16T6-2,8,8-g3

graph data
Name 16T6-2,8,8-g3
Type Hyperbolic
Degree 16
Genus 3
Galois orbit size for 16T6-2,8,8-g3-path1 1
Galois orbit size for 16T6-2,8,8-g3-path2 1
Galois orbit size for 16T6-2,8,8-g3-path3 1
Galois orbit size for 16T6-2,8,8-g3-path4 1
Galois orbit size for 16T6-2,8,8-g3-path5 1
Passport size 1
Pointed size 1

Above

32S5-2,8,8-g5 32S12-4,8,8-g9 32S4-4,8,8-g9

Below

8T2-2,4,4-g1

Belyi Curve 1: 16T6-2,8,8-g3-path1

\(-4/4097x_{1}^{8} - 65536/16785409\nu^{2}x_{1}^{7}x_{3} + 114688/16785409x_{1}^{6}x_{3}^{2} + 114688/16785409\nu^{2}x_{1}^{5}x_{3}^{3} - 71680/16785409x_{1}^{4}x_{3}^{4} - 28672/16785409\nu^{2}x_{1}^{3}x_{3}^{5} + 7168/16785409x_{1}^{2}x_{3}^{6} + 1024/16785409\nu^{2}x_{1}x_{3}^{7} + x_{2}^{2} - 64/16785409x_{3}^{8}\)

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

Belyi Curve 1 Degree: 8

Belyi Curve 1 Naive Measure: 134287370

Belyi Map 1 Numerator:

\(16785409/16384x_{2}^{2}\)

Belyi Map 1 Denominator:

\(4/4097\nu^{2}x_{1}^{7} - 7/4097x_{1}^{6} - 7/4097\nu^{2}x_{1}^{5} + 35/32776x_{1}^{4} + 7/16388\nu^{2}x_{1}^{3} - 7/65552x_{1}^{2} - 1/65552\nu^{2}x_{1} + 4097/4x_{2}^{2} + 1/1048832\)

Belyi Curve 2: 16T6-2,8,8-g3-path3

\(-256/65537x_{1}^{8} + 8192/4295098369\nu x_{1}^{7}x_{3} + 114688/4295098369x_{1}^{6}x_{3}^{2} - 917504/4295098369\nu x_{1}^{5}x_{3}^{3} - 4587520/4295098369x_{1}^{4}x_{3}^{4} + 14680064/4295098369\nu x_{1}^{3}x_{3}^{5} + 29360128/4295098369x_{1}^{2}x_{3}^{6} - 33554432/4295098369\nu x_{1}x_{3}^{7} + x_{2}^{2} - 16777216/4295098369x_{3}^{8}\)

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

Belyi Curve 2 Degree: 8

Belyi Curve 2 Naive Measure: 34360852490

Belyi Map 2 Numerator:

\(4295098369/16777216x_{2}^{2}\)

Belyi Map 2 Denominator:

\(32/65537\nu x_{1}^{7} + 448/65537x_{1}^{6} - 3584/65537\nu x_{1}^{5} - 17920/65537x_{1}^{4} + 57344/65537\nu x_{1}^{3} + 114688/65537x_{1}^{2} - 131072/65537\nu x_{1} + 65537/256x_{2}^{2} - 65536/65537\)

Belyi Curve 3: 16T6-2,8,8-g3-path4

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

Belyi Curve 3 Base Field: Rationals

Belyi Curve 3 Degree: 8

Belyi Curve 3 Naive Measure: 6

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

\(16x_{1}^{8}\)

Belyi Curve 4: 16T6-2,8,8-g3-path5

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

Belyi Curve 4 Base Field: Rationals

Belyi Curve 4 Degree: 8

Belyi Curve 4 Naive Measure: 3

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

\(4\)