32S13-4,4,8-g7

graph data
Name 32S13-4,4,8-g7
Type Hyperbolic
Degree 32
Genus 7
Galois orbit size for 32S13-4,4,8-g7-path1 1
Galois orbit size for 32S13-4,4,8-g7-path10 1
Galois orbit size for 32S13-4,4,8-g7-path11 1
Galois orbit size for 32S13-4,4,8-g7-path12 1
Galois orbit size for 32S13-4,4,8-g7-path13 1
Galois orbit size for 32S13-4,4,8-g7-path14 1
Galois orbit size for 32S13-4,4,8-g7-path15 1
Galois orbit size for 32S13-4,4,8-g7-path16 1
Galois orbit size for 32S13-4,4,8-g7-path17 1
Galois orbit size for 32S13-4,4,8-g7-path2 1
Galois orbit size for 32S13-4,4,8-g7-path3 1
Galois orbit size for 32S13-4,4,8-g7-path4 1
Galois orbit size for 32S13-4,4,8-g7-path5 1
Galois orbit size for 32S13-4,4,8-g7-path6 1
Galois orbit size for 32S13-4,4,8-g7-path7 1
Galois orbit size for 32S13-4,4,8-g7-path8 1
Galois orbit size for 32S13-4,4,8-g7-path9 1
Passport size 1
Pointed size 1

Above

64S46-8,4,16-g19 64S15-8,8,8-g21 64S21-4,4,8-g13 64S46-4,8,16-g19

Below

16T12-2,4,8-g2 16T12-4,2,8-g2 16T8-4,4,4-g3

Belyi Curve 1: 32S13-4,4,8-g7-path1

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

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

Belyi Curve 1 Degree: 14

Belyi Curve 1 Naive Measure: 13

Belyi Map 1 Numerator:

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

Belyi Map 1 Denominator:

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

Belyi Curve 2: 32S13-4,4,8-g7-path10

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

Belyi Curve 2 Base Field: Rationals

Belyi Curve 2 Degree: 16

Belyi Curve 2 Naive Measure: 3

Belyi Map 2 Numerator:

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

Belyi Map 2 Denominator:

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

Belyi Curve 3: 32S13-4,4,8-g7-path11

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

Belyi Curve 3 Base Field: Rationals

Belyi Curve 3 Degree: 16

Belyi Curve 3 Naive Measure: 3

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

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

Belyi Curve 4: 32S13-4,4,8-g7-path12

\(21845/93826423914496x_{1}^{16} - 1/18890042405425158030336\nu x_{1}^{15}x_{3} + 5/3148340400904193005056x_{1}^{14}x_{3}^{2} + 5/168661092905581768128\nu x_{1}^{13}x_{3}^{3} - 5/12973930223506289856x_{1}^{12}x_{3}^{4} - 1/270290212989714372\nu x_{1}^{11}x_{3}^{5} + 11/405435319484571558x_{1}^{10}x_{3}^{6} + 220/1419023618196000453\nu x_{1}^{9}x_{3}^{7} - 110/157669290910666717x_{1}^{8}x_{3}^{8} - 3520/1419023618196000453\nu x_{1}^{7}x_{3}^{9} + 1408/202717659742285779x_{1}^{6}x_{3}^{10} + 1024/67572553247428593\nu x_{1}^{5}x_{3}^{11} - 5120/202717659742285779x_{1}^{4}x_{3}^{12} - 81920/2635329576649715127\nu x_{1}^{3}x_{3}^{13} + 163840/6149102345516001963x_{1}^{2}x_{3}^{14} + 262144/18447307036548005889\nu x_{1}x_{3}^{15} + x_{2}^{2} - 65536/18447307036548005889x_{3}^{16}\)

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

Belyi Curve 4 Degree: 16

Belyi Curve 4 Naive Measure: 22269841419212633690445

Belyi Map 4 Numerator:

\(268435472/16777215\nu x_{1}^{15} - 250539808/1118481x_{1}^{14} - 858993920/479349\nu x_{1}^{13} + 330382592/36873x_{1}^{12} + 1762045952/61455\nu x_{1}^{11} - 10572406784/184365x_{1}^{10} - 16917004288/258111\nu x_{1}^{9} + 940474368/28679x_{1}^{8} + 23068672/258111\nu x_{1}^{7} - 46137344/184365x_{1}^{6} - 33554432/61455\nu x_{1}^{5} + 33554432/36873x_{1}^{4} + 536870912/479349\nu x_{1}^{3} - 1073741824/1118481x_{1}^{2} - 8589934592/16777215\nu x_{1} - 1568021285029272747947/87040x_{2}^{2} + 2147483648/16777215\)

Belyi Map 4 Denominator:

\(9007199254741024/281474976710655\nu x_{1}^{15} - 8406719304425024/18764998447377x_{1}^{14} - 28823037615172096/8042142191733\nu x_{1}^{13} + 11085783698143744/618626322441x_{1}^{12} + 59124179723444224/1031043870735\nu x_{1}^{11} - 354745078340927488/3093131612205x_{1}^{10} - 567592125347790848/4330384257087\nu x_{1}^{9} + 31532895853936640/481153806343x_{1}^{8} + 46137344/4330384257087\nu x_{1}^{7} - 92274688/3093131612205x_{1}^{6} - 67108864/1031043870735\nu x_{1}^{5} + 67108864/618626322441x_{1}^{4} + 1073741824/8042142191733\nu x_{1}^{3} - 2147483648/18764998447377x_{1}^{2} - 17179869184/281474976710655\nu x_{1} - 93826423914496/21845x_{2}^{2} + 4294967296/281474976710655\)

Belyi Curve 5: 32S13-4,4,8-g7-path13

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

Belyi Curve 5 Base Field: Rationals

Belyi Curve 5 Degree: 16

Belyi Curve 5 Naive Measure: 3

Belyi Map 5 Numerator:

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

Belyi Map 5 Denominator:

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

Belyi Curve 6: 32S13-4,4,8-g7-path14

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 16

Belyi Curve 6 Naive Measure: 3

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

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

Belyi Curve 7: 32S13-4,4,8-g7-path2

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

Belyi Curve 7 Base Field: \(\nu^{8}+16\), discriminant [ <2, 52> ]

Belyi Curve 7 Degree: 14

Belyi Curve 7 Naive Measure: 19

Belyi Map 7 Numerator:

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

Belyi Map 7 Denominator:

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

Belyi Curve 8: 32S13-4,4,8-g7-path3

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

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

Belyi Curve 8 Degree: 14

Belyi Curve 8 Naive Measure: 13

Belyi Map 8 Numerator:

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

Belyi Map 8 Denominator:

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

Belyi Curve 9: 32S13-4,4,8-g7-path4

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

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

Belyi Curve 9 Degree: 14

Belyi Curve 9 Naive Measure: 13

Belyi Map 9 Numerator:

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

Belyi Map 9 Denominator:

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

Belyi Curve 10: 32S13-4,4,8-g7-path5

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

Belyi Curve 10 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 10 Degree: 14

Belyi Curve 10 Naive Measure: 13

Belyi Map 10 Numerator:

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

Belyi Map 10 Denominator:

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

Belyi Curve 11: 32S13-4,4,8-g7-path6

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

Belyi Curve 11 Base Field: \(\nu^{8}+16\), discriminant [ <2, 52> ]

Belyi Curve 11 Degree: 14

Belyi Curve 11 Naive Measure: 19

Belyi Map 11 Numerator:

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

Belyi Map 11 Denominator:

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

Belyi Curve 12: 32S13-4,4,8-g7-path7

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

Belyi Curve 12 Base Field: Rationals

Belyi Curve 12 Degree: 16

Belyi Curve 12 Naive Measure: 3

Belyi Map 12 Numerator:

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

Belyi Map 12 Denominator:

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

Belyi Curve 13: 32S13-4,4,8-g7-path9

\(256/4294967295x_{1}^{16} - 16384/18446744065119617025\nu x_{1}^{15}x_{3} + 32768/1229782937674641135x_{1}^{14}x_{3}^{2} + 1835008/3689348813023923405\nu x_{1}^{13}x_{3}^{3} - 23855104/3689348813023923405x_{1}^{12}x_{3}^{4} - 381681664/6148914688373205675\nu x_{1}^{11}x_{3}^{5} + 8396996608/18446744065119617025x_{1}^{10}x_{3}^{6} + 9596567552/3689348813023923405\nu x_{1}^{9}x_{3}^{7} - 4798283776/409927645891547045x_{1}^{8}x_{3}^{8} - 153545080832/3689348813023923405\nu x_{1}^{7}x_{3}^{9} + 2149631131648/18446744065119617025x_{1}^{6}x_{3}^{10} + 1563368095744/6148914688373205675\nu x_{1}^{5}x_{3}^{11} - 1563368095744/3689348813023923405x_{1}^{4}x_{3}^{12} - 1924145348608/3689348813023923405\nu x_{1}^{3}x_{3}^{13} + 549755813888/1229782937674641135x_{1}^{2}x_{3}^{14} + 4398046511104/18446744065119617025\nu x_{1}x_{3}^{15} + x_{2}^{2} - 1099511627776/18446744065119617025x_{3}^{16}\)

Belyi Curve 13 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 13 Degree: 16

Belyi Curve 13 Naive Measure: 129537136106023833516

Belyi Map 13 Numerator:

\(1048560/65537\nu x_{1}^{15} - 14679328/65537x_{1}^{14} - 117424384/65537\nu x_{1}^{13} + 586978560/65537x_{1}^{12} + 1876840448/65537\nu x_{1}^{11} - 3741753344/65537x_{1}^{10} - 4201316352/65537\nu x_{1}^{9} + 1725792256/65537x_{1}^{8} - 1499463680/65537\nu x_{1}^{7} + 4198498304/65537x_{1}^{6} + 9160359936/65537\nu x_{1}^{5} - 15267266560/65537x_{1}^{4} - 18790481920/65537\nu x_{1}^{3} + 16106127360/65537x_{1}^{2} + 8589934592/65537\nu x_{1} + 18446181123756261375/67108864x_{2}^{2} - 2147483648/65537\)

Belyi Map 13 Denominator:

\(137438953504/4294967295\nu x_{1}^{15} - 128276356672/286331153x_{1}^{14} - 3078632564224/858993459\nu x_{1}^{13} + 15393162878464/858993459x_{1}^{12} + 82096869679104/1431655765\nu x_{1}^{11} - 492581241929728/4294967295x_{1}^{10} - 112590028144640/858993459\nu x_{1}^{9} + 18765054672896/286331153x_{1}^{8} + 599785472/858993459\nu x_{1}^{7} - 8396996608/4294967295x_{1}^{6} - 6106906624/1431655765\nu x_{1}^{5} + 6106906624/858993459x_{1}^{4} + 7516192768/858993459\nu x_{1}^{3} - 2147483648/286331153x_{1}^{2} - 17179869184/4294967295\nu x_{1} - 4294967295/256x_{2}^{2} + 4294967296/4294967295\)