32S5-4,8,8-g9

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

Above

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

Below

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

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

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

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

Belyi Curve 1 Degree: 13

Belyi Curve 1 Naive Measure: 67

Belyi Map 1 Numerator:

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

Belyi Map 1 Denominator:

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

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

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

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

Belyi Curve 2 Degree: 13

Belyi Curve 2 Naive Measure: 55

Belyi Map 2 Numerator:

\(8x_{2}^{7}x_{3}^{4} + (-4\nu + 2)x_{2}^{7} + (-32\nu + 16)x_{2}^{6}x_{3}^{4} + (8\nu + 12)x_{2}^{6} + (-64\nu - 8)x_{2}^{5}x_{3}^{4} + (-12\nu - 2)x_{2}^{5} + (-64\nu - 32)x_{2}^{4}x_{3}^{4} + (16\nu + 40)x_{2}^{4} + (-64\nu - 8)x_{2}^{3}x_{3}^{4} + (-12\nu - 2)x_{2}^{3} + (-32\nu + 16)x_{2}^{2}x_{3}^{4} + (8\nu + 12)x_{2}^{2} + 8x_{2}x_{3}^{4} + (-4\nu + 2)x_{2}2\)

Belyi Map 2 Denominator:

\(8x_{2}^{7}x_{3}^{4} + (-4\nu + 2)x_{2}^{7} + (-32\nu + 16)x_{2}^{6}x_{3}^{4} + (8\nu - 4)x_{2}^{6} + (-64\nu - 8)x_{2}^{5}x_{3}^{4} + (-12\nu - 2)x_{2}^{5} + (-64\nu - 32)x_{2}^{4}x_{3}^{4} + (16\nu + 8)x_{2}^{4} + (-64\nu - 8)x_{2}^{3}x_{3}^{4} + (-12\nu - 2)x_{2}^{3} + (-32\nu + 16)x_{2}^{2}x_{3}^{4} + (8\nu - 4)x_{2}^{2} + 8x_{2}x_{3}^{4} + (-4\nu + 2)x_{2}2\)

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

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

Belyi Curve 3 Base Field: Rationals

Belyi Curve 3 Degree: 13

Belyi Curve 3 Naive Measure: 24

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

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

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

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

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

Belyi Curve 4 Degree: 13

Belyi Curve 4 Naive Measure: 37

Belyi Map 4 Numerator:

\(-446130/49567x_{1}x_{3}^{12} - 10808397/99134x_{1}x_{3}^{8} + 12097217/99134x_{1}x_{3}^{4} + 2600146/49567x_{1} - 148710/49567x_{2}^{7}x_{3}^{14} - 2413119/99134x_{2}^{7}x_{3}^{10} + 1608675/49567x_{2}^{7}x_{3}^{6} + 157041/28324x_{2}^{7}x_{3}^{2} + 297420/49567x_{2}^{6}x_{3}^{20} + 4197639/49567x_{2}^{6}x_{3}^{16} + 11410074/49567x_{2}^{6}x_{3}^{12} - 2706372/7081x_{2}^{6}x_{3}^{8} - 18893705/198268x_{2}^{6}x_{3}^{4} - 499447/99134x_{2}^{6} - 4610010/49567x_{2}^{5}x_{3}^{18} - 10771647/14162x_{2}^{5}x_{3}^{14} + 91367357/99134x_{2}^{5}x_{3}^{10} + 61695295/198268x_{2}^{5}x_{3}^{6} + 7895795/198268x_{2}^{5}x_{3}^{2} + 1784520/49567x_{2}^{4}x_{3}^{20} + 15519684/49567x_{2}^{4}x_{3}^{16} - 40007697/99134x_{2}^{4}x_{3}^{12} - 3792513/28324x_{2}^{4}x_{3}^{8} - 109353/28324x_{2}^{4}x_{3}^{4} + 1696609/99134x_{2}^{4} + 892260/49567x_{2}^{3}x_{3}^{18} + 9470007/49567x_{2}^{3}x_{3}^{14} - 14931255/99134x_{2}^{3}x_{3}^{10} - 32391823/198268x_{2}^{3}x_{3}^{6} - 9301297/198268x_{2}^{3}x_{3}^{2} + 297420/49567x_{2}^{2}x_{3}^{20} + 472197/7081x_{2}^{2}x_{3}^{16} + 2460552/49567x_{2}^{2}x_{3}^{12} - 59360913/198268x_{2}^{2}x_{3}^{8} + 180395/2044x_{2}^{2}x_{3}^{4} + 4700845/99134x_{2}^{2} + 148710/49567x_{2}x_{3}^{18} + 3602799/99134x_{2}x_{3}^{14} + 2742161/99134x_{2}x_{3}^{10} - 4146287/49567x_{2}x_{3}^{6} - 6500365/198268x_{2}x_{3}^{2} - 148710/49567x_{3}^{16} - 3007959/99134x_{3}^{12} + 1092677/99134x_{3}^{8} + 7431503/198268x_{3}^{4} + 1300073/99134\)

Belyi Map 4 Denominator:

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

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

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

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

Belyi Curve 5 Degree: 13

Belyi Curve 5 Naive Measure: 49

Belyi Map 5 Numerator:

\(1/2x_{2}^{7}x_{3}^{4} + 1/16(3\nu + 2)x_{2}^{7} + 1/2(3\nu + 2)x_{2}^{6}x_{3}^{4} + 1/16(-6\nu + 9)x_{2}^{6} + 1/2(6\nu + 3)x_{2}^{5}x_{3}^{4} + 1/16(9\nu + 4)x_{2}^{5} + 1/2(9\nu + 4)x_{2}^{4}x_{3}^{4} + 1/16(-12\nu + 31)x_{2}^{4} + 1/2(12\nu + 7)x_{2}^{3}x_{3}^{4} + 1/16(5\nu + 10)x_{2}^{3} + 1/2(5\nu + 10)x_{2}^{2}x_{3}^{4} + 1/16(2\nu + 7)x_{2}^{2} + 1/2(-2\nu + 5)x_{2}x_{3}^{4} - 1/16\nu x_{2} - 1/2\nu x_{3}^{4} + 1/16\)

Belyi Map 5 Denominator:

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

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

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 13

Belyi Curve 6 Naive Measure: 24

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

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