32S16-16,8,16-g13

graph data
Name 32S16-16,8,16-g13
Type Hyperbolic
Degree 32
Genus 13
Galois orbit size for 32S16-16,8,16-g13-path1 1
Galois orbit size for 32S16-16,8,16-g13-path10 1
Galois orbit size for 32S16-16,8,16-g13-path11 1
Galois orbit size for 32S16-16,8,16-g13-path12 1
Galois orbit size for 32S16-16,8,16-g13-path13 1
Galois orbit size for 32S16-16,8,16-g13-path14 1
Galois orbit size for 32S16-16,8,16-g13-path15 1
Galois orbit size for 32S16-16,8,16-g13-path16 1
Galois orbit size for 32S16-16,8,16-g13-path17 1
Galois orbit size for 32S16-16,8,16-g13-path18 1
Galois orbit size for 32S16-16,8,16-g13-path2 1
Galois orbit size for 32S16-16,8,16-g13-path3 1
Galois orbit size for 32S16-16,8,16-g13-path4 1
Galois orbit size for 32S16-16,8,16-g13-path5 1
Galois orbit size for 32S16-16,8,16-g13-path6 1
Galois orbit size for 32S16-16,8,16-g13-path7 1
Galois orbit size for 32S16-16,8,16-g13-path8 1
Galois orbit size for 32S16-16,8,16-g13-path9 1
Passport size 2
Pointed size 2

Above

64S26-16,8,16-g25 64S29-16,8,16-g25 64S51-32,16,32-g29 64S50-32,16,32-g29 64S44-16,8,16-g25

Below

16T1-16,8,16-g7 16T5-8,4,8-g5

Belyi Curve 1: 32S16-16,8,16-g13-path10

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

Belyi Curve 1 Base Field: Rationals

Belyi Curve 1 Degree: 21

Belyi Curve 1 Naive Measure: 72

Belyi Map 1 Numerator:

\(32768x_{3}^{4}x_{4}^{24} - 73728x_{3}^{4}x_{4}^{20} + 57344x_{3}^{4}x_{4}^{16} - 17920x_{3}^{4}x_{4}^{12} + 1856x_{3}^{4}x_{4}^{8} - 32x_{3}^{4}x_{4}^{4} - 65536x_{3}^{3}x_{4}^{26} + 196608x_{3}^{3}x_{4}^{22} - 221184x_{3}^{3}x_{4}^{18} + 114688x_{3}^{3}x_{4}^{14} - 26624x_{3}^{3}x_{4}^{10} + 2176x_{3}^{3}x_{4}^{6} - 32x_{3}^{3}x_{4}^{2} - 131072x_{3}^{2}x_{4}^{24} + 311296x_{3}^{2}x_{4}^{20} - 266240x_{3}^{2}x_{4}^{16} + 99328x_{3}^{2}x_{4}^{12} - 15360x_{3}^{2}x_{4}^{8} + 800x_{3}^{2}x_{4}^{4} - 8x_{3}^{2} - 98304x_{3}x_{4}^{22} + 212992x_{3}x_{4}^{18} - 157696x_{3}x_{4}^{14} + 46080x_{3}x_{4}^{10} - 4480x_{3}x_{4}^{6} + 96x_{3}x_{4}^{2} - 32768x_{4}^{20} + 65536x_{4}^{16} - 43008x_{4}^{12} + 10240x_{4}^{8} - 640x_{4}^{4} + 4\)

Belyi Map 1 Denominator:

\(128x_{3}^{4}x_{4}^{8} - 32x_{3}^{4}x_{4}^{4} - 256x_{3}^{3}x_{4}^{10} + 256x_{3}^{3}x_{4}^{6} - 32x_{3}^{3}x_{4}^{2} - 512x_{3}^{2}x_{4}^{8} + 192x_{3}^{2}x_{4}^{4} - 8x_{3}^{2} - 384x_{3}x_{4}^{6} + 64x_{3}x_{4}^{2} - 128x_{4}^{4} + 4\)

Belyi Curve 2: 32S16-16,8,16-g13-path13

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

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

Belyi Curve 2 Degree: 24

Belyi Curve 2 Naive Measure: 9

Belyi Map 2 Numerator:

\(x_{3}^{2} + x_{5}^{8}\)

Belyi Map 2 Denominator:

\(x_{5}^{8}\)

Belyi Curve 3: 32S16-16,8,16-g13-path14

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

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

Belyi Curve 3 Degree: 24

Belyi Curve 3 Naive Measure: 9

Belyi Map 3 Numerator:

\(x_{3}^{2} + x_{5}^{8}\)

Belyi Map 3 Denominator:

\(x_{5}^{8}\)

Belyi Curve 4: 32S16-16,8,16-g13-path15

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

Belyi Curve 4 Base Field: Rationals

Belyi Curve 4 Degree: 27

Belyi Curve 4 Naive Measure: 53

Belyi Map 4 Numerator:

\(x_{5}^{4}\)

Belyi Map 4 Denominator:

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

Belyi Curve 5: 32S16-16,8,16-g13-path16

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

Belyi Curve 5 Base Field: Rationals

Belyi Curve 5 Degree: 27

Belyi Curve 5 Naive Measure: 53

Belyi Map 5 Numerator:

\(x_{5}^{4}\)

Belyi Map 5 Denominator:

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

Belyi Curve 6: 32S16-16,8,16-g13-path17

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 22

Belyi Curve 6 Naive Measure: 15

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

\(4\)

Belyi Curve 7: 32S16-16,8,16-g13-path18

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

Belyi Curve 7 Base Field: Rationals

Belyi Curve 7 Degree: 22

Belyi Curve 7 Naive Measure: 15

Belyi Map 7 Numerator:

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

Belyi Map 7 Denominator:

\(4\)

Belyi Curve 8: 32S16-16,8,16-g13-path9

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

Belyi Curve 8 Base Field: Rationals

Belyi Curve 8 Degree: 21

Belyi Curve 8 Naive Measure: 72

Belyi Map 8 Numerator:

\(32768x_{3}^{4}x_{4}^{24} - 73728x_{3}^{4}x_{4}^{20} + 57344x_{3}^{4}x_{4}^{16} - 17920x_{3}^{4}x_{4}^{12} + 1856x_{3}^{4}x_{4}^{8} - 32x_{3}^{4}x_{4}^{4} - 65536x_{3}^{3}x_{4}^{26} + 196608x_{3}^{3}x_{4}^{22} - 221184x_{3}^{3}x_{4}^{18} + 114688x_{3}^{3}x_{4}^{14} - 26624x_{3}^{3}x_{4}^{10} + 2176x_{3}^{3}x_{4}^{6} - 32x_{3}^{3}x_{4}^{2} - 131072x_{3}^{2}x_{4}^{24} + 311296x_{3}^{2}x_{4}^{20} - 266240x_{3}^{2}x_{4}^{16} + 99328x_{3}^{2}x_{4}^{12} - 15360x_{3}^{2}x_{4}^{8} + 800x_{3}^{2}x_{4}^{4} - 8x_{3}^{2} - 98304x_{3}x_{4}^{22} + 212992x_{3}x_{4}^{18} - 157696x_{3}x_{4}^{14} + 46080x_{3}x_{4}^{10} - 4480x_{3}x_{4}^{6} + 96x_{3}x_{4}^{2} - 32768x_{4}^{20} + 65536x_{4}^{16} - 43008x_{4}^{12} + 10240x_{4}^{8} - 640x_{4}^{4} + 4\)

Belyi Map 8 Denominator:

\(128x_{3}^{4}x_{4}^{8} - 32x_{3}^{4}x_{4}^{4} - 256x_{3}^{3}x_{4}^{10} + 256x_{3}^{3}x_{4}^{6} - 32x_{3}^{3}x_{4}^{2} - 512x_{3}^{2}x_{4}^{8} + 192x_{3}^{2}x_{4}^{4} - 8x_{3}^{2} - 384x_{3}x_{4}^{6} + 64x_{3}x_{4}^{2} - 128x_{4}^{4} + 4\)