32S5-8,8,4-g9

graph data
Name 32S5-8,8,4-g9
Type Hyperbolic
Degree 32
Genus 9
Galois orbit size for 32S5-8,8,4-g9-path1 1
Galois orbit size for 32S5-8,8,4-g9-path10 1
Galois orbit size for 32S5-8,8,4-g9-path11 1
Galois orbit size for 32S5-8,8,4-g9-path12 1
Galois orbit size for 32S5-8,8,4-g9-path13 1
Galois orbit size for 32S5-8,8,4-g9-path14 1
Galois orbit size for 32S5-8,8,4-g9-path15 1
Galois orbit size for 32S5-8,8,4-g9-path16 1
Galois orbit size for 32S5-8,8,4-g9-path17 1
Galois orbit size for 32S5-8,8,4-g9-path18 1
Galois orbit size for 32S5-8,8,4-g9-path19 1
Galois orbit size for 32S5-8,8,4-g9-path2 1
Galois orbit size for 32S5-8,8,4-g9-path20 1
Galois orbit size for 32S5-8,8,4-g9-path21 1
Galois orbit size for 32S5-8,8,4-g9-path22 1
Galois orbit size for 32S5-8,8,4-g9-path23 1
Galois orbit size for 32S5-8,8,4-g9-path3 1
Galois orbit size for 32S5-8,8,4-g9-path4 1
Galois orbit size for 32S5-8,8,4-g9-path5 1
Galois orbit size for 32S5-8,8,4-g9-path6 1
Galois orbit size for 32S5-8,8,4-g9-path7 1
Galois orbit size for 32S5-8,8,4-g9-path8 1
Galois orbit size for 32S5-8,8,4-g9-path9 1
Passport size 1
Pointed size 1

Above

64S29-16,16,8-g25 64S31-16,16,8-g25 64S17-8,8,4-g17 64S30-16,16,8-g25 64S5-8,8,4-g17 64S4-8,8,4-g17 64S7-8,8,4-g17 64S6-8,8,4-g17

Below

16T5-8,8,4-g5 16T10-4,4,2-g1 16T6-8,8,4-g5

Belyi Curve 1: 32S5-8,8,4-g9-path10

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

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

Belyi Curve 1 Degree: 18

Belyi Curve 1 Naive Measure: 30

Belyi Map 1 Numerator:

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

Belyi Map 1 Denominator:

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

Belyi Curve 2: 32S5-8,8,4-g9-path11

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

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

Belyi Curve 2 Degree: 13

Belyi Curve 2 Naive Measure: 35

Belyi Map 2 Numerator:

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

Belyi Map 2 Denominator:

\(2x_{2}^{7}x_{3}^{4} + (-2\nu - 4)x_{2}^{7} + (-4\nu - 8)x_{2}^{6}x_{3}^{4} + (-8\nu - 12)x_{2}^{6} + (16\nu + 2)x_{2}^{5}x_{3}^{4} + (-2\nu + 12)x_{2}^{5} + (-8\nu + 16)x_{2}^{4}x_{3}^{4} + (16\nu + 40)x_{2}^{4} + (-16\nu - 2)x_{2}^{3}x_{3}^{4} + (2\nu - 12)x_{2}^{3} + (-4\nu - 8)x_{2}^{2}x_{3}^{4} + (-8\nu - 12)x_{2}^{2} - 2x_{2}x_{3}^{4} + (2\nu + 4)x_{2}2\)

Belyi Curve 3: 32S5-8,8,4-g9-path2

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

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

Belyi Curve 3 Degree: 13

Belyi Curve 3 Naive Measure: 33

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

\(2x_{2}^{7}x_{3}^{4} + (-2\nu - 3)x_{2}^{7} + (-4\nu - 6)x_{2}^{6}x_{3}^{4} + (-6\nu - 9)x_{2}^{6} + (12\nu - 6)x_{2}^{5}x_{3}^{4} + (4\nu + 9)x_{2}^{5} + (8\nu + 18)x_{2}^{4}x_{3}^{4} + (12\nu + 31)x_{2}^{4} + (-24\nu + 14)x_{2}^{3}x_{3}^{4} + (-10\nu - 5)x_{2}^{3} + (-20\nu - 10)x_{2}^{2}x_{3}^{4} + (2\nu - 7)x_{2}^{2} + (-4\nu - 10)x_{2}x_{3}^{4} - x_{2} - 2x_{3}^{4} + 1\)

Belyi Curve 4: 32S5-8,8,4-g9-path3

\((-4\nu^{3} - 20\nu )x_{1}x_{3}^{2} + 12x_{1}x_{3} + (2\nu^{3} + 10\nu )x_{1} - 4x_{2}^{2}x_{4}^{2} + (-2\nu^{3} - 10\nu )x_{2}x_{3} + x_{2}x_{4}^{4} + (2\nu^{3} + 10\nu )x_{2}x_{4}^{2} + 4x_{2} - 4x_{3}^{3} + (-2\nu^{3} - 10\nu )x_{3}^{2} + x_{3}x_{4}^{4} + (\nu^{3} + 5\nu )x_{3}x_{4}^{2} + 1/2(-\nu^{3} - 5\nu )x_{4}^{4} + 2x_{4}^{2}\)
\(x_{1}x_{2}^{2} - 1/2x_{1}x_{2}x_{4}^{2} + 1/2(-\nu^{3} - 5\nu )x_{1}x_{2} - x_{1}x_{3}^{2} - 1/2x_{1}x_{3}x_{4}^{2} + 1/4(\nu^{3} + 5\nu )x_{1}x_{4}^{2} - x_{1} + x_{2}^{3}x_{3} - x_{2}x_{3}3\)
\(-2x_{1}x_{2} + 2x_{1}x_{3} + (\nu^{3} + 5\nu )x_{1} - 2x_{2}^{2}x_{3} + x_{2}x_{3}x_{4}^{2} + 2x_{2} + (-\nu^{3} - 5\nu )x_{3}^{2} + 2x_{3}3\)
\((\nu^{3} + 5\nu )x_{1}x_{3}^{2} - 2x_{1}x_{3} + (\nu^{3} + 5\nu )x_{2}x_{3} - 2x_{2} + 2x_{3}^{3} + x_{3}^{2}x_{4}^{2} + (\nu^{3} + 5\nu )x_{3}^{2} + 1/2(-\nu^{3} - 5\nu )x_{3}x_{4}^{2}\)
\(x_{1}x_{2}x_{3} + x_{2}^{2} - 1/2x_{2}x_{4}^{2} + 1/2(-\nu^{3} - 5\nu )x_{2} - x_{3}^{2} - 1/2x_{3}x_{4}^{2} + 1/4(\nu^{3} + 5\nu )x_{4}^{2} - 1\)
\(x_{2}x_{3}^{2} - x_{2} + 1/2(\nu^{3} + 5\nu )x_{3}^{2} + 1/2(\nu^{3} + 5\nu )\)
\(x_{1}^{2} - x_{2}^{2} + 1\)

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

Belyi Curve 4 Degree: 16

Belyi Curve 4 Naive Measure: 198

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

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

Belyi Curve 5: 32S5-8,8,4-g9-path8

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

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

Belyi Curve 5 Degree: 13

Belyi Curve 5 Naive Measure: 33

Belyi Map 5 Numerator:

\(2x_{2}^{7}x_{3}^{4} + (2\nu - 3)x_{2}^{7} + (4\nu - 6)x_{2}^{6}x_{3}^{4} + (6\nu + 7)x_{2}^{6} + (-12\nu - 6)x_{2}^{5}x_{3}^{4} + (-4\nu + 9)x_{2}^{5} + (-8\nu + 18)x_{2}^{4}x_{3}^{4} + (-12\nu - 1)x_{2}^{4} + (24\nu + 14)x_{2}^{3}x_{3}^{4} + (10\nu - 5)x_{2}^{3} + (20\nu - 10)x_{2}^{2}x_{3}^{4} + (-2\nu + 9)x_{2}^{2} + (4\nu - 10)x_{2}x_{3}^{4} - x_{2} - 2x_{3}^{4} + 1\)

Belyi Map 5 Denominator:

\(2x_{2}^{7}x_{3}^{4} + (2\nu - 3)x_{2}^{7} + (4\nu - 6)x_{2}^{6}x_{3}^{4} + (6\nu - 9)x_{2}^{6} + (-12\nu - 6)x_{2}^{5}x_{3}^{4} + (-4\nu + 9)x_{2}^{5} + (-8\nu + 18)x_{2}^{4}x_{3}^{4} + (-12\nu + 31)x_{2}^{4} + (24\nu + 14)x_{2}^{3}x_{3}^{4} + (10\nu - 5)x_{2}^{3} + (20\nu - 10)x_{2}^{2}x_{3}^{4} + (-2\nu - 7)x_{2}^{2} + (4\nu - 10)x_{2}x_{3}^{4} - x_{2} - 2x_{3}^{4} + 1\)

Belyi Curve 6: 32S5-8,8,4-g9-path9

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

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

Belyi Curve 6 Degree: 13

Belyi Curve 6 Naive Measure: 47

Belyi Map 6 Numerator:

\(-4x_{2}^{7}x_{3}^{4} + (-2\nu^{2} - 5)x_{2}^{7} + (8\nu^{2} + 20)x_{2}^{6}x_{3}^{4} + (-10\nu^{2} - 1)x_{2}^{6} + (-40\nu^{2} - 20)x_{2}^{5}x_{3}^{4} + (-12\nu^{2} + 7)x_{2}^{5} + (48\nu^{2} - 28)x_{2}^{4}x_{3}^{4} + (4\nu^{2} + 15)x_{2}^{4} + (16\nu^{2} + 36)x_{2}^{3}x_{3}^{4} + (6\nu^{2} - 3)x_{2}^{3} + (-24\nu^{2} + 12)x_{2}^{2}x_{3}^{4} + (-2\nu^{2} + 1)x_{2}^{2} + (-8\nu^{2} - 12)x_{2}x_{3}^{4} + x_{2} - 4x_{3}^{4} + 1\)

Belyi Map 6 Denominator:

\(-4x_{2}^{7}x_{3}^{4} + (-2\nu^{2} - 5)x_{2}^{7} + (8\nu^{2} + 20)x_{2}^{6}x_{3}^{4} + (-10\nu^{2} - 17)x_{2}^{6} + (-40\nu^{2} - 20)x_{2}^{5}x_{3}^{4} + (-12\nu^{2} + 7)x_{2}^{5} + (48\nu^{2} - 28)x_{2}^{4}x_{3}^{4} + (4\nu^{2} + 47)x_{2}^{4} + (16\nu^{2} + 36)x_{2}^{3}x_{3}^{4} + (6\nu^{2} - 3)x_{2}^{3} + (-24\nu^{2} + 12)x_{2}^{2}x_{3}^{4} + (-2\nu^{2} - 15)x_{2}^{2} + (-8\nu^{2} - 12)x_{2}x_{3}^{4} + x_{2} - 4x_{3}^{4} + 1\)