32S11-4,8,2-g3

graph data
Name 32S11-4,8,2-g3
Type Hyperbolic
Degree 32
Genus 3
Galois orbit size for 32S11-4,8,2-g3-path1 1
Galois orbit size for 32S11-4,8,2-g3-path10 1
Galois orbit size for 32S11-4,8,2-g3-path11 1
Galois orbit size for 32S11-4,8,2-g3-path2 1
Galois orbit size for 32S11-4,8,2-g3-path3 1
Galois orbit size for 32S11-4,8,2-g3-path4 1
Galois orbit size for 32S11-4,8,2-g3-path5 1
Galois orbit size for 32S11-4,8,2-g3-path6 1
Galois orbit size for 32S11-4,8,2-g3-path7 1
Galois orbit size for 32S11-4,8,2-g3-path8 1
Galois orbit size for 32S11-4,8,2-g3-path9 1
Passport size 1
Pointed size 1

Above

64S6-8,8,2-g9 64S11-8,8,4-g17 64S8-4,8,2-g5 64S10-8,8,2-g9 64S20-4,8,4-g13 64S7-8,8,4-g17 64S9-4,8,4-g13

Below

16T10-4,4,2-g1

Belyi Curve 1: 32S11-4,8,2-g3-path10

\(-256/262143\nu x_{1}^{8} + 8192/68718952449x_{1}^{7}x_{3} - 16384/9816993207\nu x_{1}^{6}x_{3}^{2} - 131072/9816993207x_{1}^{5}x_{3}^{3} + 655360/9816993207\nu x_{1}^{4}x_{3}^{4} + 2097152/9816993207x_{1}^{3}x_{3}^{5} - 4194304/9816993207\nu x_{1}^{2}x_{3}^{6} - 33554432/68718952449x_{1}x_{3}^{7} + x_{2}^{2} + 16777216/68718952449\nu x_{3}^{8}\)

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

Belyi Curve 1 Degree: 8

Belyi Curve 1 Naive Measure: 255242085526

Belyi Map 1 Numerator:

\(-4722294425687923097601/4503633987174400x_{2}^{4}\)

Belyi Map 1 Denominator:

\(-131072/262145x_{1}^{7}x_{2}^{2} - 53523734503315376467377484968274418299804912499818496/306489754683122061290103612409804135885125602516349747\nu x_{1}^{7} + 126101889076166656/18014467228696575\nu x_{1}^{6}x_{2}^{2} - 28074714117483792101775332182420216778149089265204559085568/57389111958147880540638722482262932286861607908320360305225x_{1}^{6} + 264459584685465619398656/4722366482732206260225x_{1}^{5}x_{2}^{2} + 61312290446939168518254618319117081624268636169380745445376/57389111958147880540638722482262932286861607908320360305225\nu x_{1}^{5} - 346652300922709212000730218496/1237935316882868745674162175\nu x_{1}^{4}x_{2}^{2} + 102384823924843204356873212952897179164151232461768419704832/57389111958147880540638722482262932286861607908320360305225x_{1}^{4} - 290849596537877712582426727037272064/324516077773625861597261895041025x_{1}^{3}x_{2}^{2} - 126486667987481043360836341757536085997034323594906190217216/57389111958147880540638722482262932286861607908320360305225\nu x_{1}^{3} + 152609337068177924360666839977529427623936/85069618175811604236691024951739416575\nu x_{1}^{2}x_{2}^{2} - 109177177979411507704130148280875828556496934933757205413888/57389111958147880540638722482262932286861607908320360305225x_{1}^{2} + 9183973779455176727912028608798029668262346752/4460080983492356273883779070784765175844045x_{1}x_{2}^{2} + 82535992406720427088356275857203529446451597501291741315072/80344756741407032756894211475168105201606251071648504427315\nu x_{1} - 68718952449/65536x_{2}^{4} - 6280080846004585065443259981061862643662064738893824/5845895046278183753523577484763653487456427442175\nu x_{2}^{2} + 105361889737226838544249511839436999678781020185481143386112/401723783707035163784471057375840526008031255358242522136575\)

Belyi Curve 2: 32S11-4,8,2-g3-path11

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

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

Belyi Curve 2 Degree: 8

Belyi Curve 2 Naive Measure: 3

Belyi Map 2 Numerator:

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

Belyi Map 2 Denominator:

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

Belyi Curve 3: 32S11-4,8,2-g3-path2

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

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

Belyi Curve 3 Degree: 8

Belyi Curve 3 Naive Measure: 3

Belyi Map 3 Numerator:

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

Belyi Map 3 Denominator:

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

Belyi Curve 4: 32S11-4,8,2-g3-path3

\(128/65537x_{1}^{8} + 1/4295098369(-2048\nu^{3} - 10240\nu )x_{1}^{7}x_{3} - 57344/4295098369x_{1}^{6}x_{3}^{2} + 1/4295098369(229376\nu^{3} + 1146880\nu )x_{1}^{5}x_{3}^{3} + 2293760/4295098369x_{1}^{4}x_{3}^{4} + 1/4295098369(-3670016\nu^{3} - 18350080\nu )x_{1}^{3}x_{3}^{5} - 14680064/4295098369x_{1}^{2}x_{3}^{6} + 1/4295098369(8388608\nu^{3} + 41943040\nu )x_{1}x_{3}^{7} + x_{2}^{2} + 8388608/4295098369x_{3}^{8}\)

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

Belyi Curve 4 Degree: 8

Belyi Curve 4 Naive Measure: 34360852490

Belyi Map 4 Numerator:

\(18447869999386460161/70366596710400x_{2}^{4}\)

Belyi Map 4 Denominator:

\(1/65535(-32768\nu^{3} - 163840\nu )x_{1}^{7}x_{2}^{2} + 1/17196235911533697689775860335577335152326148867(-23973505875873449359616620924640008124858630144\nu^{3} - 119867529379367246798083104623200040624293150720\nu )x_{1}^{7} - 3940512234078208/281470681677825x_{1}^{6}x_{2}^{2} - 43910801206958070900536869370712787246587683224420352/5634776602311804390497305035460303296038470829994225x_{1}^{6} + 1/2049638229457735225(114767481107913375744\nu^{3} + 573837405539566878720\nu )x_{1}^{5}x_{2}^{2} + 1/1878258867437268130165768345153434432012823609998075(15906480397314569134307803777406630734754223839248384\nu^{3} + 79532401986572845671539018887033153673771119196241920\nu )x_{1}^{5} + 676771199636814106253590528/1208944265795744340967425x_{1}^{4}x_{2}^{2} + 157823732825922840913260591374888984960530336760987648/5634776602311804390497305035460303296038470829994225x_{1}^{4} + 1/8803397816383966319331348025(-7878818353806992148703945424896\nu^{3} - 39394091769034960743519727124480\nu )x_{1}^{3}x_{2}^{2} + 1/1878258867437268130165768345153434432012823609998075(-31877544116275567905319506530426535205038848563216384\nu^{3} - 159387720581377839526597532652132676025194242816081920\nu )x_{1}^{3} - 18529697117833019806943436013387644928/5192534544231204006030166999629825x_{1}^{2}x_{2}^{2} - 159170583171986157387281605803814288253016709227610112/5634776602311804390497305035460303296038470829994225x_{1}^{2} + 1/4537375152337072225909320728729864547(9090286970839173803563236825861732696064\nu^{3} + 45451434854195869017816184129308663480320\nu )x_{1}x_{2}^{2} + 1/375651773487453626033153669030686886402564721999615(2677698393091747161007170297023967979093767604928512\nu^{3} + 13388491965458735805035851485119839895468838024642560\nu )x_{1} + 4295098369/16384x_{2}^{4} + 36715009147245839747774486416061245288005763072/22302446651903602685206436444907684961255425x_{2}^{2} + 307993666083658402891183312718102741305305859272736768/95791202239300674638454185602825156032654004109901825\)

Belyi Curve 5: 32S11-4,8,2-g3-path8

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

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

Belyi Curve 5 Degree: 8

Belyi Curve 5 Naive Measure: 134287370

Belyi Map 5 Numerator:

\(281749955297281/268304400x_{2}^{4}\)

Belyi Map 5 Denominator:

\(16384/4095\nu x_{1}^{7}x_{2}^{2} - 1380595303167693507267848568832/266365256860644908139935563479036723\nu x_{1}^{7} - 481304743936/68702695425x_{1}^{6}x_{2}^{2} + 89336747843735789449722212323426304/5453828634221704494165180662233276903425x_{1}^{6} - 656752313184256/93824981052075\nu x_{1}^{5}x_{2}^{2} + 40685165732320527053538397189046272/1817942878073901498055060220744425634475\nu x_{1}^{5} + 5053522706079660544/1153202842111053825x_{1}^{4}x_{2}^{2} - 93357383070902920915262951984267264/5453828634221704494165180662233276903425x_{1}^{4} + 915974942564270851072/524963560458776391225\nu x_{1}^{3}x_{2}^{2} - 1597360359521367043016134019121152/201993653119322388672784468971602848275\nu x_{1}^{3} - 8598673843176912944363776/19356981364796461873639425x_{1}^{2}x_{2}^{2} + 245370145178479329540646631407616/111302625188198050901330217596597487825x_{1}^{2} - 10976940908965329361584896/195816179386595319250125245\nu x_{1}x_{2}^{2} + 41585362627490380700373073362944/121196191871593433203670681382961708965\nu x_{1} + 16785409/16x_{2}^{4} + 3228751834148423843187244531696/324914849213486814301944067149825x_{2}^{2} - 125371728095392837782416671112192/5453828634221704494165180662233276903425\)

Belyi Curve 6: 32S11-4,8,2-g3-path9

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

Belyi Curve 6 Base Field: \(\nu^{4}+1\), discriminant [ <2, 8> ]

Belyi Curve 6 Degree: 8

Belyi Curve 6 Naive Measure: 134287370

Belyi Map 6 Numerator:

\(281749955297281/268304400x_{2}^{4}\)

Belyi Map 6 Denominator:

\(16384/4095\nu^{2}x_{1}^{7}x_{2}^{2} - 1380595303167693507267848568832/266365256860644908139935563479036723\nu^{2}x_{1}^{7} - 481304743936/68702695425x_{1}^{6}x_{2}^{2} + 89336747843735789449722212323426304/5453828634221704494165180662233276903425x_{1}^{6} - 656752313184256/93824981052075\nu^{2}x_{1}^{5}x_{2}^{2} + 40685165732320527053538397189046272/1817942878073901498055060220744425634475\nu^{2}x_{1}^{5} + 5053522706079660544/1153202842111053825x_{1}^{4}x_{2}^{2} - 93357383070902920915262951984267264/5453828634221704494165180662233276903425x_{1}^{4} + 915974942564270851072/524963560458776391225\nu^{2}x_{1}^{3}x_{2}^{2} - 1597360359521367043016134019121152/201993653119322388672784468971602848275\nu^{2}x_{1}^{3} - 8598673843176912944363776/19356981364796461873639425x_{1}^{2}x_{2}^{2} + 245370145178479329540646631407616/111302625188198050901330217596597487825x_{1}^{2} - 10976940908965329361584896/195816179386595319250125245\nu^{2}x_{1}x_{2}^{2} + 41585362627490380700373073362944/121196191871593433203670681382961708965\nu^{2}x_{1} + 16785409/16x_{2}^{4} + 3228751834148423843187244531696/324914849213486814301944067149825x_{2}^{2} - 125371728095392837782416671112192/5453828634221704494165180662233276903425\)