16T4-4,4,4-g3

graph data
Name 16T4-4,4,4-g3
Type Hyperbolic
Degree 16
Genus 3
Galois orbit size for 16T4-4,4,4-g3-path1 1
Galois orbit size for 16T4-4,4,4-g3-path10 1
Galois orbit size for 16T4-4,4,4-g3-path11 1
Galois orbit size for 16T4-4,4,4-g3-path12 1
Galois orbit size for 16T4-4,4,4-g3-path13 1
Galois orbit size for 16T4-4,4,4-g3-path14 1
Galois orbit size for 16T4-4,4,4-g3-path15 1
Galois orbit size for 16T4-4,4,4-g3-path2 1
Galois orbit size for 16T4-4,4,4-g3-path3 1
Galois orbit size for 16T4-4,4,4-g3-path4 1
Galois orbit size for 16T4-4,4,4-g3-path5 1
Galois orbit size for 16T4-4,4,4-g3-path6 1
Galois orbit size for 16T4-4,4,4-g3-path7 1
Galois orbit size for 16T4-4,4,4-g3-path8 1
Galois orbit size for 16T4-4,4,4-g3-path9 1
Passport size 1
Pointed size 1

Above

32S2-4,4,4-g5 32S4-8,8,4-g9 32S3-8,8,4-g9 32S4-4,8,8-g9 32S3-4,8,8-g9 32S4-8,4,8-g9 32S3-8,4,8-g9

Below

8T2-2,4,4-g1 8T2-4,2,4-g1 8T2-4,4,2-g1

Belyi Curve 1: 16T4-4,4,4-g3-path1

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

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

Belyi Curve 1 Degree: 8

Belyi Curve 1 Naive Measure: 3

Belyi Map 1 Numerator:

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

Belyi Map 1 Denominator:

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

Belyi Curve 2: 16T4-4,4,4-g3-path10

\(x_{1}^{8} - 24x_{1}^{4}x_{3}^{4} + x_{2}^{2} + 16x_{3}^{8}\)

Belyi Curve 2 Base Field: Rationals

Belyi Curve 2 Degree: 8

Belyi Curve 2 Naive Measure: 42

Belyi Map 2 Numerator:

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

Belyi Map 2 Denominator:

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

Belyi Curve 3: 16T4-4,4,4-g3-path11

\(-x_{1}^{7}x_{3} - 7x_{1}^{5}x_{3}^{3} - 7x_{1}^{3}x_{3}^{5} - x_{1}x_{3}^{7} + x_{2}^{2}\)

Belyi Curve 3 Base Field: Rationals

Belyi Curve 3 Degree: 7

Belyi Curve 3 Naive Measure: 17

Belyi Map 3 Numerator:

\(-16x_{1}^{6} - 32x_{1}^{4} - 16x_{1}^{2}\)

Belyi Map 3 Denominator:

\(-11x_{1}^{6} - x_{1}^{4} - 5x_{1}^{2} + x_{1}x_{2}^{2} + 1\)

Belyi Curve 4: 16T4-4,4,4-g3-path13

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

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

Belyi Curve 4 Degree: 8

Belyi Curve 4 Naive Measure: 3

Belyi Map 4 Numerator:

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

Belyi Map 4 Denominator:

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

Belyi Curve 5: 16T4-4,4,4-g3-path14

\(-x_{1}^{7}x_{3} - 7x_{1}^{5}x_{3}^{3} - 7x_{1}^{3}x_{3}^{5} - x_{1}x_{3}^{7} + x_{2}^{2}\)

Belyi Curve 5 Base Field: Rationals

Belyi Curve 5 Degree: 7

Belyi Curve 5 Naive Measure: 17

Belyi Map 5 Numerator:

\(5x_{1}^{6} + 31x_{1}^{4} + 11x_{1}^{2} + x_{1}x_{2}^{2} + 1\)

Belyi Map 5 Denominator:

\(-11x_{1}^{6} - x_{1}^{4} - 5x_{1}^{2} + x_{1}x_{2}^{2} + 1\)

Belyi Curve 6: 16T4-4,4,4-g3-path15

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

Belyi Curve 6 Base Field: Rationals

Belyi Curve 6 Degree: 8

Belyi Curve 6 Naive Measure: 3

Belyi Map 6 Numerator:

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

Belyi Map 6 Denominator:

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

Belyi Curve 7: 16T4-4,4,4-g3-path3

\(-1024/65535\nu x_{1}^{8} + 32768/4294836225x_{1}^{7}x_{3} - 458752/4294836225\nu x_{1}^{6}x_{3}^{2} - 3670016/4294836225x_{1}^{5}x_{3}^{3} + 3670016/858967245\nu x_{1}^{4}x_{3}^{4} + 58720256/4294836225x_{1}^{3}x_{3}^{5} - 117440512/4294836225\nu x_{1}^{2}x_{3}^{6} - 134217728/4294836225x_{1}x_{3}^{7} + x_{2}^{2} + 67108864/4294836225\nu x_{3}^{8}\)

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

Belyi Curve 7 Degree: 8

Belyi Curve 7 Naive Measure: 30922886356

Belyi Map 7 Numerator:

\(1073758208/4261413375\nu x_{1}^{7} + 6442811392/4261413375x_{1}^{6} - 17183277056/4261413375\nu x_{1}^{5} - 3439591424/852282675x_{1}^{4} + 58720256/4261413375\nu x_{1}^{3} + 117440512/4261413375x_{1}^{2} - 134217728/4261413375\nu x_{1} + 257/255\nu x_{2}^{2} - 67108864/4261413375\)

Belyi Map 7 Denominator:

\(8192/65535\nu x_{1}^{7} + 12402688/16711425x_{1}^{6} - 3538944/1856825\nu x_{1}^{5} - 1441792/983025x_{1}^{4} - 29360128/16711425\nu x_{1}^{3} - 58720256/16711425x_{1}^{2} + 67108864/16711425\nu x_{1} - 65535/1024\nu x_{2}^{2} + 33554432/16711425\)

Belyi Curve 8: 16T4-4,4,4-g3-path4

\(2048/64001x_{1}^{8} + 50266112/4096128001\nu^{2}x_{1}^{7}x_{3} + 301072384/4096128001x_{1}^{6}x_{3}^{2} - 797966336/4096128001\nu^{2}x_{1}^{5}x_{3}^{3} - 768606208/4096128001x_{1}^{4}x_{3}^{4} - 16777216/585161143\nu^{2}x_{1}^{3}x_{3}^{5} - 33554432/585161143x_{1}^{2}x_{3}^{6} + 268435456/4096128001\nu^{2}x_{1}x_{3}^{7} + x_{2}^{2} + 134217728/4096128001x_{3}^{8}\)

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

Belyi Curve 8 Degree: 8

Belyi Curve 8 Naive Measure: 25747154294

Belyi Map 8 Numerator:

\(8421376/16581375\nu^{2}x_{1}^{7}x_{2}^{2} + 1/1236474209739467185993695(321107417184750623877234688\nu^{3} + 321107417184750623877234688\nu )x_{1}^{7}x_{2} - 104073037718386712374653086650829978148226771506429952/59749100977971771070167228119821573268956577716275\nu^{2}x_{1}^{7} + 1950466015035392/270612268250625x_{1}^{6}x_{2}^{2} + 1/945902770450692397285176675(-596130506604591427907718479872\nu^{3} + 596130506604591427907718479872\nu )x_{1}^{6}x_{2} - 285960118708325009670114469497635961315877888018081644544/76180103746914008114463215852772505917919636588250625x_{1}^{6} - 114776697627767996416/1924383975589805625\nu^{2}x_{1}^{5}x_{2}^{2} + 1/1576504617417820662141961125(-1449368628047763275518671060992\nu^{3} - 1449368628047763275518671060992\nu )x_{1}^{5}x_{2} + 139253988959411784736216247239426526830675783183589441536/25393367915638002704821071950924168639306545529416875\nu^{2}x_{1}^{5} + 1/270608040000(-1052704896257\nu^{3} + 1052704896257\nu )x_{1}^{4}x_{2}^{3} - 2840363188356409463406592/8867699915164066786005x_{1}^{4}x_{2}^{2} + 1/7882523087089103310709805625(4377889650640837406089791143936\nu^{3} - 4377889650640837406089791143936\nu )x_{1}^{4}x_{2} + 261292052785088078469299674355635619134261085523593396224/76180103746914008114463215852772505917919636588250625x_{1}^{4} + 1/4228250625(-270508162624\nu^{3} - 270508162624\nu )x_{1}^{3}x_{2}^{3} + 5651675249347024841328790863872/4729513852253461986425883375\nu^{2}x_{1}^{3}x_{2}^{2} + 1/375358242242338252890943125(-74420356887269868172163940352\nu^{3} - 74420356887269868172163940352\nu )x_{1}^{3}x_{2} + 1624856118138285523854275602264130730921180476563521536/1209207995982762033562908188139246125681264072829375\nu^{2}x_{1}^{3} + 1/4228250625(1743105954176\nu^{3} - 1743105954176\nu )x_{1}^{2}x_{2}^{3} + 15326188338621512688748175005073801216/4540404240871107308898644428250625x_{1}^{2}x_{2}^{2} + 1/10134672540543132828055464375(2880591077209269604115296550912\nu^{3} - 2880591077209269604115296550912\nu )x_{1}^{2}x_{2} + 20311072494994822498562265528064550022625407558188269568/10882871963844858302066173693253215131131376655464375x_{1}^{2} + 1/90204089416875(124029105055366144\nu^{3} + 124029105055366144\nu )x_{1}x_{2}^{3} - 809989499404257052078512563818144271958016/96863470606663912958940714017489416875\nu^{2}x_{1}x_{2}^{2} + 1/70942707783801929796388250625(17840042703772477765672094400512\nu^{3} + 17840042703772477765672094400512\nu )x_{1}x_{2} - 14002365506372103431641596145059818979645189690824327168/8464455971879334234940357316974722879768848509805625\nu^{2}x_{1} + 4096128001/8456501250x_{2}^{4} + 1/17319455780308250625(-54552357879537133147136\nu^{3} + 54552357879537133147136\nu )x_{2}^{3} - 15668203604008676792005283436743857851160592384/743923077875651651194219756540000820330025x_{2}^{2} + 1/70942707783801929796388250625(-7321893531634371702642083627008\nu^{3} + 7321893531634371702642083627008\nu )x_{2} - 52573806857700657657226398766720481534472255993955745792/76180103746914008114463215852772505917919636588250625\)

Belyi Map 8 Denominator:

\(4112/255\nu^{2}x_{1}^{7}x_{2}^{2} - 52036518859193356187326543325414989074113385753214976/59749100977971771070167228119821573268956577716275\nu^{2}x_{1}^{7} + 45699346848/462407225x_{1}^{6}x_{2}^{2} - 142980059354162504835057234748817980657938944009040822272/76180103746914008114463215852772505917919636588250625x_{1}^{6} - 19270099435110993152/67919434432581375\nu^{2}x_{1}^{5}x_{2}^{2} + 69626994479705892368108123619713263415337891591794720768/25393367915638002704821071950924168639306545529416875\nu^{2}x_{1}^{5} - 458718897315392583876983552/1108462489395508348250625x_{1}^{4}x_{2}^{2} + 130646026392544039234649837177817809567130542761796698112/76180103746914008114463215852772505917919636588250625x_{1}^{4} + 42325986908433532585777341341696/70942707783801929796388250625\nu^{2}x_{1}^{3}x_{2}^{2} + 812428059069142761927137801132065365460590238281760768/1209207995982762033562908188139246125681264072829375\nu^{2}x_{1}^{3} + 510347475404145421958033305851772928/302693616058073820593242961883375x_{1}^{2}x_{2}^{2} + 10155536247497411249281132764032275011312703779094134784/10882871963844858302066173693253215131131376655464375x_{1}^{2} - 1214407725300731807220346411289988535549952/290590411819991738876822142052468250625\nu^{2}x_{1}x_{2}^{2} - 7001182753186051715820798072529909489822594845412163584/8464455971879334234940357316974722879768848509805625\nu^{2}x_{1} + 4096128001/4194304x_{2}^{4} - 65278032000021878801487692115363237536394969088/6199358982297097093285164637833340169416875x_{2}^{2} - 26286903428850328828613199383360240767236127996977872896/76180103746914008114463215852772505917919636588250625\)

Belyi Curve 9: 16T4-4,4,4-g3-path5

\(-2x_{1}^{7}x_{3} - 14x_{1}^{5}x_{3}^{3} - 14x_{1}^{3}x_{3}^{5} - 2x_{1}x_{3}^{7} + x_{2}^{2}\)

Belyi Curve 9 Base Field: Rationals

Belyi Curve 9 Degree: 7

Belyi Curve 9 Naive Measure: 33

Belyi Map 9 Numerator:

\(11x_{1}^{6} + x_{1}^{4} + 5x_{1}^{2} - 1/2x_{1}x_{2}^{2} - 1\)

Belyi Map 9 Denominator:

\(16x_{1}^{6} + 32x_{1}^{4} + 16x_{1}^{2}\)

Belyi Curve 10: 16T4-4,4,4-g3-path6

\(-x_{1}^{7}x_{3} - 7x_{1}^{5}x_{3}^{3} - 7x_{1}^{3}x_{3}^{5} - x_{1}x_{3}^{7} + x_{2}^{2}\)

Belyi Curve 10 Base Field: Rationals

Belyi Curve 10 Degree: 7

Belyi Curve 10 Naive Measure: 17

Belyi Map 10 Numerator:

\(5x_{1}^{6} + 31x_{1}^{4} + 11x_{1}^{2} + x_{1}x_{2}^{2} + 1\)

Belyi Map 10 Denominator:

\(16x_{1}^{6} + 32x_{1}^{4} + 16x_{1}^{2}\)

Belyi Curve 11: 16T4-4,4,4-g3-path8

\(-1024/65535x_{1}^{8} - 32768/4294836225\nu x_{1}^{7}x_{3} - 458752/4294836225x_{1}^{6}x_{3}^{2} + 3670016/4294836225\nu x_{1}^{5}x_{3}^{3} + 3670016/858967245x_{1}^{4}x_{3}^{4} - 58720256/4294836225\nu x_{1}^{3}x_{3}^{5} - 117440512/4294836225x_{1}^{2}x_{3}^{6} + 134217728/4294836225\nu x_{1}x_{3}^{7} + x_{2}^{2} + 67108864/4294836225x_{3}^{8}\)

Belyi Curve 11 Base Field: \(\nu^{2}+1\), discriminant [ <2, 2> ]

Belyi Curve 11 Degree: 8

Belyi Curve 11 Naive Measure: 30922886356

Belyi Map 11 Numerator:

\(-2105344/16581375\nu x_{1}^{7} - 12763136/16581375x_{1}^{6} + 35258368/16581375\nu x_{1}^{5} + 1703936/663255x_{1}^{4} - 29360128/16581375\nu x_{1}^{3} - 58720256/16581375x_{1}^{2} + 67108864/16581375\nu x_{1} + 16974593/261120x_{2}^{2} + 33554432/16581375\)

Belyi Map 11 Denominator:

\(8192/65535\nu x_{1}^{7} + 12402688/16711425x_{1}^{6} - 3538944/1856825\nu x_{1}^{5} - 1441792/983025x_{1}^{4} - 29360128/16711425\nu x_{1}^{3} - 58720256/16711425x_{1}^{2} + 67108864/16711425\nu x_{1} + 65535/1024x_{2}^{2} + 33554432/16711425\)

Belyi Curve 12: 16T4-4,4,4-g3-path9

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

Belyi Curve 12 Base Field: Rationals

Belyi Curve 12 Degree: 8

Belyi Curve 12 Naive Measure: 3

Belyi Map 12 Numerator:

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

Belyi Map 12 Denominator:

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