32S13-4,8,4-g7

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

Above

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

Below

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

Belyi Curve 1: 32S13-4,8,4-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: 32S13-4,8,4-g7-path10

\(128/1099511627775\nu x_{1}^{16} + 8192/1208925819612430151450625x_{1}^{15}x_{3} + 16384/80595054640828676763375\nu x_{1}^{14}x_{3}^{2} - 917504/241785163922486030290125x_{1}^{13}x_{3}^{3} - 11927552/241785163922486030290125\nu x_{1}^{12}x_{3}^{4} + 190840832/402975273204143383816875x_{1}^{11}x_{3}^{5} + 381681664/109902347237493650131875\nu x_{1}^{10}x_{3}^{6} - 436207616/21980469447498730026375x_{1}^{9}x_{3}^{7} - 218103808/2442274383055414447375\nu x_{1}^{8}x_{3}^{8} + 6979321856/21980469447498730026375x_{1}^{7}x_{3}^{9} + 97710505984/109902347237493650131875\nu x_{1}^{6}x_{3}^{10} - 781684047872/402975273204143383816875x_{1}^{5}x_{3}^{11} - 781684047872/241785163922486030290125\nu x_{1}^{4}x_{3}^{12} + 962072674304/241785163922486030290125x_{1}^{3}x_{3}^{13} + 274877906944/80595054640828676763375\nu x_{1}^{2}x_{3}^{14} - 2199023255552/1208925819612430151450625x_{1}x_{3}^{15} + x_{2}^{2} - 549755813888/1208925819612430151450625\nu x_{3}^{16}\)

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

Belyi Curve 2 Degree: 16

Belyi Curve 2 Naive Measure: 5827266677971318383064526

Belyi Map 2 Numerator:

\(70368811286528/1152918206075109375\nu x_{1}^{15} - 65677700366336/76861213738340625x_{1}^{14} - 1576273398726656/230583641215021875\nu x_{1}^{13} + 7881487252717568/230583641215021875x_{1}^{12} + 4670742574661632/42700674299078125\nu x_{1}^{11} - 22933647891890176/104810746006828125x_{1}^{10} - 5247694121467904/20962149201365625\nu x_{1}^{9} + 884144787685376/6987383067121875x_{1}^{8} + 114349209288704/20962149201365625\nu x_{1}^{7} - 1600888930041856/104810746006828125x_{1}^{6} - 12807111440334848/384306068691703125\nu x_{1}^{5} + 12807111440334848/230583641215021875x_{1}^{4} + 15762598695796736/230583641215021875\nu x_{1}^{3} - 4503599627370496/76861213738340625x_{1}^{2} - 36028797018963968/1152918206075109375\nu x_{1} + 1152924803144876033/134217600\nu x_{2}^{2} + 9007199254740992/1152918206075109375\)

Belyi Map 2 Denominator:

\(-67108864/1099511627775\nu x_{1}^{15} + 985159331479552/1152920405094170625x_{1}^{14} + 7881231702163456/1152920405094170625\nu x_{1}^{13} - 2627037147693056/76861360339611375x_{1}^{12} - 126091529616883712/1152920405094170625\nu x_{1}^{11} + 252133031454703616/1152920405094170625x_{1}^{10} + 95945858540896256/384306801698056875\nu x_{1}^{9} - 142346486183624704/1152920405094170625x_{1}^{8} + 114349209288704/20962189183530375\nu x_{1}^{7} - 1600888930041856/104810945917651875x_{1}^{6} - 12807111440334848/384306801698056875\nu x_{1}^{5} + 12807111440334848/230584081018834125x_{1}^{4} + 15762598695796736/230584081018834125\nu x_{1}^{3} - 4503599627370496/76861360339611375x_{1}^{2} - 36028797018963968/1152920405094170625\nu x_{1} + 1099511627775/128\nu x_{2}^{2} + 9007199254740992/1152920405094170625\)

Belyi Curve 3: 32S13-4,8,4-g7-path14

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

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:

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

Belyi Curve 4: 32S13-4,8,4-g7-path16

\(-21845/375305695657984x_{1}^{16} + 1/75560169621700632121344\nu x_{1}^{15}x_{3} - 5/12593361603616772020224x_{1}^{14}x_{3}^{2} - 5/674644371622327072512\nu x_{1}^{13}x_{3}^{3} + 5/51895720894025159424x_{1}^{12}x_{3}^{4} + 1/1081160851958857488\nu x_{1}^{11}x_{3}^{5} - 11/1621741277938286232x_{1}^{10}x_{3}^{6} - 55/1419023618196000453\nu x_{1}^{9}x_{3}^{7} + 55/315338581821333434x_{1}^{8}x_{3}^{8} + 880/1419023618196000453\nu x_{1}^{7}x_{3}^{9} - 352/202717659742285779x_{1}^{6}x_{3}^{10} - 256/67572553247428593\nu x_{1}^{5}x_{3}^{11} + 1280/202717659742285779x_{1}^{4}x_{3}^{12} + 20480/2635329576649715127\nu x_{1}^{3}x_{3}^{13} - 40960/6149102345516001963x_{1}^{2}x_{3}^{14} - 65536/18447307036548005889\nu x_{1}x_{3}^{15} + x_{2}^{2} + 16384/18447307036548005889x_{3}^{16}\)

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

Belyi Curve 4 Degree: 16

Belyi Curve 4 Naive Measure: 88932080034955556238568

Belyi Map 4 Numerator:

\(18014399583223808/4722365638444765413375\nu x_{1}^{15} - 16813441901658112/314824375896317694225x_{1}^{14} - 11529221230821376/26984946505398659505\nu x_{1}^{13} + 22171600434495488/10378825578999484425x_{1}^{12} + 118248902154518528/17298042631665807375\nu x_{1}^{11} - 709502209020133376/51894127894997422125x_{1}^{10} - 1135280940050808832/72651779052996390975\nu x_{1}^{9} + 2337561720651776/298978514621384325x_{1}^{8} + 1548112371908608/72651779052996390975\nu x_{1}^{7} - 3096224743817216/51894127894997422125x_{1}^{6} - 2251799813685248/17298042631665807375\nu x_{1}^{5} + 2251799813685248/10378825578999484425x_{1}^{4} + 36028797018963968/134924732526993297525\nu x_{1}^{3} - 72057594037927936/314824375896317694225x_{1}^{2} - 576460752303423488/4722365638444765413375\nu x_{1} + 18742755539770811875328/1090938405375x_{2}^{2} + 144115188075855872/4722365638444765413375\)

Belyi Map 4 Denominator:

\(-1073741824/281474976710655\nu x_{1}^{15} + 252201529740623872/4722366201394651725825x_{1}^{14} + 288230221532889088/674623743056378817975\nu x_{1}^{13} - 96076648884994048/44974916203758587865x_{1}^{12} - 4611664852828553216/674623743056378817975\nu x_{1}^{11} + 9223215356447817728/674623743056378817975x_{1}^{10} + 29524086229041152/1889702361502461675\nu x_{1}^{9} - 36865188917143404544/4722366201394651725825x_{1}^{8} + 1548112371908608/72651787713763872705\nu x_{1}^{7} - 3096224743817216/51894134081259909075x_{1}^{6} - 2251799813685248/17298044693753303025\nu x_{1}^{5} + 2251799813685248/10378826816251981815x_{1}^{4} + 36028797018963968/134924748611275763595\nu x_{1}^{3} - 72057594037927936/314824413426310115055x_{1}^{2} - 576460752303423488/4722366201394651725825\nu x_{1} + 375305695657984/21845x_{2}^{2} + 144115188075855872/4722366201394651725825\)

Belyi Curve 5: 32S13-4,8,4-g7-path17

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

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:

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

Belyi Curve 6: 32S13-4,8,4-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 6 Base Field: \(\nu^{8}+16\), discriminant [ <2, 52> ]

Belyi Curve 6 Degree: 14

Belyi Curve 6 Naive Measure: 19

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

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

Belyi Curve 7: 32S13-4,8,4-g7-path3

\(-144x_{1}^{3} - 576x_{1}^{2} - 252x_{1}x_{3}^{4} - 1080x_{1}x_{3}^{2} - 684x_{1} + x_{2}^{2}x_{3}^{4} - 2x_{2}^{2}x_{3}^{2} + x_{2}^{2} - 108x_{3}^{4} - 288x_{3}^{2} + 396\)
\(-36x_{1}^{3} - 72x_{1}^{2} + x_{1}x_{2}^{2}x_{3}^{2} - x_{1}x_{2}^{2} - 108x_{1}x_{3}^{2} - 72x_{1} - 2x_{2}^{2}x_{3}^{2} - 2x_{2}^{2} - 36x_{3}^{2} + 36\)
\(x_{1}^{4} + 2x_{1}^{3} + 5x_{1}^{2} + 7x_{1}x_{3}^{2} + 5x_{1} - 1/36x_{2}^{2}x_{3}^{2} + 1/36x_{2}^{2} + 3x_{3}^{2} - 3\)
\(x_{1}^{2}x_{3}^{2} - x_{1}^{2} - 2x_{1}x_{3}^{2} - 2x_{1} - x_{3}^{2} + 1\)

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

Belyi Curve 7 Degree: 14

Belyi Curve 7 Naive Measure: 4004

Belyi Map 7 Numerator:

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

Belyi Map 7 Denominator:

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

Belyi Curve 8: 32S13-4,8,4-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 8 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 8 Degree: 14

Belyi Curve 8 Naive Measure: 13

Belyi Map 8 Numerator:

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

Belyi Map 8 Denominator:

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

Belyi Curve 9: 32S13-4,8,4-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 9 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 9 Degree: 14

Belyi Curve 9 Naive Measure: 13

Belyi Map 9 Numerator:

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

Belyi Map 9 Denominator:

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

Belyi Curve 10: 32S13-4,8,4-g7-path6

\(x_{1}^{5} + x_{1} + 8/9x_{2}^{2}\)
\(x_{1}^{2}x_{3}^{2} - x_{1}^{2} + x_{3}^{2} + 1\)
\(-9/4x_{1}^{3} + 9/4x_{1}x_{3}^{2} + x_{2}^{2}x_{3}^{2} - x_{2}^{2}\)

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

Belyi Curve 10 Degree: 14

Belyi Curve 10 Naive Measure: 35

Belyi Map 10 Numerator:

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

Belyi Map 10 Denominator:

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

Belyi Curve 11: 32S13-4,8,4-g7-path7

\(x_{1}^{16} - 24x_{1}^{8}x_{3}^{8} + x_{2}^{2} + 16x_{3}^{16}\)

Belyi Curve 11 Base Field: Rationals

Belyi Curve 11 Degree: 16

Belyi Curve 11 Naive Measure: 42

Belyi Map 11 Numerator:

\(-32x_{1}^{8}x_{2}^{2} - 896x_{1}^{8}x_{2} - 6144x_{1}^{8} + x_{2}^{4} + 32x_{2}^{3} + 256x_{2}^{2} + 512x_{2} + 4096\)

Belyi Map 11 Denominator:

\(-32x_{1}^{8}x_{2}^{2} + 6144x_{1}^{8} + x_{2}^{4} - 256x_{2}^{2} - 4096\)

Belyi Curve 12: 32S13-4,8,4-g7-path8

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

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

Belyi Curve 12 Degree: 16

Belyi Curve 12 Naive Measure: 22269841419212633690445

Belyi Map 12 Numerator:

\(18014399583223808/4722365638444765413375\nu x_{1}^{15} - 16813441901658112/314824375896317694225x_{1}^{14} - 11529221230821376/26984946505398659505\nu x_{1}^{13} + 22171600434495488/10378825578999484425x_{1}^{12} + 118248902154518528/17298042631665807375\nu x_{1}^{11} - 709502209020133376/51894127894997422125x_{1}^{10} - 1135280940050808832/72651779052996390975\nu x_{1}^{9} + 2337561720651776/298978514621384325x_{1}^{8} + 1548112371908608/72651779052996390975\nu x_{1}^{7} - 3096224743817216/51894127894997422125x_{1}^{6} - 2251799813685248/17298042631665807375\nu x_{1}^{5} + 2251799813685248/10378825578999484425x_{1}^{4} + 36028797018963968/134924732526993297525\nu x_{1}^{3} - 72057594037927936/314824375896317694225x_{1}^{2} - 576460752303423488/4722365638444765413375\nu x_{1} - 4685688884942702968832/1090938405375x_{2}^{2} + 144115188075855872/4722365638444765413375\)

Belyi Map 12 Denominator:

\(-1073741824/281474976710655\nu x_{1}^{15} + 252201529740623872/4722366201394651725825x_{1}^{14} + 288230221532889088/674623743056378817975\nu x_{1}^{13} - 96076648884994048/44974916203758587865x_{1}^{12} - 4611664852828553216/674623743056378817975\nu x_{1}^{11} + 9223215356447817728/674623743056378817975x_{1}^{10} + 29524086229041152/1889702361502461675\nu x_{1}^{9} - 36865188917143404544/4722366201394651725825x_{1}^{8} + 1548112371908608/72651787713763872705\nu x_{1}^{7} - 3096224743817216/51894134081259909075x_{1}^{6} - 2251799813685248/17298044693753303025\nu x_{1}^{5} + 2251799813685248/10378826816251981815x_{1}^{4} + 36028797018963968/134924748611275763595\nu x_{1}^{3} - 72057594037927936/314824413426310115055x_{1}^{2} - 576460752303423488/4722366201394651725825\nu x_{1} - 93826423914496/21845x_{2}^{2} + 144115188075855872/4722366201394651725825\)

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

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

Belyi Curve 13 Base Field: Rationals

Belyi Curve 13 Degree: 16

Belyi Curve 13 Naive Measure: 3

Belyi Map 13 Numerator:

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

Belyi Map 13 Denominator:

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