32S12-8,8,4-g9

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

Above

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

Below

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

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

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

Belyi Map 1 Denominator:

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

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

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

Belyi Map 2 Denominator:

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

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

\(-4096/68669157375x_{1}^{3}x_{2} - 13628416/281200199450625x_{1}^{2}x_{2} + 59015168/79204722845259375x_{1}^{2}x_{3}^{2} - 240581632/18278012964290625x_{1}x_{2} + 33980416/79204722845259375x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{6} + 4/4095x_{2}^{2}x_{3}^{2} - 181184/8122034375x_{2}x_{3}^{4} - 155694976/6092670988096875x_{2} + 12004352/169724406096984375x_{3}^{2}\)
\(-64/3353805x_{1}^{2}x_{3}^{2} + x_{1}x_{2}x_{3}^{4} + 4/4095x_{1}x_{2} - 32/3353805x_{1}x_{3}^{2} + 317/260x_{2}x_{3}^{4} + 323/257985x_{2} - 8/5589675x_{3}^{2}\)
\(x_{1}^{3}x_{3}^{2} + 3/4x_{1}^{2}x_{3}^{2} + 1/4x_{1}x_{3}^{2} + 16769025/1024x_{2}x_{3}^{4} + 4095/256x_{2} + 1/32x_{3}^{2}\)
\(x_{1}^{4} + 128/63x_{1}^{3} + 32/21x_{1}^{2} + 32/63x_{1} + 4095/4x_{2}x_{3}^{2} + 4/63\)

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

Belyi Curve 3 Degree: 20

Belyi Curve 3 Naive Measure: 352785812759874322

Belyi Map 3 Numerator:

\(32644096/62523502209x_{1}^{3} + 856714496/1354675881195x_{1}^{2} + 3491080696761677775/124284928x_{1}x_{3}^{16} + 11903044154487885/201963008x_{1}x_{3}^{12} + 238884400559/7767808x_{1}x_{3}^{8} + 887308288/493166447865x_{1}x_{3}^{4} + 1085882624/4064027643585x_{1} + 116267104412943608573978068255270604765625/1042577541300224x_{2}^{3}x_{3}^{22} + 145607751839701876526542946626900171875/260644385325056x_{2}^{3}x_{3}^{18} + 36447674536977571795766086078228125/32580548165632x_{2}^{3}x_{3}^{14} + 9117924424237213343100227851875/8145137041408x_{2}^{3}x_{3}^{10} + 2279686892799384906649300125/4072568520704x_{2}^{3}x_{3}^{6} + 113932816341913129026375/1018142130176x_{2}^{3}x_{3}^{2} - 19105522176981425446889706721795425/32580548165632x_{2}^{2}x_{3}^{20} - 6420709836091241301057395068575/2036284260352x_{2}^{2}x_{3}^{16} - 6168233549154452830503178275/1018142130176x_{2}^{2}x_{3}^{12} - 637206571793347682428575/127267766272x_{2}^{2}x_{3}^{8} - 4049567414050211051765/2672623091712x_{2}^{2}x_{3}^{4} + 7380295/121372x_{2}^{2} + 4778247419998485890116073715/1018142130176x_{2}x_{3}^{18} + 3541246205290072256679975/254535532544x_{2}x_{3}^{14} + 868685812757558899929/63633883136x_{2}x_{3}^{10} + 1480724397961882385/334077886464x_{2}x_{3}^{6} + 608231064832/477992095623x_{2}x_{3}^{2} + 3477338160277719452841/413620240384x_{3}^{16} + 916277046260355927/51702530048x_{3}^{12} + 10815644810607029/1163306926080x_{3}^{8} + 238600192/493166447865x_{3}^{4} + 157025296/4064027643585\)

Belyi Map 3 Denominator:

\(-4095/4x_{2}^{2}\)

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

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

Belyi Map 4 Denominator:

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

Belyi Curve 5: 32S12-8,8,4-g9-path19

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

Belyi Curve 5 Base Field: Rationals

Belyi Curve 5 Degree: 17

Belyi Curve 5 Naive Measure: 20

Belyi Map 5 Numerator:

\(-384x_{1}^{2} + 200/3x_{1}x_{3}^{12} - 64x_{1}x_{3}^{4} - 5/6x_{2}^{3}x_{3}^{6} + 5/6x_{2}^{2}x_{3}^{20} - 35/3x_{2}^{2}x_{3}^{12} + 7/3x_{2}^{2}x_{3}^{4} + 8x_{2}^{2} + 5/2x_{2}x_{3}^{18} - 248/3x_{2}x_{3}^{10} + 144x_{2}x_{3}^{2} - 10/3x_{3}^{16} + 8/3x_{3}^{8} - 64\)

Belyi Map 5 Denominator:

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

Belyi Curve 6: 32S12-8,8,4-g9-path25

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 17

Belyi Curve 6 Naive Measure: 20

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

\(-384x_{1}^{2} + 200/3x_{1}x_{3}^{12} - 64x_{1}x_{3}^{4} - 5/6x_{2}^{3}x_{3}^{6} + 5/6x_{2}^{2}x_{3}^{20} - 35/3x_{2}^{2}x_{3}^{12} + 13/3x_{2}^{2}x_{3}^{4} + 5/2x_{2}x_{3}^{18} - 248/3x_{2}x_{3}^{10} + 144x_{2}x_{3}^{2} - 10/3x_{3}^{16} + 8/3x_{3}^{8} - 64\)

Belyi Curve 7: 32S12-8,8,4-g9-path26

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

Belyi Curve 7 Base Field: Rationals

Belyi Curve 7 Degree: 17

Belyi Curve 7 Naive Measure: 20

Belyi Map 7 Numerator:

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

Belyi Map 7 Denominator:

\(-384x_{1}^{2} + 200/3x_{1}x_{3}^{12} - 64x_{1}x_{3}^{4} - 5/6x_{2}^{3}x_{3}^{6} + 5/6x_{2}^{2}x_{3}^{20} - 35/3x_{2}^{2}x_{3}^{12} + 13/3x_{2}^{2}x_{3}^{4} + 5/2x_{2}x_{3}^{18} - 248/3x_{2}x_{3}^{10} + 144x_{2}x_{3}^{2} - 10/3x_{3}^{16} + 8/3x_{3}^{8} - 64\)

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

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

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

Belyi Curve 8 Degree: 20

Belyi Curve 8 Naive Measure: 7

Belyi Map 8 Numerator:

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

Belyi Map 8 Denominator:

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

Belyi Curve 9: 32S12-8,8,4-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 9 Base Field: Rationals

Belyi Curve 9 Degree: 14

Belyi Curve 9 Naive Measure: 20

Belyi Map 9 Numerator:

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

Belyi Map 9 Denominator:

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

Belyi Curve 10: 32S12-8,8,4-g9-path30

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

Belyi Curve 10 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 10 Degree: 20

Belyi Curve 10 Naive Measure: 7

Belyi Map 10 Numerator:

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

Belyi Map 10 Denominator:

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

Belyi Curve 11: 32S12-8,8,4-g9-path31

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

Belyi Curve 11 Base Field: Rationals

Belyi Curve 11 Degree: 17

Belyi Curve 11 Naive Measure: 20

Belyi Map 11 Numerator:

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

Belyi Map 11 Denominator:

\(-384x_{1}^{2} + 200/3x_{1}x_{3}^{12} - 64x_{1}x_{3}^{4} - 5/6x_{2}^{3}x_{3}^{6} + 5/6x_{2}^{2}x_{3}^{20} - 35/3x_{2}^{2}x_{3}^{12} + 13/3x_{2}^{2}x_{3}^{4} + 5/2x_{2}x_{3}^{18} - 248/3x_{2}x_{3}^{10} + 144x_{2}x_{3}^{2} - 10/3x_{3}^{16} + 8/3x_{3}^{8} - 64\)

Belyi Curve 12: 32S12-8,8,4-g9-path32

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

Belyi Curve 12 Base Field: Rationals

Belyi Curve 12 Degree: 17

Belyi Curve 12 Naive Measure: 20

Belyi Map 12 Numerator:

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

Belyi Map 12 Denominator:

\(-384x_{1}^{2} + 200/3x_{1}x_{3}^{12} - 64x_{1}x_{3}^{4} - 5/6x_{2}^{3}x_{3}^{6} + 5/6x_{2}^{2}x_{3}^{20} - 35/3x_{2}^{2}x_{3}^{12} + 13/3x_{2}^{2}x_{3}^{4} + 5/2x_{2}x_{3}^{18} - 248/3x_{2}x_{3}^{10} + 144x_{2}x_{3}^{2} - 10/3x_{3}^{16} + 8/3x_{3}^{8} - 64\)

Belyi Curve 13: 32S12-8,8,4-g9-path33

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

Belyi Curve 13 Base Field: Rationals

Belyi Curve 13 Degree: 20

Belyi Curve 13 Naive Measure: 7

Belyi Map 13 Numerator:

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

Belyi Map 13 Denominator:

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

Belyi Curve 14: 32S12-8,8,4-g9-path34

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

Belyi Curve 14 Base Field: Rationals

Belyi Curve 14 Degree: 20

Belyi Curve 14 Naive Measure: 7

Belyi Map 14 Numerator:

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

Belyi Map 14 Denominator:

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

Belyi Curve 15: 32S12-8,8,4-g9-path35

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

Belyi Curve 15 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 15 Degree: 20

Belyi Curve 15 Naive Measure: 533619437993393453952291

Belyi Map 15 Numerator:

\(283669637825025/67108864x_{2}^{2}x_{3}^{4}\)

Belyi Map 15 Denominator:

\(238487171235840/1121154893057\nu x_{1}^{3} + 456246712860672/1121154893057x_{1}^{2} - 87908126033502821544089336875/1264171614208\nu x_{1}x_{3}^{16} - 3959150185683270914826175/7407255552\nu x_{1}x_{3}^{12} - 19806186669265104555/19289728\nu x_{1}x_{3}^{8} + 37956639260672/9953650349\nu x_{1}x_{3}^{4} - 482229834743808/1121154893057\nu x_{1} - 48648785070521822264699777761111603530912361570669555078125/1528289704839955192279637358026948608x_{2}^{3}x_{3}^{22} - 3690662799594599370999170866380153385669216031664609375/5969881659531074969842333429792768x_{2}^{3}x_{3}^{18} - 55995440848330608516594235592973905915376863390625/11659925116271630800473307480064x_{2}^{3}x_{3}^{14} - 849545696564039176434681299818583061955809375/45546582485436057814348857344x_{2}^{3}x_{3}^{10} - 348341090734776481737887659495018258125/9617099342364032477691904x_{2}^{3}x_{3}^{6} - 39105834525852053957496892203001875/1389971389326051569041408x_{2}^{3}x_{3}^{2} - 59392330783360669800757175366294068059561474211125/45546582485436057814348857344x_{2}^{2}x_{3}^{20} - 853793152424805539100734687371158855276892875/44479084458433650209325056x_{2}^{2}x_{3}^{16} - 36753731894536627269814971017825762793375/347492847331512892260352x_{2}^{2}x_{3}^{12} - 175445857962791389540988531375452875/678696967444361117696x_{2}^{2}x_{3}^{8} - 2506956145331732526006438460725/10604640116318142464x_{2}^{2}x_{3}^{4} + 627653632705875/19752681472x_{2}^{2} - 28617634596624626528097827192807135888175/1357393934888722235392x_{2}x_{3}^{18} - 1309769918052320117633726450474293875/5302320058159071232x_{2}x_{3}^{14} - 19973074768016470573729411269005/20712187727183872x_{2}x_{3}^{10} - 101481232436467834081878225/80906983309312x_{2}x_{3}^{6} + 48950046512686080/2558088139693x_{2}x_{3}^{2} - 13424330342031299317377823871885/121360474963968x_{3}^{16} - 201921695062624849792158835/237032177664x_{3}^{12} - 3036065512903273239437/1851813888x_{3}^{8} + 70049305985024/9953650349x_{3}^{4} - 210671853633536/1121154893057\)

Belyi Curve 16: 32S12-8,8,4-g9-path36

\(-536870912/281462092005375\nu x_{1}^{3}x_{2} - 206704964796416/18445618199572250625x_{1}^{2}x_{2} - 84837120930545139712/406104878821182527176875x_{1}^{2}x_{3}^{2} + 131182723720019968/4740523877290068410625\nu x_{1}x_{2} + 96719305597408772096/135368292940394175725625\nu x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{6} + 256/65535x_{2}^{2}x_{3}^{2} - 9975933960192/95298165909845x_{2}x_{3}^{4} - 207711687950729216/526724875254452045625x_{2} + 941384052409534251008/1218314636463547581530625x_{3}^{2}\)
\(8388608/858967245\nu x_{1}^{2}x_{3}^{2} + x_{1}x_{2}x_{3}^{4} + 256/65535x_{1}x_{2} + 33554432/858967245x_{1}x_{3}^{2} + 1526/257\nu x_{2}x_{3}^{4} + 395776/16711425\nu x_{2} - 67108864/1431612075\nu x_{3}^{2}\)
\(x_{1}^{3}x_{3}^{2} - 6\nu x_{1}^{2}x_{3}^{2} - 16x_{1}x_{3}^{2} - 4294836225/2097152\nu x_{2}x_{3}^{4} - 65535/8192\nu x_{2} + 16\nu x_{3}^{2}\)
\(x_{1}^{4} + 16/255\nu x_{1}^{3} + 32/85x_{1}^{2} - 256/255\nu x_{1} + 65535/256x_{2}x_{3}^{2} - 256/255\)

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

Belyi Curve 16 Degree: 20

Belyi Curve 16 Naive Measure: 1765073502972636096413995

Belyi Map 16 Numerator:

\(-254155241291776/1078203909375\nu x_{1}^{3} - 19606093692928/39933478125x_{1}^{2} + 50794726029639695231500882125/780836798464\nu x_{1}x_{3}^{16} + 788298175305559587370995/1525071872\nu x_{1}x_{3}^{12} + 12229994049261633357/11914624\nu x_{1}x_{3}^{8} + 22966398091264/6052722075\nu x_{1}x_{3}^{4} + 644958562287616/1078203909375\nu x_{1} - 28227970215531461873443790080009979801779071499940373046875/943973766568354219036140995728113664x_{2}^{3}x_{3}^{22} - 2166475210182033659948090793237809003182743003014765625/3687397525657633668109925764562944x_{2}^{3}x_{3}^{18} - 33253900088099830265862421766648325929649801234375/7201948292300065758027198758912x_{2}^{3}x_{3}^{14} - 510406767458335641890162372307173849549165625/28132610516797131867293745152x_{2}^{3}x_{3}^{10} - 7833852831644173420531573441232430354375/219786019662477592713232384x_{2}^{3}x_{3}^{6} - 24046380170655572243102748749641125/858539139306553096536064x_{2}^{3}x_{3}^{2} + 27613827161546183698031443198648450081681069921875/28132610516797131867293745152x_{2}^{2}x_{3}^{20} + 450015018362987988391555886372987633707678125/27473252457809699089154048x_{2}^{2}x_{3}^{16} + 21953627902278049594064616946193267237625/214634784826638274134016x_{2}^{2}x_{3}^{12} + 118762682549342273092673855700474285/419208564114527879168x_{2}^{2}x_{3}^{8} + 9619330624166796516202401132447/32750669071447490560x_{2}^{2}x_{3}^{4} - 411776551704165/12200574976x_{2}^{2} - 17056000261936670508391981204208314715625/838417128229055758336x_{2}x_{3}^{18} - 781777673898701847996375022265965125/3275066907144749056x_{2}x_{3}^{14} - 11939265678790251317209328853435/12793230106034176x_{2}x_{3}^{10} - 303758248107598847439277683/249867775508480x_{2}x_{3}^{6} - 30539955577947136/1543444129125x_{2}x_{3}^{2} + 2595419551988521905749310491625/24986777550848x_{3}^{16} + 40204236506201853438088215/48802299904x_{3}^{12} + 3112709130403807148941/1906339840x_{3}^{8} + 42414412660736/6052722075x_{3}^{4} + 353609539649536/1078203909375\)

Belyi Map 16 Denominator:

\(-254155241291776/1078203909375\nu x_{1}^{3} - 19606093692928/39933478125x_{1}^{2} + 50794726029639695231500882125/780836798464\nu x_{1}x_{3}^{16} + 788298175305559587370995/1525071872\nu x_{1}x_{3}^{12} + 12229994049261633357/11914624\nu x_{1}x_{3}^{8} + 22966398091264/6052722075\nu x_{1}x_{3}^{4} + 644958562287616/1078203909375\nu x_{1} - 28227970215531461873443790080009979801779071499940373046875/943973766568354219036140995728113664x_{2}^{3}x_{3}^{22} - 2166475210182033659948090793237809003182743003014765625/3687397525657633668109925764562944x_{2}^{3}x_{3}^{18} - 33253900088099830265862421766648325929649801234375/7201948292300065758027198758912x_{2}^{3}x_{3}^{14} - 510406767458335641890162372307173849549165625/28132610516797131867293745152x_{2}^{3}x_{3}^{10} - 7833852831644173420531573441232430354375/219786019662477592713232384x_{2}^{3}x_{3}^{6} - 24046380170655572243102748749641125/858539139306553096536064x_{2}^{3}x_{3}^{2} + 27613827161546183698031443198648450081681069921875/28132610516797131867293745152x_{2}^{2}x_{3}^{20} + 450015018362987988391555886372987633707678125/27473252457809699089154048x_{2}^{2}x_{3}^{16} + 21953627902278049594064616946193267237625/214634784826638274134016x_{2}^{2}x_{3}^{12} + 118762682549342273092673855700474285/419208564114527879168x_{2}^{2}x_{3}^{8} + 9619194333141755984854535532447/32750669071447490560x_{2}^{2}x_{3}^{4} - 411776551704165/12200574976x_{2}^{2} - 17056000261936670508391981204208314715625/838417128229055758336x_{2}x_{3}^{18} - 781777673898701847996375022265965125/3275066907144749056x_{2}x_{3}^{14} - 11939265678790251317209328853435/12793230106034176x_{2}x_{3}^{10} - 303758248107598847439277683/249867775508480x_{2}x_{3}^{6} - 30539955577947136/1543444129125x_{2}x_{3}^{2} + 2595419551988521905749310491625/24986777550848x_{3}^{16} + 40204236506201853438088215/48802299904x_{3}^{12} + 3112709130403807148941/1906339840x_{3}^{8} + 42414412660736/6052722075x_{3}^{4} + 353609539649536/1078203909375\)

Belyi Curve 17: 32S12-8,8,4-g9-path37

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

Belyi Curve 17 Base Field: Rationals

Belyi Curve 17 Degree: 20

Belyi Curve 17 Naive Measure: 7

Belyi Map 17 Numerator:

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

Belyi Map 17 Denominator:

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

Belyi Curve 18: 32S12-8,8,4-g9-path38

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

Belyi Curve 18 Base Field: Rationals

Belyi Curve 18 Degree: 20

Belyi Curve 18 Naive Measure: 7

Belyi Map 18 Numerator:

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

Belyi Map 18 Denominator:

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

Belyi Curve 19: 32S12-8,8,4-g9-path39

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

Belyi Curve 19 Base Field: Rationals

Belyi Curve 19 Degree: 20

Belyi Curve 19 Naive Measure: 7

Belyi Map 19 Numerator:

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

Belyi Map 19 Denominator:

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

Belyi Curve 20: 32S12-8,8,4-g9-path40

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

Belyi Curve 20 Base Field: Rationals

Belyi Curve 20 Degree: 20

Belyi Curve 20 Naive Measure: 7

Belyi Map 20 Numerator:

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

Belyi Map 20 Denominator:

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

Belyi Curve 21: 32S12-8,8,4-g9-path5

\(-4096/68669157375x_{1}^{3}x_{2} - 13628416/281200199450625x_{1}^{2}x_{2} + 59015168/79204722845259375x_{1}^{2}x_{3}^{2} - 240581632/18278012964290625x_{1}x_{2} + 33980416/79204722845259375x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{6} + 4/4095x_{2}^{2}x_{3}^{2} - 181184/8122034375x_{2}x_{3}^{4} - 155694976/6092670988096875x_{2} + 12004352/169724406096984375x_{3}^{2}\)
\(-64/3353805x_{1}^{2}x_{3}^{2} + x_{1}x_{2}x_{3}^{4} + 4/4095x_{1}x_{2} - 32/3353805x_{1}x_{3}^{2} + 317/260x_{2}x_{3}^{4} + 323/257985x_{2} - 8/5589675x_{3}^{2}\)
\(x_{1}^{3}x_{3}^{2} + 3/4x_{1}^{2}x_{3}^{2} + 1/4x_{1}x_{3}^{2} + 16769025/1024x_{2}x_{3}^{4} + 4095/256x_{2} + 1/32x_{3}^{2}\)
\(x_{1}^{4} + 128/63x_{1}^{3} + 32/21x_{1}^{2} + 32/63x_{1} + 4095/4x_{2}x_{3}^{2} + 4/63\)

Belyi Curve 21 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 21 Degree: 20

Belyi Curve 21 Naive Measure: 352785812759874322

Belyi Map 21 Numerator:

\(32644096/62523502209x_{1}^{3} + 856714496/1354675881195x_{1}^{2} + 3491080696761677775/124284928x_{1}x_{3}^{16} + 11903044154487885/201963008x_{1}x_{3}^{12} + 238884400559/7767808x_{1}x_{3}^{8} + 887308288/493166447865x_{1}x_{3}^{4} + 1085882624/4064027643585x_{1} + 116267104412943608573978068255270604765625/1042577541300224x_{2}^{3}x_{3}^{22} + 145607751839701876526542946626900171875/260644385325056x_{2}^{3}x_{3}^{18} + 36447674536977571795766086078228125/32580548165632x_{2}^{3}x_{3}^{14} + 9117924424237213343100227851875/8145137041408x_{2}^{3}x_{3}^{10} + 2279686892799384906649300125/4072568520704x_{2}^{3}x_{3}^{6} + 113932816341913129026375/1018142130176x_{2}^{3}x_{3}^{2} - 19105522176981425446889706721795425/32580548165632x_{2}^{2}x_{3}^{20} - 6420709836091241301057395068575/2036284260352x_{2}^{2}x_{3}^{16} - 6168233549154452830503178275/1018142130176x_{2}^{2}x_{3}^{12} - 637206571793347682428575/127267766272x_{2}^{2}x_{3}^{8} - 4049567414050211051765/2672623091712x_{2}^{2}x_{3}^{4} + 7380295/121372x_{2}^{2} + 4778247419998485890116073715/1018142130176x_{2}x_{3}^{18} + 3541246205290072256679975/254535532544x_{2}x_{3}^{14} + 868685812757558899929/63633883136x_{2}x_{3}^{10} + 1480724397961882385/334077886464x_{2}x_{3}^{6} + 608231064832/477992095623x_{2}x_{3}^{2} + 3477338160277719452841/413620240384x_{3}^{16} + 916277046260355927/51702530048x_{3}^{12} + 10815644810607029/1163306926080x_{3}^{8} + 238600192/493166447865x_{3}^{4} + 157025296/4064027643585\)

Belyi Map 21 Denominator:

\(-4095/4x_{2}^{2}\)

Belyi Curve 22: 32S12-8,8,4-g9-path6

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

Belyi Curve 22 Base Field: Rationals

Belyi Curve 22 Degree: 14

Belyi Curve 22 Naive Measure: 20

Belyi Map 22 Numerator:

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

Belyi Map 22 Denominator:

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

Belyi Curve 23: 32S12-8,8,4-g9-path7

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

Belyi Curve 23 Base Field: Rationals

Belyi Curve 23 Degree: 17

Belyi Curve 23 Naive Measure: 20

Belyi Map 23 Numerator:

\(-384x_{1}^{2} + 200/3x_{1}x_{3}^{12} - 64x_{1}x_{3}^{4} - 5/6x_{2}^{3}x_{3}^{6} + 5/6x_{2}^{2}x_{3}^{20} - 35/3x_{2}^{2}x_{3}^{12} + 7/3x_{2}^{2}x_{3}^{4} + 8x_{2}^{2} + 5/2x_{2}x_{3}^{18} - 248/3x_{2}x_{3}^{10} + 144x_{2}x_{3}^{2} - 10/3x_{3}^{16} + 8/3x_{3}^{8} - 64\)

Belyi Map 23 Denominator:

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