Torsion group Z/9Z, rank = 3


Dujella (2001)

y2 + xy + y = x3 - x2 - 41368267697099x + 102411668493915101147

	Torsion points:

O, [4139439, -1450978598], [3716955, -16222934], [125841, 9859340566],
[125841, -9859466408], [3716955, 12505978], [4139439, 1446839158],
[3673027, -136629582], [3673027, 132956554] 

	Independent points of infinite order:

P1 = [3025089, 2223808552]
P2 = [3729331, 51928842]
P3 = [-2838653, -14033112846]

MacLeod (2004)

y2 + xy = x3 - 8474716297178877x + 301161814193541330007281

	Torsion points:

O, [54971430, 37527848871], [54971430, -37582820301]
[97121076, 627791491455], [97121076, -627888612531]
[44033514, 115602693279], [44033514, -115646726793]
[-48061038, 772973605455], [-48061038, -772925544417]

	Independent points of infinite order:

P1 = [48324174, 66878807739]
P2 = [-28998456, 722877139959]
P3 = [56476542, 51708701655]

MacLeod (2004)

y2 + xy = x3 - 266439437015692920x + 52937734034316909274523712

	Torsion points:

O, [299767944, 71440300848], [299767944, -71740068792]
[388314864, 2833307282568], [388314864, -2833695597432]
[-117667536, 9091803588168], [-117667536, -9091685920632]
[284826864, 394302098568], [284826864, -394586925432]

	Independent points of infinite order:

P1 = [34127184, 6624532209288]
P2 = [-41137698, 7989314741940]
P3 = [136178364, 4379408300568]

Eroshkin (2006)

y2 + xy = x3 - 146942284829128414437x + 685550722977646815589024721409

	Torsion points:

O, [9320725554, -354530634917529], [7065700782, -7035827601873], 
[26358690498, 3889183445544327], [26358690498, -3889209804234825], 
[7065700782, 7028761901091], [9320725554, 354521314191975], 
[7359613170, 52319779154727], [7359613170, -52327138767897]

	Independent points of infinite order:

P1 = [6892237338, 13821416904687]
P2 = [7199027430, 28385662942203]
P3 = [8536669710, 230778029410287]

Eroshkin - Dujella (2007)

y2 + xy = x3 - 45537718416564775527x + 118466673743669180600288571849

	Torsion points:

O, [6668767728, 333707218157565], [3999585114, 17740057042701], 
[-3222909018, 481409735328837], [3999585114, -17744056627815], 
[6668767728, -333713886925293], [-3222909018, -481406512419819], 
[3344642910, -59789381298003], [3344642910, 59786036655093]

	Independent points of infinite order:

P1 = [3659272008, 28815206764365]
P2 = [3716972154, 23609390724045]
P3 = [-9155761747020/1369, -17827975645783497621/50653]

Eroshkin (2009)

y2 + xy = x3 - 1888354378647802297898219715x + 245047137264557236198562303343041637914817

	Torsion points:

O, [62396739661074, 608401581036672153213], [260317700712174, 4170609361522189749513], 
[260317700712174, -4170609621839890461687], [62396739661074, -608401643433411814287], 
[-60038087407746, -376840410128498410647], [-60038087407746, 376840470166585818393], 
[1218369110574, 492694867636140952713], [1218369110574, -492694868854510063287]

	Independent points of infinite order:

P1 = [-21952176546426, -525282688894521544287]
P2 = [-9491414606226, -511971907561087956087]
P3 = [33826260533488413/784, 10841062120907970329503005/21952]

Eroshkin (2009)

y2 + xy = x3 - 1120394540235646336605x + 20279431254320501002290372230577

	Torsion points:

O, [83608361634, -22606622047015377], [-30047373006, 5178454433221503], 
[-30047373006, -5178424385848497], [83608361634, 22606538438653743], 
[25545105894, 2885876195864403], [25545105894, -2885901740970297], 
[6202880994, 3683530208515503], [6202880994, -3683536411396497]

	Independent points of infinite order:

P1 = [31607576994, 4055064251251503]
P2 = [28585928994, 3407485803427503]
P3 = [59588810274, 12849326954471343]

Eroshkin (2009)

y2 + xy = x3 - 175119063305608654696002x + 28206363730630407175298748363753924

	Torsion points:

O, [245834649060, -3601136639616978], [245834649060, 3600890804967918], 
[242136091428, 360199813578990], [279763585332, 33329560482239406], 
[279763585332, -33329840245824738], [242136091428, -360441949670418], 
[659174148864, 446307335391042510], [659174148864, -446307994565191374]

	Independent points of infinite order:

P1 = [243181113252, 1315002813469422]
P2 = [-14217656376, -175194948342565074]
P3 = [445875470117952/1849, 22371128743527533130/79507]

van Beek (2015)

y2 + xy = x3 - 573307517816083917344340x + 167075794327277663399911932573090192

	Torsion points:

O, [1543728752424, -1720727975488158612], [563762191824, -151809351700043412], 
[440358995304, -2710647987862932], [1543728752424, 1720726431759406188], 
[440358995304, 2710207628867628], [563762191824, 151808787937851588], 
[455682838824, -21225283500971412], [455682838824, 21224827818132588]

	Independent points of infinite order:

P1 = [155959510824, 285405764690580588]
P2 = [-1401858098243856/1681, -17599389554238178723452/68921]
P3 = [4043670173485024/9801, 26942785163980552610812/970299]

van Beek (2015)

y2 + xy = x3 - 20720292010739233893383802x + 36302950298543685334430534235913526244

	Torsion points:

O, [2628135181434, -191811478844580], [1517363104152, -2890728440432015694], 
[2673605383428, 128218994515341030], [2673605383428, -128221668120724458], 
[1517363104152, 2890726923068911542], [2628135181434, 189183343663146], 
[2626349901420, -4837604597836458], [2626349901420, 4834978247935038]

	Independent points of infinite order:

P1 = [-46903846698170/9, -44704844132533389266/27]
P2 = [2941305718812, 896823085619174166]
P3 = [308423490629187/4, 5407297836439220108511/8]

van Beek (2015)

y2 + xy = x3 - 32174355462579650355x + 416162791151353397699582660577

	Torsion points:

O, [-7314925506, -510003971822247], [30463417404, 5263697953455963], 
[7403909394, 764073095956653], [7403909394, -764080499866047], 
[30463417404, -5263728416873367], [-7314925506, 510011286747753], 
[208975494, -639881550884247], [208975494, 639881341908753]

	Independent points of infinite order:

P1 = [166911654, 640934520337713]
P2 = [27386058757386/169, 143234884774188178041/2197]
P3 = [67307010402, 17411660303958525]

van Beek (2015)

y2 + xy = x3 - 127315383961088129848470x + 17485155070163945330435232719141412

	Torsion points:

O, [37061947494, 113214410051151708], [206097616764, 84414785826618], 
[204758083764, -981124776221382], [204758083764, 980920018137618], 
[206097616764, -84620883443382], [37061947494, -113214447113099202], 
[221895342864, 12650810798428218], [221895342864, -12651032693771082]

	Independent points of infinite order:

P1 = [10291765410186/49, 1090296499085007474/343]
P2 = [-55979284852881/196, -479609533124397157833/2744]
P3 = [7648653205054401024924/24891688441, 337302481673686984286164322772798/3927186577025011]

van Beek (2015)

y2 + xy = x3 + 12759458432333437998x + 14894378294123135181710531844

	Torsion points:

O, [698129820, 155377882636542], [698129820, -155378580766362], 
[4122896328, 370918987584030], [18794825802, 2625613077571506], 
[18794825802, -2625631872397308], [4122896328, -370923110480358], 
[-860023116, -57313109433330], [-860023116, 57313969456446]

	Independent points of infinite order:

P1 = [12910976424, 1527018386963130]
P2 = [146943571912614/34225, 2442559503582664255368/6331625]
P3 = [372643046813127/128164, 12696523508868798426999/45882712]

van Beek (2015)

y2 + xy = x3 - 1950873533612743056660x + 33177096191505561806432962227600

	Torsion points:

O, [37945966170, -3713181131409960], [25853437260, -144071223623460], 
[-16152189480, -7776493602281460], [37945966170, 3713143185443790], 
[-16152189480, 7776509754470940], [25853437260, 144045370186200], 
[23313823920, -605518967501460], [23313823920, 605495653677540]

	Independent points of infinite order:

P1 = [24748026030, 232628604513480]
P2 = [6507559170, 4556009858342040]
P3 = [-500066052304/25, -1001533122077024684/125]

High rank curves with prescribed torsion Andrej Dujella home page