32S6-4,4,4-g5

graph data
Name 32S6-4,4,4-g5
Type Hyperbolic
Degree 32
Genus 5
Galois orbit size for 32S6-4,4,4-g5-path1 1
Galois orbit size for 32S6-4,4,4-g5-path10 1
Galois orbit size for 32S6-4,4,4-g5-path11 1
Galois orbit size for 32S6-4,4,4-g5-path12 1
Galois orbit size for 32S6-4,4,4-g5-path13 1
Galois orbit size for 32S6-4,4,4-g5-path14 1
Galois orbit size for 32S6-4,4,4-g5-path15 1
Galois orbit size for 32S6-4,4,4-g5-path16 1
Galois orbit size for 32S6-4,4,4-g5-path17 1
Galois orbit size for 32S6-4,4,4-g5-path18 1
Galois orbit size for 32S6-4,4,4-g5-path19 1
Galois orbit size for 32S6-4,4,4-g5-path2 1
Galois orbit size for 32S6-4,4,4-g5-path20 1
Galois orbit size for 32S6-4,4,4-g5-path21 1
Galois orbit size for 32S6-4,4,4-g5-path22 1
Galois orbit size for 32S6-4,4,4-g5-path23 1
Galois orbit size for 32S6-4,4,4-g5-path24 1
Galois orbit size for 32S6-4,4,4-g5-path25 1
Galois orbit size for 32S6-4,4,4-g5-path26 1
Galois orbit size for 32S6-4,4,4-g5-path27 1
Galois orbit size for 32S6-4,4,4-g5-path28 1
Galois orbit size for 32S6-4,4,4-g5-path29 1
Galois orbit size for 32S6-4,4,4-g5-path3 1
Galois orbit size for 32S6-4,4,4-g5-path30 1
Galois orbit size for 32S6-4,4,4-g5-path31 1
Galois orbit size for 32S6-4,4,4-g5-path32 1
Galois orbit size for 32S6-4,4,4-g5-path33 1
Galois orbit size for 32S6-4,4,4-g5-path4 1
Galois orbit size for 32S6-4,4,4-g5-path5 1
Galois orbit size for 32S6-4,4,4-g5-path6 1
Galois orbit size for 32S6-4,4,4-g5-path7 1
Galois orbit size for 32S6-4,4,4-g5-path8 1
Galois orbit size for 32S6-4,4,4-g5-path9 1
Passport size 3
Pointed size 3

Above

64S37-8,8,4-g17 64S36-8,8,4-g17 64S37-8,4,8-g17 64S36-8,4,8-g17 64S5-8,8,4-g17 64S4-8,8,4-g17 64S9-4,4,8-g13 64S8-4,4,8-g13 64S33-8,4,4-g13 64S32-8,4,4-g13 64S9-8,4,4-g13 64S8-8,4,4-g13 64S35-4,4,4-g9 64S5-8,4,8-g17 64S4-8,4,8-g17 64S37-4,8,8-g17 64S36-4,8,8-g17 64S34-4,4,4-g9 64S33-4,8,4-g13 64S32-4,8,4-g13 64S33-4,4,8-g13 64S32-4,4,8-g13 64S9-4,8,4-g13 64S8-4,8,4-g13 64S5-4,8,8-g17 64S4-4,8,8-g17 64S23-4,4,4-g9

Below

16T10-4,4,2-g1 16T10-2,4,4-g1 16T10-4,2,4-g1

Belyi Curve 1: 32S6-4,4,4-g5-path10

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

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

Belyi Curve 1 Degree: 10

Belyi Curve 1 Naive Measure: 16

Belyi Map 1 Numerator:

\(-1/16x_{2}^{8} + 3/4x_{2}^{6} - 19/8x_{2}^{4} + 3/4x_{2}^{2} - 1/16\)

Belyi Map 1 Denominator:

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

Belyi Curve 2: 32S6-4,4,4-g5-path11

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

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

Belyi Curve 2 Degree: 10

Belyi Curve 2 Naive Measure: 22

Belyi Map 2 Numerator:

\(x_{2}^{8} + 12x_{2}^{6} + 38x_{2}^{4} + 12x_{2}^{2} + 1\)

Belyi Map 2 Denominator:

\(x_{2}^{8} - 4x_{2}^{6} + 6x_{2}^{4} - 4x_{2}^{2} + 1\)

Belyi Curve 3: 32S6-4,4,4-g5-path13

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

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

Belyi Curve 3 Degree: 10

Belyi Curve 3 Naive Measure: 22

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

\(x_{2}^{8} - 4x_{2}^{6} + 6x_{2}^{4} - 4x_{2}^{2} + 1\)

Belyi Curve 4: 32S6-4,4,4-g5-path14

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

Belyi Curve 4 Base Field: Rationals

Belyi Curve 4 Degree: 12

Belyi Curve 4 Naive Measure: 33

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

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

Belyi Curve 5: 32S6-4,4,4-g5-path19

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

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

Belyi Curve 5 Degree: 10

Belyi Curve 5 Naive Measure: 22

Belyi Map 5 Numerator:

\(1/16x_{2}^{8} + 1/4x_{2}^{6} + 3/8x_{2}^{4} + 1/4x_{2}^{2} + 1/16\)

Belyi Map 5 Denominator:

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

Belyi Curve 6: 32S6-4,4,4-g5-path2

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 10

Belyi Curve 6 Naive Measure: 37

Belyi Map 6 Numerator:

\(x_{2}^{8} + 12x_{2}^{6} + 38x_{2}^{4} + 12x_{2}^{2} + 1\)

Belyi Map 6 Denominator:

\(x_{2}^{8} - 4x_{2}^{6} + 6x_{2}^{4} - 4x_{2}^{2} + 1\)

Belyi Curve 7: 32S6-4,4,4-g5-path20

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

Belyi Curve 7 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 7 Degree: 10

Belyi Curve 7 Naive Measure: 16

Belyi Map 7 Numerator:

\(1/16x_{2}^{8} + 1/4x_{2}^{6} + 3/8x_{2}^{4} + 1/4x_{2}^{2} + 1/16\)

Belyi Map 7 Denominator:

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

Belyi Curve 8: 32S6-4,4,4-g5-path21

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

Belyi Curve 8 Base Field: Rationals

Belyi Curve 8 Degree: 10

Belyi Curve 8 Naive Measure: 37

Belyi Map 8 Numerator:

\(-x_{2}^{8} + 4x_{2}^{6} - 6x_{2}^{4} + 4x_{2}^{2} - 1\)

Belyi Map 8 Denominator:

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

Belyi Curve 9: 32S6-4,4,4-g5-path22

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

Belyi Curve 9 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 9 Degree: 10

Belyi Curve 9 Naive Measure: 22

Belyi Map 9 Numerator:

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

Belyi Map 9 Denominator:

\(x_{2}^{8} - 4x_{2}^{6} + 6x_{2}^{4} - 4x_{2}^{2} + 1\)

Belyi Curve 10: 32S6-4,4,4-g5-path24

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

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

Belyi Curve 10 Degree: 10

Belyi Curve 10 Naive Measure: 22

Belyi Map 10 Numerator:

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

Belyi Map 10 Denominator:

\(x_{2}^{8} - 12x_{2}^{6} + 38x_{2}^{4} - 12x_{2}^{2} + 1\)

Belyi Curve 11: 32S6-4,4,4-g5-path25

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

Belyi Curve 11 Base Field: \(\nu^{4}+6\nu^{2}+1\), discriminant [ <2, 14> ]

Belyi Curve 11 Degree: 14

Belyi Curve 11 Naive Measure: 60

Belyi Map 11 Numerator:

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

Belyi Map 11 Denominator:

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

Belyi Curve 12: 32S6-4,4,4-g5-path3

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

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

Belyi Curve 12 Degree: 10

Belyi Curve 12 Naive Measure: 22

Belyi Map 12 Numerator:

\(-2x_{1}x_{3}^{2} - 2x_{1} - 1/16x_{2}^{6}x_{3}^{4} + 1/4x_{2}^{6}x_{3}^{2} - 3/16x_{2}^{6} - 41/64x_{2}^{4}x_{3}^{4} + 9/32x_{2}^{4}x_{3}^{2} + 119/64x_{2}^{4} - 53/32x_{2}^{2}x_{3}^{4} - 13/4x_{2}^{2}x_{3}^{2} - 115/32x_{2}^{2} - 17/64x_{3}^{4} + 15/32x_{3}^{2} - 17/64\)

Belyi Map 12 Denominator:

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

Belyi Curve 13: 32S6-4,4,4-g5-path30

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

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

Belyi Curve 13 Degree: 10

Belyi Curve 13 Naive Measure: 22

Belyi Map 13 Numerator:

\(x_{2}^{8} + 4x_{2}^{6} + 6x_{2}^{4} + 4x_{2}^{2} + 1\)

Belyi Map 13 Denominator:

\(x_{2}^{8} - 12x_{2}^{6} + 38x_{2}^{4} - 12x_{2}^{2} + 1\)

Belyi Curve 14: 32S6-4,4,4-g5-path31

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

Belyi Curve 14 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 14 Degree: 10

Belyi Curve 14 Naive Measure: 16

Belyi Map 14 Numerator:

\(x_{2}^{8} + 4x_{2}^{6} + 6x_{2}^{4} + 4x_{2}^{2} + 1\)

Belyi Map 14 Denominator:

\(x_{2}^{8} - 12x_{2}^{6} + 38x_{2}^{4} - 12x_{2}^{2} + 1\)

Belyi Curve 15: 32S6-4,4,4-g5-path32

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

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

Belyi Curve 15 Degree: 12

Belyi Curve 15 Naive Measure: 9

Belyi Map 15 Numerator:

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

Belyi Map 15 Denominator:

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

Belyi Curve 16: 32S6-4,4,4-g5-path33

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

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

Belyi Curve 16 Degree: 10

Belyi Curve 16 Naive Measure: 22

Belyi Map 16 Numerator:

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

Belyi Map 16 Denominator:

\(x_{2}^{8} - 12x_{2}^{6} + 38x_{2}^{4} - 12x_{2}^{2} + 1\)

Belyi Curve 17: 32S6-4,4,4-g5-path8

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

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

Belyi Curve 17 Degree: 10

Belyi Curve 17 Naive Measure: 22

Belyi Map 17 Numerator:

\(1/16x_{2}^{8} + 3/4x_{2}^{6} + 19/8x_{2}^{4} + 3/4x_{2}^{2} + 1/16\)

Belyi Map 17 Denominator:

\(x_{2}^{6} + 2x_{2}^{4} + x_{2}^{2}\)

Belyi Curve 18: 32S6-4,4,4-g5-path9

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

Belyi Curve 18 Base Field: Rationals

Belyi Curve 18 Degree: 10

Belyi Curve 18 Naive Measure: 37

Belyi Map 18 Numerator:

\(x_{2}^{8} + 12x_{2}^{6} + 38x_{2}^{4} + 12x_{2}^{2} + 1\)

Belyi Map 18 Denominator:

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