32S11-4,4,8-g7

graph data
Name 32S11-4,4,8-g7
Type Hyperbolic
Degree 32
Genus 7
Galois orbit size for 32S11-4,4,8-g7-path1 1
Galois orbit size for 32S11-4,4,8-g7-path10 1
Galois orbit size for 32S11-4,4,8-g7-path11 1
Galois orbit size for 32S11-4,4,8-g7-path12 1
Galois orbit size for 32S11-4,4,8-g7-path13 1
Galois orbit size for 32S11-4,4,8-g7-path14 1
Galois orbit size for 32S11-4,4,8-g7-path15 1
Galois orbit size for 32S11-4,4,8-g7-path16 1
Galois orbit size for 32S11-4,4,8-g7-path17 1
Galois orbit size for 32S11-4,4,8-g7-path18 1
Galois orbit size for 32S11-4,4,8-g7-path19 1
Galois orbit size for 32S11-4,4,8-g7-path2 1
Galois orbit size for 32S11-4,4,8-g7-path20 1
Galois orbit size for 32S11-4,4,8-g7-path21 1
Galois orbit size for 32S11-4,4,8-g7-path22 1
Galois orbit size for 32S11-4,4,8-g7-path3 1
Galois orbit size for 32S11-4,4,8-g7-path4 1
Galois orbit size for 32S11-4,4,8-g7-path5 1
Galois orbit size for 32S11-4,4,8-g7-path6 1
Galois orbit size for 32S11-4,4,8-g7-path7 1
Galois orbit size for 32S11-4,4,8-g7-path8 1
Galois orbit size for 32S11-4,4,8-g7-path9 1
Passport size 2
Pointed size 2

Above

64S11-8,4,8-g17 64S10-8,4,8-g17 64S20-4,4,8-g13 64S9-4,4,8-g13 64S8-4,4,8-g13 64S7-4,8,8-g17 64S6-4,8,8-g17 64S7-8,4,8-g17 64S6-8,4,8-g17 64S11-4,8,8-g17 64S10-4,8,8-g17

Below

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

Belyi Curve 1: 32S11-4,4,8-g7-path10

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

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

Belyi Curve 1 Degree: 11

Belyi Curve 1 Naive Measure: 23

Belyi Map 1 Numerator:

\(-1/2x_{2}^{5}x_{3}^{12} + 15/4x_{2}^{5}x_{3}^{8} - 35/8x_{2}^{5}x_{3}^{4} - 3/16x_{2}^{5} + 3/2x_{2}^{4}x_{3}^{12} - 27/4x_{2}^{4}x_{3}^{8} - 3/8x_{2}^{4}x_{3}^{4} - 13/16x_{2}^{4} + x_{2}^{3}x_{3}^{12} - 15/2x_{2}^{3}x_{3}^{8} + 35/4x_{2}^{3}x_{3}^{4} - 5/8x_{2}^{3} - 3x_{2}^{2}x_{3}^{12} + 27/2x_{2}^{2}x_{3}^{8} - 13/4x_{2}^{2}x_{3}^{4} + 5/8x_{2}^{2} - 5/2x_{2}x_{3}^{12} + 43/4x_{2}x_{3}^{8} + 17/8x_{2}x_{3}^{4} + 1/16x_{2} - 1/2x_{3}^{12} + 9/4x_{3}^{8} + 1/8x_{3}^{4} - 1/16\)

Belyi Map 1 Denominator:

\(2x_{2}^{5}x_{3}^{4} - 3x_{2}^{5} - 6x_{2}^{4}x_{3}^{4} - 4x_{2}^{3}x_{3}^{4} + 6x_{2}^{3} + 12x_{2}^{2}x_{3}^{4} - 4x_{2}^{2} + 10x_{2}x_{3}^{4} + x_{2} + 2x_{3}^{4}\)

Belyi Curve 2: 32S11-4,4,8-g7-path11

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

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

Belyi Curve 2 Degree: 12

Belyi Curve 2 Naive Measure: 34

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: 32S11-4,4,8-g7-path13

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

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

Belyi Curve 3 Degree: 12

Belyi Curve 3 Naive Measure: 34

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: 32S11-4,4,8-g7-path14

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

Belyi Curve 4 Base Field: Rationals

Belyi Curve 4 Degree: 12

Belyi Curve 4 Naive Measure: 40

Belyi Map 4 Numerator:

\(32x_{3}^{5}x_{4}^{6} + 8x_{3}^{5}x_{4}^{2} + 48x_{3}^{4}x_{4}^{8} - 32x_{3}^{4}x_{4}^{4} + 3x_{3}^{4} + 96x_{3}^{3}x_{4}^{6} - 8x_{3}^{3}x_{4}^{2} + 32x_{3}^{2}x_{4}^{4} - 64x_{3}x_{4}^{6} + 48x_{4}^{8} - 1\)

Belyi Map 4 Denominator:

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

Belyi Curve 5: 32S11-4,4,8-g7-path19

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

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

Belyi Curve 5 Degree: 11

Belyi Curve 5 Naive Measure: 22

Belyi Map 5 Numerator:

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

Belyi Map 5 Denominator:

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

Belyi Curve 6: 32S11-4,4,8-g7-path2

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 12

Belyi Curve 6 Naive Measure: 104

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: 32S11-4,4,8-g7-path20

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

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

Belyi Curve 7 Degree: 12

Belyi Curve 7 Naive Measure: 14

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: 32S11-4,4,8-g7-path21

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

Belyi Curve 8 Base Field: Rationals

Belyi Curve 8 Degree: 12

Belyi Curve 8 Naive Measure: 19

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: 32S11-4,4,8-g7-path22

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

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

Belyi Curve 9 Degree: 12

Belyi Curve 9 Naive Measure: 16

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: 32S11-4,4,8-g7-path3

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

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

Belyi Curve 10 Degree: 11

Belyi Curve 10 Naive Measure: 32

Belyi Map 10 Numerator:

\(1/16x_{1}x_{3}^{9} + 1/2x_{1}x_{3}^{7} - 41/16x_{1}x_{3}^{5} + 7/4x_{1}x_{3}^{3} + x_{1}x_{3} + 1/8x_{2}^{4}x_{3}^{6} - 1/4x_{2}^{4}x_{3}^{4} - 23/8x_{2}^{4}x_{3}^{2} + 1/2x_{2}^{4} + 1/16x_{2}^{2}x_{3}^{9} + 3/8x_{2}^{2}x_{3}^{7} - 31/16x_{2}^{2}x_{3}^{5} + 1/4x_{2}^{2}x_{3}^{3} - x_{2}^{2}x_{3} - 1/2x_{3}^{8} - 3/2x_{3}^{6} - 1/2x_{3}^{4} - 1/4\)

Belyi Map 10 Denominator:

\(-1/4x_{1}x_{3}^{9} - 3/2x_{1}x_{3}^{7} + 9/4x_{1}x_{3}^{5} - 7/2x_{1}x_{3}^{3} + 2x_{1}x_{3} - 1/2x_{2}^{4}x_{3}^{6} + 2x_{2}^{4}x_{3}^{4} + 11/2x_{2}^{4}x_{3}^{2} - x_{2}^{4} - 1/4x_{2}^{2}x_{3}^{9} - x_{2}^{2}x_{3}^{7} + 27/4x_{2}^{2}x_{3}^{5} - 1/2x_{2}^{2}x_{3}^{3} + 2x_{2}^{2}x_{3} + x_{3}^{8} + 4x_{3}^{6} - 3x_{3}^{4}\)

Belyi Curve 11: 32S11-4,4,8-g7-path8

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

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

Belyi Curve 11 Degree: 12

Belyi Curve 11 Naive Measure: 27

Belyi Map 11 Numerator:

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

Belyi Map 11 Denominator:

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

Belyi Curve 12: 32S11-4,4,8-g7-path9

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

Belyi Curve 12 Base Field: Rationals

Belyi Curve 12 Degree: 12

Belyi Curve 12 Naive Measure: 19

Belyi Map 12 Numerator:

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

Belyi Map 12 Denominator:

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