32S11-8,4,4-g7

graph data
Name 32S11-8,4,4-g7
Type Hyperbolic
Degree 32
Genus 7
Galois orbit size for 32S11-8,4,4-g7-path1 1
Galois orbit size for 32S11-8,4,4-g7-path10 1
Galois orbit size for 32S11-8,4,4-g7-path11 1
Galois orbit size for 32S11-8,4,4-g7-path12 1
Galois orbit size for 32S11-8,4,4-g7-path13 1
Galois orbit size for 32S11-8,4,4-g7-path14 1
Galois orbit size for 32S11-8,4,4-g7-path15 1
Galois orbit size for 32S11-8,4,4-g7-path16 1
Galois orbit size for 32S11-8,4,4-g7-path17 1
Galois orbit size for 32S11-8,4,4-g7-path18 1
Galois orbit size for 32S11-8,4,4-g7-path19 1
Galois orbit size for 32S11-8,4,4-g7-path2 1
Galois orbit size for 32S11-8,4,4-g7-path20 1
Galois orbit size for 32S11-8,4,4-g7-path21 1
Galois orbit size for 32S11-8,4,4-g7-path22 1
Galois orbit size for 32S11-8,4,4-g7-path3 1
Galois orbit size for 32S11-8,4,4-g7-path4 1
Galois orbit size for 32S11-8,4,4-g7-path5 1
Galois orbit size for 32S11-8,4,4-g7-path6 1
Galois orbit size for 32S11-8,4,4-g7-path7 1
Galois orbit size for 32S11-8,4,4-g7-path8 1
Galois orbit size for 32S11-8,4,4-g7-path9 1
Passport size 2
Pointed size 2

Above

64S11-8,4,8-g17 64S10-8,4,8-g17 64S11-8,8,4-g17 64S10-8,8,4-g17 64S7-8,4,8-g17 64S6-8,4,8-g17 64S20-8,4,4-g13 64S9-8,4,4-g13 64S8-8,4,4-g13 64S7-8,8,4-g17 64S6-8,8,4-g17

Below

16T10-4,4,2-g1 16T10-4,2,4-g1

Belyi Curve 1: 32S11-8,4,4-g7-path10

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

Belyi Curve 1 Base Field: Rationals

Belyi Curve 1 Degree: 12

Belyi Curve 1 Naive Measure: 104

Belyi Map 1 Numerator:

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

Belyi Map 1 Denominator:

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

Belyi Curve 2: 32S11-8,4,4-g7-path11

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

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

Belyi Curve 2 Degree: 11

Belyi Curve 2 Naive Measure: 26

Belyi Map 2 Numerator:

\(-32x_{2}^{5}x_{3}^{4} + 80\nu x_{2}^{5} + 160\nu x_{2}^{4}x_{3}^{4} - 128x_{2}^{4} + 192x_{2}^{3}x_{3}^{4} + 32\nu x_{2}^{3} + 64\nu x_{2}^{2}x_{3}^{4} - 64x_{2}^{2} + 96x_{2}x_{3}^{4} + 16\nu x_{2} + 32\nu x_{3}^{4}\)

Belyi Map 2 Denominator:

\(8x_{2}^{5}x_{3}^{12} - 100\nu x_{2}^{5}x_{3}^{8} - 246x_{2}^{5}x_{3}^{4} + 83\nu x_{2}^{5} - 40\nu x_{2}^{4}x_{3}^{12} - 300x_{2}^{4}x_{3}^{8} + 102\nu x_{2}^{4}x_{3}^{4} - 139x_{2}^{4} - 48x_{2}^{3}x_{3}^{12} + 344\nu x_{2}^{3}x_{3}^{8} + 196x_{2}^{3}x_{3}^{4} + 30\nu x_{2}^{3} - 16\nu x_{2}^{2}x_{3}^{12} - 120x_{2}^{2}x_{3}^{8} + 28\nu x_{2}^{2}x_{3}^{4} - 78x_{2}^{2} - 24x_{2}x_{3}^{12} + 172\nu x_{2}x_{3}^{8} + 98x_{2}x_{3}^{4} + 23\nu x_{2} - 8\nu x_{3}^{12} - 60x_{3}^{8} + 46\nu x_{3}^{4} + 1\)

Belyi Curve 3: 32S11-8,4,4-g7-path13

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

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

Belyi Curve 3 Degree: 11

Belyi Curve 3 Naive Measure: 23

Belyi Map 3 Numerator:

\(-32x_{2}^{5}x_{3}^{4} - 48x_{2}^{5} - 96x_{2}^{4}x_{3}^{4} + 64x_{2}^{3}x_{3}^{4} + 96x_{2}^{3} + 192x_{2}^{2}x_{3}^{4} + 64x_{2}^{2} - 160x_{2}x_{3}^{4} + 16x_{2} + 32x_{3}^{4}\)

Belyi Map 3 Denominator:

\(8x_{2}^{5}x_{3}^{12} + 60x_{2}^{5}x_{3}^{8} + 70x_{2}^{5}x_{3}^{4} - 3x_{2}^{5} + 24x_{2}^{4}x_{3}^{12} + 108x_{2}^{4}x_{3}^{8} - 6x_{2}^{4}x_{3}^{4} + 13x_{2}^{4} - 16x_{2}^{3}x_{3}^{12} - 120x_{2}^{3}x_{3}^{8} - 140x_{2}^{3}x_{3}^{4} - 10x_{2}^{3} - 48x_{2}^{2}x_{3}^{12} - 216x_{2}^{2}x_{3}^{8} - 52x_{2}^{2}x_{3}^{4} - 10x_{2}^{2} + 40x_{2}x_{3}^{12} + 172x_{2}x_{3}^{8} - 34x_{2}x_{3}^{4} + x_{2} - 8x_{3}^{12} - 36x_{3}^{8} + 2x_{3}^{4} + 1\)

Belyi Curve 4: 32S11-8,4,4-g7-path14

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

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

Belyi Curve 4 Degree: 18

Belyi Curve 4 Naive Measure: 58

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

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

Belyi Curve 5: 32S11-8,4,4-g7-path19

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

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

Belyi Curve 5 Degree: 12

Belyi Curve 5 Naive Measure: 16

Belyi Map 5 Numerator:

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

Belyi Map 5 Denominator:

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

Belyi Curve 6: 32S11-8,4,4-g7-path2

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

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

Belyi Curve 6 Degree: 12

Belyi Curve 6 Naive Measure: 27

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

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

Belyi Curve 7: 32S11-8,4,4-g7-path20

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

Belyi Curve 7 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 7 Degree: 12

Belyi Curve 7 Naive Measure: 16

Belyi Map 7 Numerator:

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

Belyi Map 7 Denominator:

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

Belyi Curve 8: 32S11-8,4,4-g7-path21

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

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

Belyi Curve 8 Degree: 16

Belyi Curve 8 Naive Measure: 13

Belyi Map 8 Numerator:

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

Belyi Map 8 Denominator:

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

Belyi Curve 9: 32S11-8,4,4-g7-path22

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

Belyi Curve 9 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 9 Degree: 11

Belyi Curve 9 Naive Measure: 26

Belyi Map 9 Numerator:

\(-32x_{2}^{5}x_{3}^{4} - 80x_{2}^{5} - 160x_{2}^{4}x_{3}^{4} + 128x_{2}^{4} - 192x_{2}^{3}x_{3}^{4} + 32x_{2}^{3} + 64x_{2}^{2}x_{3}^{4} - 64x_{2}^{2} + 96x_{2}x_{3}^{4} - 16x_{2} - 32x_{3}^{4}\)

Belyi Map 9 Denominator:

\(8x_{2}^{5}x_{3}^{12} + 100x_{2}^{5}x_{3}^{8} + 214x_{2}^{5}x_{3}^{4} + 3x_{2}^{5} + 40x_{2}^{4}x_{3}^{12} + 300x_{2}^{4}x_{3}^{8} - 58x_{2}^{4}x_{3}^{4} - 11x_{2}^{4} + 48x_{2}^{3}x_{3}^{12} + 344x_{2}^{3}x_{3}^{8} + 4x_{2}^{3}x_{3}^{4} + 2x_{2}^{3} - 16x_{2}^{2}x_{3}^{12} - 120x_{2}^{2}x_{3}^{8} + 36x_{2}^{2}x_{3}^{4} + 14x_{2}^{2} - 24x_{2}x_{3}^{12} - 172x_{2}x_{3}^{8} - 2x_{2}x_{3}^{4} + 7x_{2} + 8x_{3}^{12} + 60x_{3}^{8} + 14x_{3}^{4} + 1\)

Belyi Curve 10: 32S11-8,4,4-g7-path3

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

Belyi Curve 10 Base Field: Rationals

Belyi Curve 10 Degree: 14

Belyi Curve 10 Naive Measure: 17

Belyi Map 10 Numerator:

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

Belyi Map 10 Denominator:

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

Belyi Curve 11: 32S11-8,4,4-g7-path8

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

Belyi Curve 11 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 11 Degree: 12

Belyi Curve 11 Naive Measure: 16

Belyi Map 11 Numerator:

\(1/16x_{2}^{8} + 1/4x_{2}^{6} + 3/8x_{2}^{4} + 1/4x_{2}^{2} + 1/16\)

Belyi Map 11 Denominator:

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

Belyi Curve 12: 32S11-8,4,4-g7-path9

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

Belyi Curve 12 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 12 Degree: 12

Belyi Curve 12 Naive Measure: 16

Belyi Map 12 Numerator:

\(1/16x_{2}^{8} + 1/4x_{2}^{6} + 3/8x_{2}^{4} + 1/4x_{2}^{2} + 1/16\)

Belyi Map 12 Denominator:

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