32S11-4,8,4-g7

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

Above

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

Below

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

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

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

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

Belyi Curve 1 Degree: 12

Belyi Curve 1 Naive Measure: 16

Belyi Map 1 Numerator:

\(-1/16x_{2}^{8} + 3/4x_{2}^{6} - 19/8x_{2}^{4} + 3/4x_{2}^{2} - 1/16\)

Belyi Map 1 Denominator:

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

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

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

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

Belyi Curve 2 Degree: 11

Belyi Curve 2 Naive Measure: 23

Belyi Map 2 Numerator:

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

Belyi Map 2 Denominator:

\(8x_{2}^{5}x_{3}^{12} + 60\nu x_{2}^{5}x_{3}^{8} - 102x_{2}^{5}x_{3}^{4} - 45\nu x_{2}^{5} + 24\nu x_{2}^{4}x_{3}^{12} - 108x_{2}^{4}x_{3}^{8} - 90\nu x_{2}^{4}x_{3}^{4} + 13x_{2}^{4} + 16x_{2}^{3}x_{3}^{12} + 120\nu x_{2}^{3}x_{3}^{8} - 204x_{2}^{3}x_{3}^{4} - 106\nu x_{2}^{3} + 48\nu x_{2}^{2}x_{3}^{12} - 216x_{2}^{2}x_{3}^{8} - 244\nu x_{2}^{2}x_{3}^{4} + 74x_{2}^{2} + 40x_{2}x_{3}^{12} + 172\nu x_{2}x_{3}^{8} - 126x_{2}x_{3}^{4} + 15\nu x_{2} - 8\nu x_{3}^{12} + 36x_{3}^{8} + 30\nu x_{3}^{4} + 1\)

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

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

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

Belyi Curve 3 Degree: 12

Belyi Curve 3 Naive Measure: 16

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

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

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

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

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

Belyi Curve 4 Degree: 16

Belyi Curve 4 Naive Measure: 50

Belyi Map 4 Numerator:

\(1/32(-\nu^{3} - 5\nu )x_{3}^{5}x_{4}^{8} + 1/16(-\nu^{3} - 5\nu )x_{3}^{5}x_{4}^{6} + 1/32(-\nu^{3} - 5\nu )x_{3}^{5}x_{4}^{4} + 1/8x_{3}^{4}x_{4}^{8} + 3/8x_{3}^{4}x_{4}^{6} + 7/16x_{3}^{4}x_{4}^{4} + 1/8x_{3}^{4}x_{4}^{2} + 1/32(\nu^{3} + 5\nu )x_{3}^{3}x_{4}^{8} + 1/8(\nu^{3} + 5\nu )x_{3}^{3}x_{4}^{6} + 1/8(\nu^{3} + 5\nu )x_{3}^{3}x_{4}^{4} + 1/32(-\nu^{3} - 5\nu )x_{3}^{3} - 1/16x_{3}^{2}x_{4}^{8} + 1/8x_{3}^{2}x_{4}^{6} - 1/8x_{3}^{2}x_{4}^{2} - 1/16x_{3}^{2} + 1/16(-\nu^{3} - 5\nu )x_{3}x_{4}^{6} + 1/32(-3\nu^{3} - 15\nu )x_{3}x_{4}^{4} + 1/4(\nu^{3} + 5\nu )x_{3}x_{4}^{2} + 1/32(\nu^{3} + 5\nu )x_{3} + 15/16x_{4}^{4} + 1/8\)

Belyi Map 4 Denominator:

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

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

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

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

Belyi Curve 5 Degree: 11

Belyi Curve 5 Naive Measure: 31

Belyi Map 5 Numerator:

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

Belyi Map 5 Denominator:

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

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

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 12

Belyi Curve 6 Naive Measure: 19

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

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

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

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

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

Belyi Curve 7 Degree: 11

Belyi Curve 7 Naive Measure: 22

Belyi Map 7 Numerator:

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

Belyi Map 7 Denominator:

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

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

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

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

Belyi Curve 8 Degree: 14

Belyi Curve 8 Naive Measure: 17

Belyi Map 8 Numerator:

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

Belyi Map 8 Denominator:

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

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

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

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

Belyi Curve 9 Degree: 12

Belyi Curve 9 Naive Measure: 16

Belyi Map 9 Numerator:

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

Belyi Map 9 Denominator:

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

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

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

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

Belyi Curve 10 Degree: 12

Belyi Curve 10 Naive Measure: 103

Belyi Map 10 Numerator:

\(136270082448/845185x_{1}x_{3}^{16} - 3867572052/845185x_{1}x_{3}^{14} + 63174592281/3380740x_{1}x_{3}^{12} + 174578343147/13522960x_{1}x_{3}^{10} - 19242770671/108183680x_{1}x_{3}^{8} - 11460116761/10818368x_{1}x_{3}^{6} - 22296549719/108183680x_{1}x_{3}^{4} + 161296562857/3461877760x_{1}x_{3}^{2} - 10014148055/5539004416x_{1} - x_{2}^{6}x_{3}^{4} - 1/2x_{2}^{6}x_{3}^{2} - 1/16x_{2}^{6} + 22711680408/845185x_{2}^{5}x_{3}^{16} + 7872284811/845185x_{2}^{5}x_{3}^{14} + 50550983023/13522960x_{2}^{5}x_{3}^{12} + 105578523881/13522960x_{2}^{5}x_{3}^{10} + 76417098721/19669760x_{2}^{5}x_{3}^{8} + 120377908253/216367360x_{2}^{5}x_{3}^{6} - 23658539559/692375552x_{2}^{5}x_{3}^{4} - 40575296017/3461877760x_{2}^{5}x_{3}^{2} - 10047877987/11078008832x_{2}^{5} + 11355840204/169037x_{2}^{4}x_{3}^{16} + 45039344157/1690370x_{2}^{4}x_{3}^{14} + 287382149789/13522960x_{2}^{4}x_{3}^{12} + 25057387299/27045920x_{2}^{4}x_{3}^{10} + 2840807898361/216367360x_{2}^{4}x_{3}^{8} + 1813936511383/432734720x_{2}^{4}x_{3}^{6} - 2137569931969/3461877760x_{2}^{4}x_{3}^{4} - 117273423031/6923755520x_{2}^{4}x_{3}^{2} + 34281022307/11078008832x_{2}^{4} + 136270082448/845185x_{2}^{3}x_{3}^{16} + 37268550273/169037x_{2}^{3}x_{3}^{14} - 476113833399/6761480x_{2}^{3}x_{3}^{12} + 222387388115/2704592x_{2}^{3}x_{3}^{10} + 1192767438701/54091840x_{2}^{3}x_{3}^{8} - 1491145077677/216367360x_{2}^{3}x_{3}^{6} + 839473790093/1730938880x_{2}^{3}x_{3}^{4} - 10014148055/692375552x_{2}^{3}x_{3}^{2} + 272540164896/845185x_{2}^{2}x_{3}^{16} - 135488346399/845185x_{2}^{2}x_{3}^{14} + 58386644181/614680x_{2}^{2}x_{3}^{12} - 411190004421/27045920x_{2}^{2}x_{3}^{10} - 1404956738813/108183680x_{2}^{2}x_{3}^{8} + 283758257169/108183680x_{2}^{2}x_{3}^{6} - 1065623172719/1730938880x_{2}^{2}x_{3}^{4} + 462588362919/6923755520x_{2}^{2}x_{3}^{2} - 10360335831/5539004416x_{2}^{2} + 22711680408/845185x_{2}x_{3}^{16} + 4586971416/76835x_{2}x_{3}^{14} + 19857282559/13522960x_{2}x_{3}^{12} + 42225888475/2704592x_{2}x_{3}^{10} + 343480614215/43273472x_{2}x_{3}^{8} + 32627692561/54091840x_{2}x_{3}^{6} + 329009914229/3461877760x_{2}x_{3}^{4} - 133961670707/3461877760x_{2}x_{3}^{2} - 10047877987/11078008832x_{2} - 11355840204/845185x_{3}^{16} - 41939805423/1690370x_{3}^{14} + 32925251173/2704592x_{3}^{12} - 5961877021/676148x_{3}^{10} - 259611326783/216367360x_{3}^{8} + 485467975673/432734720x_{3}^{6} - 310114062861/3461877760x_{3}^{4} + 13260495627/3461877760x_{3}^{2} - 10031013021/11078008832\)

Belyi Map 10 Denominator:

\(4x_{2}^{4}\)

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

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

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:

\(1/16x_{2}^{8} + 3/4x_{2}^{6} + 19/8x_{2}^{4} + 3/4x_{2}^{2} + 1/16\)

Belyi Map 11 Denominator:

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

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

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

Belyi Curve 12 Base Field: Rationals

Belyi Curve 12 Degree: 12

Belyi Curve 12 Naive Measure: 104

Belyi Map 12 Numerator:

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

Belyi Map 12 Denominator:

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