32S8-8,8,4-g9

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

Above

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

Below

16T10-4,4,2-g1

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

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

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

Belyi Curve 1 Degree: 16

Belyi Curve 1 Naive Measure: 15

Belyi Map 1 Numerator:

\(-5/32\nu x_{3}^{7}x_{4}^{8} + 1/16\nu x_{3}^{7}x_{4}^{4} - 23/64x_{3}^{6}x_{4}^{8} + 11/64x_{3}^{6}x_{4}^{4} + 9/16\nu x_{3}^{5}x_{4}^{8} - 1/8\nu x_{3}^{5} + 23/32x_{3}^{4}x_{4}^{8} - x_{3}^{4}x_{4}^{4} + 5/32x_{3}^{4} - 13/32\nu x_{3}^{3}x_{4}^{8} + 3/8\nu x_{3}^{3}x_{4}^{4} - 7/64x_{3}^{2}x_{4}^{8} + 27/64x_{3}^{2}x_{4}^{4} - 1/16\nu x_{3}x_{4}^{4} - 1/8\nu x_{3} + 7/32x_{4}^{4} + 5/32\)

Belyi Map 1 Denominator:

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

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

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

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

Belyi Curve 2 Degree: 13

Belyi Curve 2 Naive Measure: 24

Belyi Map 2 Numerator:

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

Belyi Map 2 Denominator:

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

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

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

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

Belyi Curve 3 Degree: 14

Belyi Curve 3 Naive Measure: 30

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

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

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

\(8x_{2}^{3} + (-4\nu^{3} - 20\nu )x_{2}^{2} + x_{2}x_{4}^{4} - 16x_{2} + (4\nu^{3} + 20\nu )x_{3}^{2} + 1/2(-\nu^{3} - 5\nu )x_{4}^{4} + 4\nu^{3} + 20\nu \)
\(1/2(-\nu^{3} - 7\nu )x_{1}x_{4}^{2} + 4x_{2}^{2}x_{3} + (-2\nu^{3} - 10\nu )x_{2}x_{3} - 4x_{3}^{3} + x_{3}x_{4}^{4} - 4x_{3}3\)
\(1/8(\nu^{3} + 7\nu )x_{1}x_{2}x_{4}^{2} + 1/8(\nu^{2} + 3)x_{1}x_{4}^{2} + x_{2}^{3}x_{3} - x_{2}x_{3}3\)
\(x_{1}x_{3}x_{4}^{2} + (-\nu^{3} - 7\nu )x_{2}^{2} + (-\nu^{2} - 3)x_{2} + (\nu^{3} + 7\nu )x_{3}^{2} + \nu^{3} + 7\nu \)
\((-\nu^{3} - 7\nu )x_{1}x_{2} + x_{2}x_{3}x_{4}^{2} + 1/2(\nu^{3} + 5\nu )x_{3}x_{4}^{2}\)
\(x_{1}x_{2}x_{3} + 1/8(\nu^{3} + 7\nu )x_{2}x_{4}^{2} + 1/8(\nu^{2} + 3)x_{4}^{2}\)
\(x_{1}x_{3}^{2} + x_{1} + 1/4(\nu^{3} + 7\nu )x_{3}x_{4}^{2}\)
\(x_{2}x_{3}^{2} - x_{2} + 1/2(\nu^{3} + 5\nu )x_{3}^{2} + 1/2(\nu^{3} + 5\nu )\)
\(x_{1}^{2} - x_{2}^{2} + 1\)

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

Belyi Curve 4 Degree: 14

Belyi Curve 4 Naive Measure: 217

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

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

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

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

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

Belyi Curve 5 Degree: 13

Belyi Curve 5 Naive Measure: 32

Belyi Map 5 Numerator:

\(2x_{2}^{7}x_{3}^{4} + 8\nu x_{2}^{7}x_{3}^{2} + 2x_{2}^{7} + 4x_{2}^{6}x_{3}^{4} - 16\nu x_{2}^{6}x_{3}^{2} + 12x_{2}^{6} + 2x_{2}^{5}x_{3}^{4} - 8\nu x_{2}^{5}x_{3}^{2} - 14x_{2}^{5} + 8x_{2}^{4}x_{3}^{4} - 8x_{2}^{4} - 2x_{2}^{3}x_{3}^{4} - 8\nu x_{2}^{3}x_{3}^{2} + 14x_{2}^{3} + 4x_{2}^{2}x_{3}^{4} + 16\nu x_{2}^{2}x_{3}^{2} + 12x_{2}^{2} - 2x_{2}x_{3}^{4} + 8\nu x_{2}x_{3}^{2} - 2x_{2}2\)

Belyi Map 5 Denominator:

\(2x_{2}^{7}x_{3}^{4} + 8\nu x_{2}^{7}x_{3}^{2} + 2x_{2}^{7} + 4x_{2}^{6}x_{3}^{4} - 16\nu x_{2}^{6}x_{3}^{2} - 4x_{2}^{6} + 2x_{2}^{5}x_{3}^{4} - 8\nu x_{2}^{5}x_{3}^{2} - 14x_{2}^{5} + 8x_{2}^{4}x_{3}^{4} + 24x_{2}^{4} - 2x_{2}^{3}x_{3}^{4} - 8\nu x_{2}^{3}x_{3}^{2} + 14x_{2}^{3} + 4x_{2}^{2}x_{3}^{4} + 16\nu x_{2}^{2}x_{3}^{2} - 4x_{2}^{2} - 2x_{2}x_{3}^{4} + 8\nu x_{2}x_{3}^{2} - 2x_{2}2\)

Belyi Curve 6: 32S8-8,8,4-g9-path9

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

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

Belyi Curve 6 Degree: 13

Belyi Curve 6 Naive Measure: 62

Belyi Map 6 Numerator:

\(-4x_{2}^{7}x_{3}^{4} + 8x_{2}^{7}x_{3}^{2} + x_{2}^{7} - 4x_{2}^{6}x_{3}^{4} + (4\nu^{3} - 4\nu + 9)x_{2}^{6} + (-16\nu^{3} + 16\nu + 20)x_{2}^{5}x_{3}^{4} - 56x_{2}^{5}x_{3}^{2} + (-4\nu^{3} + 4\nu - 5)x_{2}^{5} + (-16\nu^{3} + 16\nu + 20)x_{2}^{4}x_{3}^{4} + (-8\nu^{3} + 8\nu - 5)x_{2}^{4} + (32\nu^{3} - 32\nu - 44)x_{2}^{3}x_{3}^{4} + 56x_{2}^{3}x_{3}^{2} + (8\nu^{3} - 8\nu + 11)x_{2}^{3} + (32\nu^{3} - 32\nu - 44)x_{2}^{2}x_{3}^{4} + (-12\nu^{3} + 12\nu - 13)x_{2}^{2} + (48\nu^{3} - 48\nu - 68)x_{2}x_{3}^{4} - 8x_{2}x_{3}^{2} + (12\nu^{3} - 12\nu + 17)x_{2} + (48\nu^{3} - 48\nu - 68)x_{3}^{4} + 1\)

Belyi Map 6 Denominator:

\(-4x_{2}^{7}x_{3}^{4} + 8x_{2}^{7}x_{3}^{2} + x_{2}^{7} - 4x_{2}^{6}x_{3}^{4} + (4\nu^{3} - 4\nu - 7)x_{2}^{6} + (-16\nu^{3} + 16\nu + 20)x_{2}^{5}x_{3}^{4} - 56x_{2}^{5}x_{3}^{2} + (-4\nu^{3} + 4\nu - 5)x_{2}^{5} + (-16\nu^{3} + 16\nu + 20)x_{2}^{4}x_{3}^{4} + (-8\nu^{3} + 8\nu + 27)x_{2}^{4} + (32\nu^{3} - 32\nu - 44)x_{2}^{3}x_{3}^{4} + 56x_{2}^{3}x_{3}^{2} + (8\nu^{3} - 8\nu + 11)x_{2}^{3} + (32\nu^{3} - 32\nu - 44)x_{2}^{2}x_{3}^{4} + (-12\nu^{3} + 12\nu - 29)x_{2}^{2} + (48\nu^{3} - 48\nu - 68)x_{2}x_{3}^{4} - 8x_{2}x_{3}^{2} + (12\nu^{3} - 12\nu + 17)x_{2} + (48\nu^{3} - 48\nu - 68)x_{3}^{4} + 1\)