32S8-8,4,8-g9

graph data
Name 32S8-8,4,8-g9
Type Hyperbolic
Degree 32
Genus 9
Galois orbit size for 32S8-8,4,8-g9-path1 1
Galois orbit size for 32S8-8,4,8-g9-path10 1
Galois orbit size for 32S8-8,4,8-g9-path11 1
Galois orbit size for 32S8-8,4,8-g9-path2 1
Galois orbit size for 32S8-8,4,8-g9-path3 1
Galois orbit size for 32S8-8,4,8-g9-path4 1
Galois orbit size for 32S8-8,4,8-g9-path5 1
Galois orbit size for 32S8-8,4,8-g9-path6 1
Galois orbit size for 32S8-8,4,8-g9-path7 1
Galois orbit size for 32S8-8,4,8-g9-path8 1
Galois orbit size for 32S8-8,4,8-g9-path9 1
Passport size 1
Pointed size 1

Above

64S13-8,4,8-g17 64S24-8,4,8-g17 64S5-8,4,8-g17 64S11-8,4,8-g17

Below

16T10-4,2,4-g1

Belyi Curve 1: 32S8-8,4,8-g9-path10

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

Belyi Curve 1 Base Field: Rationals

Belyi Curve 1 Degree: 13

Belyi Curve 1 Naive Measure: 24

Belyi Map 1 Numerator:

\(-2x_{2}^{7}x_{3}^{4} + 16x_{2}^{6} + 2x_{2}^{5}x_{3}^{4} + 32x_{2}^{4} + 2x_{2}^{3}x_{3}^{4} + 16x_{2}^{2} - 2x_{2}x_{3}^{4}\)

Belyi Map 1 Denominator:

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

Belyi Curve 2: 32S8-8,4,8-g9-path11

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

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

Belyi Curve 2 Degree: 13

Belyi Curve 2 Naive Measure: 30

Belyi Map 2 Numerator:

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

Belyi Map 2 Denominator:

\(4\nu x_{2}^{7}x_{3}^{4} - 16x_{2}^{6} - 4\nu x_{2}^{5}x_{3}^{4} - 32x_{2}^{4} - 4\nu x_{2}^{3}x_{3}^{4} - 16x_{2}^{2} + 4\nu x_{2}x_{3}^{4}\)

Belyi Curve 3: 32S8-8,4,8-g9-path2

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

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

Belyi Curve 3 Degree: 13

Belyi Curve 3 Naive Measure: 52

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

\(-2x_{2}^{7}x_{3}^{4} + (4\nu + 4)x_{2}^{7}x_{3}^{2} - 3\nu x_{2}^{7} + 6\nu x_{2}^{6}x_{3}^{4} - 5x_{2}^{6} - 2x_{2}^{5}x_{3}^{4} + (28\nu + 28)x_{2}^{5}x_{3}^{2} - 3\nu x_{2}^{5} + 6\nu x_{2}^{4}x_{3}^{4} + 3x_{2}^{4} - 6x_{2}^{3}x_{3}^{4} + (28\nu + 28)x_{2}^{3}x_{3}^{2} + 7\nu x_{2}^{3} - 14\nu x_{2}^{2}x_{3}^{4} + x_{2}^{2} + 10x_{2}x_{3}^{4} + (4\nu + 4)x_{2}x_{3}^{2} - \nu x_{2} + 2\nu x_{3}^{4} + 1\)

Belyi Curve 4: 32S8-8,4,8-g9-path3

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

Belyi Curve 4 Base Field: Rationals

Belyi Curve 4 Degree: 16

Belyi Curve 4 Naive Measure: 11

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

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

Belyi Curve 5: 32S8-8,4,8-g9-path8

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

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

Belyi Curve 5 Degree: 14

Belyi Curve 5 Naive Measure: 30

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: 32S8-8,4,8-g9-path9

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

Belyi Curve 6 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 6 Degree: 13

Belyi Curve 6 Naive Measure: 67

Belyi Map 6 Numerator:

\(-1/4\nu^{2}x_{2}^{7}x_{3}^{4} + 1/2x_{2}^{7}x_{3}^{2} + 1/16\nu^{2}x_{2}^{7} + 1/4x_{2}^{6}x_{3}^{4} + 1/2(-\nu^{3} - \nu )x_{2}^{6}x_{3}^{2} + 1/16(2\nu^{3} - 2\nu + 13)x_{2}^{6} + 1/4(2\nu^{3} + 3\nu^{2} + 2\nu )x_{2}^{5}x_{3}^{4} + 1/2(-3\nu^{3} + 3\nu - 3)x_{2}^{5}x_{3}^{2} + 1/16(2\nu^{3} - 9\nu^{2} + 2\nu )x_{2}^{5} + 1/4(2\nu^{3} - 2\nu - 3)x_{2}^{4}x_{3}^{4} + (2\nu^{3} - 2\nu^{2} + 2\nu )x_{2}^{4}x_{3}^{2} + 1/16(-12\nu^{3} + 12\nu - 13)x_{2}^{4} + 1/4\nu^{2}x_{2}^{3}x_{3}^{4} + 1/2(4\nu^{3} - 4\nu + 3)x_{2}^{3}x_{3}^{2} + 1/16(-12\nu^{3} + 19\nu^{2} - 12\nu )x_{2}^{3} - 1/4x_{2}^{2}x_{3}^{4} + 1/2(-3\nu^{3} + 4\nu^{2} - 3\nu )x_{2}^{2}x_{3}^{2} + 1/16(2\nu^{3} - 2\nu + 7)x_{2}^{2} + 1/4(-2\nu^{3} - 3\nu^{2} - 2\nu )x_{2}x_{3}^{4} + 1/2(-\nu^{3} + \nu - 1)x_{2}x_{3}^{2} + 1/16(2\nu^{3} - 3\nu^{2} + 2\nu )x_{2} + 1/4(-2\nu^{3} + 2\nu + 3)x_{3}^{4} + 1/16\)

Belyi Map 6 Denominator:

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