32S8-4,8,8-g9

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

Above

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

Below

16T10-2,4,4-g1

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

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

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

Belyi Curve 1 Degree: 13

Belyi Curve 1 Naive Measure: 37

Belyi Map 1 Numerator:

\(-1/8x_{2}^{7}x_{3}^{4} + 1/8\nu^{2}x_{2}^{7} + 1/4\nu^{2}x_{2}^{6}x_{3}^{4} + 1/4(\nu^{3} - \nu + 1)x_{2}^{6} + 1/8(4\nu^{3} - 4\nu + 7)x_{2}^{5}x_{3}^{4} + 1/8(-4\nu^{3} - 7\nu^{2} - 4\nu )x_{2}^{5} + 1/2(2\nu^{3} - 3\nu^{2} + 2\nu )x_{2}^{4}x_{3}^{4} + 1/4(-5\nu^{3} + 5\nu - 2)x_{2}^{4} + 1/8(-16\nu^{3} + 16\nu - 23)x_{2}^{3}x_{3}^{4} + 1/8(16\nu^{3} + 23\nu^{2} + 16\nu )x_{2}^{3} + 1/4(-12\nu^{3} + 17\nu^{2} - 12\nu )x_{2}^{2}x_{3}^{4} + 1/4(7\nu^{3} - 7\nu - 7)x_{2}^{2} + 1/8(12\nu^{3} - 12\nu + 17)x_{2}x_{3}^{4} + 1/8(-12\nu^{3} - 17\nu^{2} - 12\nu )x_{2} + 1/4(-3\nu^{3} + 3\nu + 4)\)

Belyi Map 1 Denominator:

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

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

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

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

Belyi Curve 2 Degree: 13

Belyi Curve 2 Naive Measure: 35

Belyi Map 2 Numerator:

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

Belyi Map 2 Denominator:

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

Belyi Curve 3: 32S8-4,8,8-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: 32S8-4,8,8-g9-path3

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

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

Belyi Curve 4 Degree: 13

Belyi Curve 4 Naive Measure: 43

Belyi Map 4 Numerator:

\(28552320/49567x_{1}x_{3}^{12} + 86467176/49567x_{1}x_{3}^{8} - 24194434/49567x_{1}x_{3}^{4} - 2600146/49567x_{1} - 19034880/49567\nu x_{2}^{7}x_{3}^{14} - 38609904/49567\nu x_{2}^{7}x_{3}^{10} + 12869400/49567\nu x_{2}^{7}x_{3}^{6} + 157041/14162\nu x_{2}^{7}x_{3}^{2} - 304558080/49567x_{2}^{6}x_{3}^{20} - 1074595584/49567x_{2}^{6}x_{3}^{16} - 730244736/49567x_{2}^{6}x_{3}^{12} + 43301952/7081x_{2}^{6}x_{3}^{8} + 18893705/49567x_{2}^{6}x_{3}^{4} + 499447/99134x_{2}^{6} + 2360325120/49567\nu x_{2}^{5}x_{3}^{18} + 689385408/7081\nu x_{2}^{5}x_{3}^{14} - 1461877712/49567\nu x_{2}^{5}x_{3}^{10} - 123390590/49567\nu x_{2}^{5}x_{3}^{6} - 7895795/99134\nu x_{2}^{5}x_{3}^{2} + 1827348480/49567x_{2}^{4}x_{3}^{20} + 3973039104/49567x_{2}^{4}x_{3}^{16} - 1280246304/49567x_{2}^{4}x_{3}^{12} - 15170052/7081x_{2}^{4}x_{3}^{8} - 109353/7081x_{2}^{4}x_{3}^{4} + 1696609/99134x_{2}^{4} + 456837120/49567\nu x_{2}^{3}x_{3}^{18} + 1212160896/49567\nu x_{2}^{3}x_{3}^{14} - 238900080/49567\nu x_{2}^{3}x_{3}^{10} - 64783646/49567\nu x_{2}^{3}x_{3}^{6} - 9301297/99134\nu x_{2}^{3}x_{3}^{2} - 304558080/49567x_{2}^{2}x_{3}^{20} - 120882432/7081x_{2}^{2}x_{3}^{16} - 157475328/49567x_{2}^{2}x_{3}^{12} + 237443652/49567x_{2}^{2}x_{3}^{8} - 180395/511x_{2}^{2}x_{3}^{4} - 4700845/99134x_{2}^{2} - 76139520/49567\nu x_{2}x_{3}^{18} - 230579136/49567\nu x_{2}x_{3}^{14} - 43874576/49567\nu x_{2}x_{3}^{10} + 33170296/49567\nu x_{2}x_{3}^{6} + 6500365/99134\nu x_{2}x_{3}^{2} - 38069760/49567x_{3}^{16} - 96254688/49567x_{3}^{12} + 8741416/49567x_{3}^{8} + 7431503/49567x_{3}^{4} + 1300073/99134\)

Belyi Map 4 Denominator:

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

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

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

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

Belyi Curve 5 Degree: 13

Belyi Curve 5 Naive Measure: 37

Belyi Map 5 Numerator:

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

Belyi Map 5 Denominator:

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

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

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