16T8-4,4,4-g3

graph data
Name 16T8-4,4,4-g3
Type Hyperbolic
Degree 16
Genus 3
Galois orbit size for 16T8-4,4,4-g3-path1 1
Galois orbit size for 16T8-4,4,4-g3-path10 1
Galois orbit size for 16T8-4,4,4-g3-path11 1
Galois orbit size for 16T8-4,4,4-g3-path12 1
Galois orbit size for 16T8-4,4,4-g3-path13 1
Galois orbit size for 16T8-4,4,4-g3-path14 1
Galois orbit size for 16T8-4,4,4-g3-path15 1
Galois orbit size for 16T8-4,4,4-g3-path16 1
Galois orbit size for 16T8-4,4,4-g3-path17 1
Galois orbit size for 16T8-4,4,4-g3-path18 1
Galois orbit size for 16T8-4,4,4-g3-path19 1
Galois orbit size for 16T8-4,4,4-g3-path2 1
Galois orbit size for 16T8-4,4,4-g3-path20 1
Galois orbit size for 16T8-4,4,4-g3-path21 1
Galois orbit size for 16T8-4,4,4-g3-path22 1
Galois orbit size for 16T8-4,4,4-g3-path23 1
Galois orbit size for 16T8-4,4,4-g3-path24 1
Galois orbit size for 16T8-4,4,4-g3-path25 1
Galois orbit size for 16T8-4,4,4-g3-path26 1
Galois orbit size for 16T8-4,4,4-g3-path27 1
Galois orbit size for 16T8-4,4,4-g3-path28 1
Galois orbit size for 16T8-4,4,4-g3-path29 1
Galois orbit size for 16T8-4,4,4-g3-path3 1
Galois orbit size for 16T8-4,4,4-g3-path30 1
Galois orbit size for 16T8-4,4,4-g3-path31 1
Galois orbit size for 16T8-4,4,4-g3-path32 1
Galois orbit size for 16T8-4,4,4-g3-path33 1
Galois orbit size for 16T8-4,4,4-g3-path4 1
Galois orbit size for 16T8-4,4,4-g3-path5 1
Galois orbit size for 16T8-4,4,4-g3-path6 1
Galois orbit size for 16T8-4,4,4-g3-path7 1
Galois orbit size for 16T8-4,4,4-g3-path8 1
Galois orbit size for 16T8-4,4,4-g3-path9 1
Passport size 3
Pointed size 3

Above

32S14-4,4,8-g7 32S13-4,8,4-g7 32S12-4,8,8-g9 32S14-8,4,4-g7 32S13-4,4,8-g7 32S2-4,4,4-g5 32S12-8,8,4-g9 32S13-8,4,4-g7 32S12-8,4,8-g9 32S15-8,8,8-g11 32S14-4,8,4-g7

Below

8T2-4,2,4-g1 8T4-2,4,2-g0 8T2-4,4,2-g1 8T4-4,2,2-g0 8T5-4,4,4-g2 8T2-2,4,4-g1 8T4-2,2,4-g0

Belyi Curve 1: 16T8-4,4,4-g3-path1

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

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:

\(-4x_{1}^{4}\)

Belyi Map 1 Denominator:

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

Belyi Curve 2: 16T8-4,4,4-g3-path10

\(x_{1}^{8} - 24x_{1}^{4}x_{3}^{4} + x_{2}^{2} + 16x_{3}^{8}\)

Belyi Curve 2 Base Field: Rationals

Belyi Curve 2 Degree: 8

Belyi Curve 2 Naive Measure: 42

Belyi Map 2 Numerator:

\(-32x_{1}^{4}x_{2}^{2} - 896x_{1}^{4}x_{2} - 6144x_{1}^{4} + x_{2}^{4} + 32x_{2}^{3} + 256x_{2}^{2} + 512x_{2} + 4096\)

Belyi Map 2 Denominator:

\(-32x_{1}^{4}x_{2}^{2} + 6144x_{1}^{4} + x_{2}^{4} - 256x_{2}^{2} - 4096\)

Belyi Curve 3: 16T8-4,4,4-g3-path11

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

Belyi Curve 3 Base Field: Rationals

Belyi Curve 3 Degree: 7

Belyi Curve 3 Naive Measure: 17

Belyi Map 3 Numerator:

\(-16x_{1}^{6} - 32x_{1}^{4} - 16x_{1}^{2}\)

Belyi Map 3 Denominator:

\(-11x_{1}^{6} - x_{1}^{4} - 5x_{1}^{2} + x_{1}x_{2}^{2} + 1\)

Belyi Curve 4: 16T8-4,4,4-g3-path13

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

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

Belyi Curve 4 Degree: 8

Belyi Curve 4 Naive Measure: 3

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

\(x_{1}^{4}\)

Belyi Curve 5: 16T8-4,4,4-g3-path14

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

Belyi Curve 5 Base Field: Rationals

Belyi Curve 5 Degree: 7

Belyi Curve 5 Naive Measure: 17

Belyi Map 5 Numerator:

\(5x_{1}^{6} + 31x_{1}^{4} + 11x_{1}^{2} + x_{1}x_{2}^{2} + 1\)

Belyi Map 5 Denominator:

\(-11x_{1}^{6} - x_{1}^{4} - 5x_{1}^{2} + x_{1}x_{2}^{2} + 1\)

Belyi Curve 6: 16T8-4,4,4-g3-path15

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 8

Belyi Curve 6 Naive Measure: 3

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

\(4x_{1}^{4}\)

Belyi Curve 7: 16T8-4,4,4-g3-path16

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

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

Belyi Curve 7 Degree: 8

Belyi Curve 7 Naive Measure: 30922886356

Belyi Map 7 Numerator:

\(-2056/255\nu x_{1}^{7} - 12464/255x_{1}^{6} + 34432/255\nu x_{1}^{5} + 8320/51x_{1}^{4} - 28672/255\nu x_{1}^{3} - 57344/255x_{1}^{2} + 65536/255\nu x_{1} - 4328521215/262144x_{2}^{2} + 32768/255\)

Belyi Map 7 Denominator:

\(-1048592/65535\nu x_{1}^{7} - 6291808/65535x_{1}^{6} + 16780544/65535\nu x_{1}^{5} + 3358976/13107x_{1}^{4} - 57344/65535\nu x_{1}^{3} - 114688/65535x_{1}^{2} + 131072/65535\nu x_{1} - 65535/256x_{2}^{2} + 65536/65535\)

Belyi Curve 8: 16T8-4,4,4-g3-path17

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

Belyi Curve 8 Base Field: Rationals

Belyi Curve 8 Degree: 8

Belyi Curve 8 Naive Measure: 3

Belyi Map 8 Numerator:

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

Belyi Map 8 Denominator:

\(4x_{1}^{4}\)

Belyi Curve 9: 16T8-4,4,4-g3-path18

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

Belyi Curve 9 Base Field: Rationals

Belyi Curve 9 Degree: 8

Belyi Curve 9 Naive Measure: 3

Belyi Map 9 Numerator:

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

Belyi Map 9 Denominator:

\(4x_{1}^{4}\)

Belyi Curve 10: 16T8-4,4,4-g3-path19

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

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

Belyi Curve 10 Degree: 8

Belyi Curve 10 Naive Measure: 30922886356

Belyi Map 10 Numerator:

\(-2105344/16581375\nu x_{1}^{7} - 12763136/16581375x_{1}^{6} + 35258368/16581375\nu x_{1}^{5} + 1703936/663255x_{1}^{4} - 29360128/16581375\nu x_{1}^{3} - 58720256/16581375x_{1}^{2} + 67108864/16581375\nu x_{1} - 16974593/65280x_{2}^{2} + 33554432/16581375\)

Belyi Map 10 Denominator:

\(8192/65535\nu x_{1}^{7} + 12402688/16711425x_{1}^{6} - 3538944/1856825\nu x_{1}^{5} - 1441792/983025x_{1}^{4} - 29360128/16711425\nu x_{1}^{3} - 58720256/16711425x_{1}^{2} + 67108864/16711425\nu x_{1} - 65535/256x_{2}^{2} + 33554432/16711425\)

Belyi Curve 11: 16T8-4,4,4-g3-path20

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

Belyi Curve 11 Base Field: Rationals

Belyi Curve 11 Degree: 8

Belyi Curve 11 Naive Measure: 3

Belyi Map 11 Numerator:

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

Belyi Map 11 Denominator:

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

Belyi Curve 12: 16T8-4,4,4-g3-path21

\(2\nu x_{1}^{8} + x_{2}^{2} - 512\nu x_{3}^{8}\)

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

Belyi Curve 12 Degree: 8

Belyi Curve 12 Naive Measure: 515

Belyi Map 12 Numerator:

\(32x_{1}^{4} + 1/2\nu x_{2}^{2} + 512\)

Belyi Map 12 Denominator:

\(-32x_{1}^{4} + 1/2\nu x_{2}^{2} + 512\)

Belyi Curve 13: 16T8-4,4,4-g3-path22

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

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

Belyi Curve 13 Degree: 8

Belyi Curve 13 Naive Measure: 30922886356

Belyi Map 13 Numerator:

\(1073758208/4261413375\nu x_{1}^{7} + 6442811392/4261413375x_{1}^{6} - 17183277056/4261413375\nu x_{1}^{5} - 3439591424/852282675x_{1}^{4} + 58720256/4261413375\nu x_{1}^{3} + 117440512/4261413375x_{1}^{2} - 134217728/4261413375\nu x_{1} + 1028/255x_{2}^{2} - 67108864/4261413375\)

Belyi Map 13 Denominator:

\(8192/65535\nu x_{1}^{7} + 12402688/16711425x_{1}^{6} - 3538944/1856825\nu x_{1}^{5} - 1441792/983025x_{1}^{4} - 29360128/16711425\nu x_{1}^{3} - 58720256/16711425x_{1}^{2} + 67108864/16711425\nu x_{1} - 65535/256x_{2}^{2} + 33554432/16711425\)

Belyi Curve 14: 16T8-4,4,4-g3-path23

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

Belyi Curve 14 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 14 Degree: 8

Belyi Curve 14 Naive Measure: 30922886356

Belyi Map 14 Numerator:

\(1073758208/4261413375\nu x_{1}^{7} + 6442811392/4261413375x_{1}^{6} - 17183277056/4261413375\nu x_{1}^{5} - 3439591424/852282675x_{1}^{4} + 58720256/4261413375\nu x_{1}^{3} + 117440512/4261413375x_{1}^{2} - 134217728/4261413375\nu x_{1} + 1028/255x_{2}^{2} - 67108864/4261413375\)

Belyi Map 14 Denominator:

\(8192/65535\nu x_{1}^{7} + 12402688/16711425x_{1}^{6} - 3538944/1856825\nu x_{1}^{5} - 1441792/983025x_{1}^{4} - 29360128/16711425\nu x_{1}^{3} - 58720256/16711425x_{1}^{2} + 67108864/16711425\nu x_{1} - 65535/256x_{2}^{2} + 33554432/16711425\)

Belyi Curve 15: 16T8-4,4,4-g3-path24

\(2x_{1}^{7}x_{3} + 14x_{1}^{5}x_{3}^{3} + 14x_{1}^{3}x_{3}^{5} + 2x_{1}x_{3}^{7} + x_{2}^{2}\)

Belyi Curve 15 Base Field: Rationals

Belyi Curve 15 Degree: 7

Belyi Curve 15 Naive Measure: 33

Belyi Map 15 Numerator:

\(11x_{1}^{6} + x_{1}^{4} + 5x_{1}^{2} + 1/2x_{1}x_{2}^{2} - 1\)

Belyi Map 15 Denominator:

\(16x_{1}^{6} + 32x_{1}^{4} + 16x_{1}^{2}\)

Belyi Curve 16: 16T8-4,4,4-g3-path3

\(-1024/65535\nu x_{1}^{8} + 32768/4294836225x_{1}^{7}x_{3} - 458752/4294836225\nu x_{1}^{6}x_{3}^{2} - 3670016/4294836225x_{1}^{5}x_{3}^{3} + 3670016/858967245\nu x_{1}^{4}x_{3}^{4} + 58720256/4294836225x_{1}^{3}x_{3}^{5} - 117440512/4294836225\nu x_{1}^{2}x_{3}^{6} - 134217728/4294836225x_{1}x_{3}^{7} + x_{2}^{2} + 67108864/4294836225\nu x_{3}^{8}\)

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

Belyi Curve 16 Degree: 8

Belyi Curve 16 Naive Measure: 30922886356

Belyi Map 16 Numerator:

\(1073758208/4261413375\nu x_{1}^{7} + 6442811392/4261413375x_{1}^{6} - 17183277056/4261413375\nu x_{1}^{5} - 3439591424/852282675x_{1}^{4} + 58720256/4261413375\nu x_{1}^{3} + 117440512/4261413375x_{1}^{2} - 134217728/4261413375\nu x_{1} + 257/255\nu x_{2}^{2} - 67108864/4261413375\)

Belyi Map 16 Denominator:

\(8192/65535\nu x_{1}^{7} + 12402688/16711425x_{1}^{6} - 3538944/1856825\nu x_{1}^{5} - 1441792/983025x_{1}^{4} - 29360128/16711425\nu x_{1}^{3} - 58720256/16711425x_{1}^{2} + 67108864/16711425\nu x_{1} - 65535/1024\nu x_{2}^{2} + 33554432/16711425\)

Belyi Curve 17: 16T8-4,4,4-g3-path4

\(1024/65535\nu^{2}x_{1}^{8} + 32768/4294836225x_{1}^{7}x_{3} + 458752/4294836225\nu^{2}x_{1}^{6}x_{3}^{2} - 3670016/4294836225x_{1}^{5}x_{3}^{3} - 3670016/858967245\nu^{2}x_{1}^{4}x_{3}^{4} + 58720256/4294836225x_{1}^{3}x_{3}^{5} + 117440512/4294836225\nu^{2}x_{1}^{2}x_{3}^{6} - 134217728/4294836225x_{1}x_{3}^{7} + x_{2}^{2} - 67108864/4294836225\nu^{2}x_{3}^{8}\)

Belyi Curve 17 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 17 Degree: 8

Belyi Curve 17 Naive Measure: 30922886356

Belyi Map 17 Numerator:

\(-1073758208/4261413375\nu^{2}x_{1}^{7} + 6442811392/4261413375x_{1}^{6} + 17183277056/4261413375\nu^{2}x_{1}^{5} - 3439591424/852282675x_{1}^{4} - 58720256/4261413375\nu^{2}x_{1}^{3} + 117440512/4261413375x_{1}^{2} + 134217728/4261413375\nu^{2}x_{1} - 257/255\nu^{2}x_{2}^{2} - 67108864/4261413375\)

Belyi Map 17 Denominator:

\(-8192/65535\nu^{2}x_{1}^{7} + 12402688/16711425x_{1}^{6} + 3538944/1856825\nu^{2}x_{1}^{5} - 1441792/983025x_{1}^{4} + 29360128/16711425\nu^{2}x_{1}^{3} - 58720256/16711425x_{1}^{2} - 67108864/16711425\nu^{2}x_{1} + 65535/1024\nu^{2}x_{2}^{2} + 33554432/16711425\)

Belyi Curve 18: 16T8-4,4,4-g3-path5

\(-2x_{1}^{7}x_{3} - 14x_{1}^{5}x_{3}^{3} - 14x_{1}^{3}x_{3}^{5} - 2x_{1}x_{3}^{7} + x_{2}^{2}\)

Belyi Curve 18 Base Field: Rationals

Belyi Curve 18 Degree: 7

Belyi Curve 18 Naive Measure: 33

Belyi Map 18 Numerator:

\(11x_{1}^{6} + x_{1}^{4} + 5x_{1}^{2} - 1/2x_{1}x_{2}^{2} - 1\)

Belyi Map 18 Denominator:

\(16x_{1}^{6} + 32x_{1}^{4} + 16x_{1}^{2}\)

Belyi Curve 19: 16T8-4,4,4-g3-path6

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

Belyi Curve 19 Base Field: Rationals

Belyi Curve 19 Degree: 7

Belyi Curve 19 Naive Measure: 17

Belyi Map 19 Numerator:

\(5x_{1}^{6} + 31x_{1}^{4} + 11x_{1}^{2} + x_{1}x_{2}^{2} + 1\)

Belyi Map 19 Denominator:

\(16x_{1}^{6} + 32x_{1}^{4} + 16x_{1}^{2}\)

Belyi Curve 20: 16T8-4,4,4-g3-path8

\(-1024/65535x_{1}^{8} - 32768/4294836225\nu x_{1}^{7}x_{3} - 458752/4294836225x_{1}^{6}x_{3}^{2} + 3670016/4294836225\nu x_{1}^{5}x_{3}^{3} + 3670016/858967245x_{1}^{4}x_{3}^{4} - 58720256/4294836225\nu x_{1}^{3}x_{3}^{5} - 117440512/4294836225x_{1}^{2}x_{3}^{6} + 134217728/4294836225\nu x_{1}x_{3}^{7} + x_{2}^{2} + 67108864/4294836225x_{3}^{8}\)

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

Belyi Curve 20 Degree: 8

Belyi Curve 20 Naive Measure: 30922886356

Belyi Map 20 Numerator:

\(-2105344/16581375\nu x_{1}^{7} - 12763136/16581375x_{1}^{6} + 35258368/16581375\nu x_{1}^{5} + 1703936/663255x_{1}^{4} - 29360128/16581375\nu x_{1}^{3} - 58720256/16581375x_{1}^{2} + 67108864/16581375\nu x_{1} + 16974593/261120x_{2}^{2} + 33554432/16581375\)

Belyi Map 20 Denominator:

\(8192/65535\nu x_{1}^{7} + 12402688/16711425x_{1}^{6} - 3538944/1856825\nu x_{1}^{5} - 1441792/983025x_{1}^{4} - 29360128/16711425\nu x_{1}^{3} - 58720256/16711425x_{1}^{2} + 67108864/16711425\nu x_{1} + 65535/1024x_{2}^{2} + 33554432/16711425\)

Belyi Curve 21: 16T8-4,4,4-g3-path9

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

Belyi Curve 21 Base Field: Rationals

Belyi Curve 21 Degree: 8

Belyi Curve 21 Naive Measure: 3

Belyi Map 21 Numerator:

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

Belyi Map 21 Denominator:

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