyi Curve 21 Base Field: Rationals

Belyi Curve 21 Degree: 20

Belyi Curve 21 Naive Measure: 7

Belyi Map 21 Numerator:

\(1\)

Belyi Map 21 Denominator:

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