Torsion group Z/3Z, rank = 14

Elkies (2018)

y2 + xy + y = x3 - 18867784760635242349892340171680238008255587838x 
            + 997538055692091226068752063841965542145691025244956491303628897332656

	Torsion points:

O, [79595398712780367992550, 141712462567168788023667802119037], 
[79595398712780367992550, -141712462646764186736448170111588]

	Independent points of infinite order:

P1 = [12864256042996859681925, 27512668636643519258048327414154037]
P2 = [79318784030712572297850, 4651924309564719427822985851537]
P3 = [79336902537034132657950, 14824810026698869991976958468012]
P4 = [79315263837171944584362, 1127922585398596499789708582548]
P5 = [79507759827058070939525, 98883656314865201492304020804037]
P6 = [79478240571896535857925, 84452732814512106743755527804037]
P7 = [70940904022155789777690, 4007291237835963069650624862069172]
P8 = [63074263073401074937725, 7641906126570214399100355852414037]
P9 = [80622777748625913134925, 644590688365435089407297943261037]
P10 = [79235130495136906316925, 33647493872935940195847354654037]
P11 = [9144028966628125493325, 28736301909404056080997883298324037]
P12 = [-72399779656408399006200, -44542805882853035305521480502944713]
P13 = [-82480606624979486358075, -44639048587678819021651441428095963]
P14 = [86685829156539722432925, 3655590592921252290248091129138037]

Elkies - Klagsbrun (2020)

y2 + y = x3 + x2 - 24791434545122407462176574410341851245x 
            + 47463673909255798147742316846176526846664068435304032999

	Torsion points:

O, [3249307859596751381, 1102210625320961087278315187], 
[3249307859596751381, -1102210625320961087278315188]

	Independent points of infinite order:

P1 = [3045140793366361281, 455519346190208924256293612]
P2 = [-528216676545090209, 7772486334965581899275176132]
P3 = [2203920316828156637, 1878923129661099866370805107]
P4 = [2949565141361554893, 27319813023963716607265779]
P5 = [2082360557357435101, 2206479862158999933236306572]
P6 = [3418925429587448731, 1633279503594310209884293012]
P7 = [3008880827154416631, 331201878306635437696852187]
P8 = [-731297975613141829, 8074806749345022438871193067]
P9 = [2787230928945927881, 131376375535040254079887187]
P10 = [3756970281368687881, 2711451964147031600152527187]
P11 = [202374556341695155861/81, 771315701029785672999504179948/729]
P12 = [87853802467691849341/36, 264253862993427486286969158017/216]
P13 = [2799466327651303381, 18387620641344154427691187]
P14 = [3617621200737169131, 2263211419824045143602502187]

Torsion group Z/3Z, rank = 13

Eroshkin (2007)

y2 + xy = x3 - 560715933702165990261993692150795879540x 
         + 5299428030171662962897867758309003693598430128674403539600

	Torsion points:

O, [8593602267036083080, 33399039471607927923677923660],
[8593602267036083080, -33399039480201530190714006740]

	Independent points of infinite order:

P1 = [30978765264695144680, 132887030666136666891811891660]
P2 = [42714959134973152720, 243484714136201166251098454860]
P3 = [17517044808100174120, 29195754303760059255051062860]
P4 = [14407536544275712180, 14545360491234198754332073660]
P5 = [27024189809840210680, 99410687206608213450615379660]
P6 = [15223373261754620200, 17072594325815595384787766860]
P7 = [-19625425810319741720, 93513850432355415588023731660]
P8 = [18799275311571587070, 37447008534737807545137468570]
P9 = [-18348119874544858160, 97008001583008247645310994660]
P10 = [16004432727240519730, 20612929621400491642269765910]
P11 = [-27235206268166134070, 19202914916421997002398631160]
P12 = [13326569619771689680, 13920236659741078875971506660]
P13 = [27069452184006616180, 99781951498337473669562238160]

Eroshkin (2007)

y2 + xy = x3 - 73262771788012628080963454016709537315x 
         + 240598790801630018163569184912325717758514948008409652225

	Torsion points:

O, [5896914810510119630, 3692079207152305353879997385],
[5896914810510119630, -3692079213049220164390117015]

	Independent points of infinite order:

P1 = [4457502222769039370, 1611641339515815829274369045]
P2 = [4198800350419611170, 2647181824163160663752575085]
P3 = [4663784697212755040, 598710342059502067618068575]
P4 = [4600449196285678970, 960025506026782647465074945]
P5 = [4554776706911186990, 1181769967892808613450633385]
P6 = [2380239538380692222, 8927552162407067648923077497]
P7 = [23702110592754993710, 108718564570356475210335730985]
P8 = [4305435373222259510, 2231488968639382184715604505]
P9 = [4618909929490933310, 863707409335431200909470985]
P10 = [4372416906624676220, 1963518441135919270046813945]
P11 = [36249530734715794730, 212644711673400126644106584885]
P12 = [4693924773486814190, 360230832576196960473000425]
P13 = [4450103380274798480, 1642908737129617992291509135]

Eroshkin (2007)

y2 + xy = x3 - 7032750154590180472810630714591592198580x 
         + 234825110521164672374627005794918283861649641835073217760400

	Torsion points:

O, [30796046079264381960, 217832673380546979749336183820],
[30796046079264381960, -217832673411343025828600565780]

	Independent points of infinite order:

P1 = [55900848433614700560, 127957972604422146757975646220]
P2 = [60064561238865449310, 170599150318472748808550606970]
P3 = [69812369394041538000, 290002371439954245595102699020]
P4 = [61612368528157270560, 188165990173054207603274843220]
P5 = [3822294292797501180, 456069855621822797983703355720]
P6 = [52372587674097006360, 100766688720822135191405667420]
P7 = [51554975084878426920, 96335957098298726629044580620]
P8 = [51147073430501124120, 94458713634598078961223670620]
P9 = [68932207591190829072, 278538666893854221091708241892]
P10 = [47727681973487919720, 88818038537044422433497220620]
P11 = [53338169456226155160, 107033016967755131766827494620]
P12 = [64102710537148435560, 217749619349148168747134366220]
P13 = [52406676754698809160, 100969677496651189732961320620]

Eroshkin (2008)

y2 + xy = x3 - 245159698188178088219881294961406816115x 
         + 1510191009902655798002552220643158891490617937867360088417

	Torsion points:


O, [6148183244268310574, 15339705160042210643860741913], 
[6148183244268310574, -15339705166190393888129052487]

	Independent points of infinite order:

P1 = [10833013559463040274, 11210358084599745780316885613]
P2 = [10706258660637427034, 10613283545662027955006055473]
P3 = [10395143048176178894, 9220066684034404984174781273]
P4 = [10137602529625035794, 8167669882323985109476036973]
P5 = [10038848157700788524, 7795499127400476443202567113]
P6 = [10824343292658261344, 11169050746067025317933688623]
P7 = [-4069189724279614606, -49400540986606610630851023667]
P8 = [-7414688387566494706, -54040095999701376974332025767]
P9 = [-976881606269713816, -41818067551922275505813314057]
P10 = [11184169214867230724, 12932997956751474213414498713]
P11 = [10145221804179288974, 8197181181207665106642796313]
P12 = [-8725386066273219586, -54635334476631791259074142967]
P13 = [13085600454673829294, -23298390072192192872367107527]

Eroshkin (2009)

y2 + xy = x3 - 35822192130572784206480514296239908919425x 
         + 2609719568750620065454923921391767461604324824175741297455625

	Torsion points:


O, [104605984638686332650, 84532082161048799807719188075], 
[104605984638686332650, -84532082265654784446405520725]

	Independent points of infinite order:

P1 = [111765301087195517610, 46430435456204326675146597675]
P2 = [111116582336682124050, 34996683196066023464195962875]
P3 = [111913032824849252130, 49063192420997854527679793475]
P4 = [110123909838304503990, 18512376609554805482691445335]
P5 = [109146709187032115370, 10506129421757959855129586475]
P6 = [110382143345238886970, 22567063509763234684093270475]
P7 = [110245179394644403100, 20382223547410442956289336075]
P8 = [108522553923729999720, 17018033523451040060797113555]
P9 = [108512500742043656106, 17163334387154750309786061867]
P10 = [109741057847530716000, 13297340647117901361820923075]
P11 = [108902034932485796850, 12260441074498263650721717075]
P12 = [110291739877102075140, 21117334974942773428569817185]
P13 = [111294342929434080810, 38103895579546047175336101675]

Elkies (2018)

y2 + xy + y = x3 - 314261798247699281583485627025593248x 
            + 67755045622702867235372184283705632291457304269577006

	Torsion points:

O, [362443692537696970, 38280682383727391288231327], 
[362443692537696970, -38280682746171083825928298]

	Independent points of infinite order:

P1 = [331064222522331745, 108002572991933200465427]
P2 = [60302653368352270, 221412534637578854030048702]
P3 = [340457177749093345, 15004107520711561275138827]
P4 = [278817716186182945, 42525146455634850545742227]
P5 = [336710803989195745, 10677513423307188160561427]
P6 = [296643657499084225, 25202472452361902409344747]
P7 = [177447905093399842, 132579729686645884210002263]
P8 = [331615767977756020, 2887605493923435665798402]
P9 = [-47959331959652030, -287604798938943632624433673]
P10 = [240433217677549720, 78070900353018751873632077]
P11 = [331548647063699980, 2700485784265218800691617]
P12 = [340960186243818900, 15563361748866798001200322]
P13 = [343605917424468145, 18457352478941937160416827]

Elkies (2019)

y2 + xy + y = x3 - 2557499874808783579775134722531513x 
         + 49997987268990292225606516653085266862246125287656

	Torsion points:


O, [23520615568468400, 1689992802320795588198487], 
[23520615568468400, -1689992825841411156666888]

	Independent points of infinite order:

P1 = [-28465592926448050, -9986658729964666636804638]
P2 = [28249320354054050, 542210705414253924282462]
P3 = [-15291469173181975, -9248260821777444455292138]
P4 = [36017445535938650, 2146447264917226301003487]
P5 = [33946963722917750, 1516241635846977320628762]
P6 = [11870345050407425, 4616510943415456790163462]
P7 = [13537794354454250, 4225656968223993402287862]
P8 = [-23701351493317975, -9864069141152846242092138]
P9 = [30543013939705940, 614058994056949780912212]
P10 = [-9337716026000044, -8547807428694198848172201]
P11 = [8677307945457380, 5334710580871357201569732]
P12 = [10929571604341625, 4832308665083601374170512]
P13 = [18782079081299525, 2930616063900065076042612]

Torsion group Z/3Z, rank = 12

Eroshkin (2006)

y2 = x3 + x2 - 298337765420974027925404974199074153225x 
    + 2209525297419800283762159062541471723368027978017989315000

	Torsion points:

O, [4932343693390245825, 29291893581651695450789307975], 
[4932343693390245825, -29291893581651695450789307975]

	Independent points of infinite order:

P1 = [12582816306572774955, 21161259527567477756392809855]
P2 = [-10238901114928946460, 64736234016763296633829049790]
P3 = [-7930737095588324325, 63849411541423127439301106625]
P4 = [11495083131989372685, 17292569185912354737409456065]
P5 = [-4400110027279034376, 58626392520949825159787439240]
P6 = [710843695600020675, 44696901079924332333043574625]
P7 = [-6124232965906747285, 61700230054196535588270161535]
P8 = [14925305818114923000, 32887307382139198660372885800]
P9 = [-10825479394428034875, 64579608463195877622328856175]
P10 = [-12802868599160550075, 62694036529804395978303380625]
P11 = [-4669378572351124797, 59167308170559298781522768031]
P12 = [-1894227661029731335, 52610343752060548173493438665]

Eroshkin (2006)

y2 + xy + y = x3 - 81089552755699140908030202688368478038x 
             + 281263936268193464077589295595406966386314342335769150988 

	Torsion points:

O, [4625450842703735354, 2269090345561701691033219560], 
[4625450842703735354, -2269090350187152533736954915]

	Independent points of infinite order:

P1 = [5741172515907340367, 2224852147461708059604916497]
P2 = [-6283260873451047176, -23296171078888629764646328180]
P3 = [-4377070887290532121, -23501903154705392080910055465]
P4 = [2577560963520055004, 9453862857802006685518121535]
P5 = [2012165467991458804, 11235889541275053343944971060]
P6 = [5052438574023597248, 733645881558773670043750083]
P7 = [2616398712495877619, 9328022441806695951924810555]
P8 = [3958074780969203204, 4723772959038114089554236135]
P9 = [6205498548772362149, 4126195518911771323234166295]
P10 = [-5276925921661264561, -23711315833456709412101226735]
P11 = [5589234149317131623, 1624971859309398399443882583]
P12 = [3832164828660767909, 5176158037521448409935173015]

Eroshkin (2006)

y2 + xy + y = x3 - 227204253761112793208000940016487352978x 
             + 1318089867386926711140924601590234088488769981566633802648

	Torsion points:

O, [9146520430865863874, 2269090343301166896952155300], 
[9146520430865863874, -2269090352447687327818019175]

	Independent points of infinite order:

P1 = [8145244832551235237, 2802136845278504101116818655]
P2 = [-2614487012653179451, -43522879055509397420848043025]
P3 = [7955909699716289909, 3748942862747705666913785655]
P4 = [-1305804858417159701, -40156540233256298925358896025]
P5 = [8120355012367564949, 2927152226216520118893211875]
P6 = [8806012412343667349, 442884651697222550228202375]
P7 = [8760832457062556699, 67105415368154448552857175]
P8 = [1723413480817630199, 30522807816313203931685157675]
P9 = [8209263705352116389, 2479721714605586058212981115]
P10 = [8221419028699601399, 2418353623765018460259967275]
P11 = [5015537671883797799, 17455867889605497871382497725]
P12 = [-4345785245986300288, -47152909934632349816986147545]

Eroshkin (2006)

y2 + xy + y = x3 - 2061966282312748415211387989253532899918x 
             + 36079330409584890827203029261000439483009603580628519854608

	Torsion points:

O, [22898259700584047894, 29498174510918422610714110140], 
[22898259700584047894, -29498174533816682311298158035]

	Independent points of infinite order:

P1 = [32108239270224712709, 54542283280378129719770629785]
P2 = [26962907377126366529, -9204040020243165214050504975]
P3 = [29251246559798375369, 28154024182174591974149785065]
P4 = [25493183133877482137, -9018143487402250631685901203]
P5 = [26108053542756440369, -6438078003708091583745780435]
P6 = [26021870731648428569, -6595568383037854439551154835]
P7 = [-49871157374901107131, -121966418059392151469935087935]
P8 = [18483472222146972869, 65434732112323934885849947065]
P9 = [25935621501012332069, -6834967387480344717682947735]
P10 = [26752295552864584817, -7951360755245895938491306323]
P11 = [25026126671698282244, 12261311424145861787090253315]
P12 = [24898439153453520769, -13226275131687532126996461235]

Eroshkin (2006)

y2 + xy + y = x3 - 69993120101005305009860092170583378x 
             + 7639258120193164015870793100033493923516729761682248
	Torsion points:

O, [84445978180236274, 48278517997658438832968825], 
[84445978180236274, -48278518082104417013205100]

	Independent points of infinite order:

P1 = [171777816398125539, 26167670979268269042638245]
P2 = [169740137584324909, 25477845876425289117707135]
P3 = [149332975795951549, 22740998019085332334924775]
P4 = [235735302889811599, -65111340369741228518429275]
P5 = [140624626342500187, 24029021933798151088331855]
P6 = [29075000559302299, 75025241273474528659949525]
P7 = [147635813086825357, 22884193456036867524732515]
P8 = [160487006141243383, 23233334933109112425738323]
P9 = [457571992790515204, 267236203879055542974358715]
P10 = [-21296509469728001, 95499782305256485473234875]
P11 = [26669222671152124, 76102328670260618345498000]
P12 = [161956158824337094, -23484430563596793664294915]

Eroshkin (2006)

y2 + xy + y = x3 - 51012395091150795883443767656668298x 
             + 76434974688414573638384904319711864744626666727159256

	Torsion points:

O, [2839813206318970, 276206683329981852839709377], 
[2839813206318970, -276206683332821666046028348]

	Independent points of infinite order:

P1 = [457917641278676257, 386128984291844301517862723]
P2 = [270901045199761935, 287221831349545531747622527]
P3 = [-397254005700385953, 184415027696538474481219633]
P4 = [331187106878779291, 309623241474837135248838809]
P5 = [272065748698447585, 287566503142636601677121387]
P6 = [-419789818786899680, 154507705600056059101944152]
P7 = [768976913240434780, 701372420641720230475880867]
P8 = [95702143817093695, 269127303497606029751993777]
P9 = [-351066191606211200, 225999296032216298947768472]
P10 = [-221402129973000095, 277265825716404359585558867]
P11 = [329196849140580055, 308734681486552150048900727]
P12 = [183025514774816392, 270609415057442326162910063]

Eroshkin (2006)

y2 + xy = x3 - 15075739125931590120183633247999865x 
         + 719481272914731335170056334451666685290651887623417

	Torsion points:

O, [53124380663862874, 8277655824923614118134963], 
[53124380663862874, -8277655878047994781997837]

	Independent points of infinite order:

P1 = [89804331152227354, 9479988398821069756793203]
P2 = [-62701899104078726, -37659586597332022817662637]
P3 = [-13211903879226806, -30271344415900117440882077]
P4 = [-10977722414604866, -29726345592837952853574677]
P5 = [86749319841487474, 8031086365180078649558563]
P6 = [76996051410701074, 3895081239712605109656163]
P7 = [76763044815074074, 3814822332867371404251763]
P8 = [76160708262893074, 3615100101120986165178763]
P9 = [90468440519765674, 9800261165197851637034563]
P10 = [6275050034321434, 25002546738193716541475443]
P11 = [12762046847775514, 23003533123328924393108083]
P12 = [-17295822293641526, -31225864445576588088555437]

Eroshkin (2007)

y2 + xy = x3 - 420123886323242988442406904384921148985x 
         +14547746280073821301806145504809500759831594126734723275897 

	Torsion points:

O, [1026250449144166234, 118817821851820512660319757683], 
[1026250449144166234, -118817821852846763109463923917]

	Independent points of infinite order:

P1 = [30660270495414485914, 174610677242694886680521951923]
P2 = [-27751044875610449636, 69533982916346344707296560273]
P3 = [80265229592403732634, 705645602324169346229354703283]
P4 = [25462531704105141634, 142683984969460557005691585283]
P5 = [62209773870344426314, 478714153470297354327191030563]
P6 = [77369542854586528534, 667218561061651312960497227983]
P7 = [7611285268038426634, 108586361707872928552354808083]
P8 = [1204020729424715528314, 41772437377979978301951301099123]
P9 = [-3756642886530059846, 126779282893774465971084696403]
P10 = [476226013166193361114, 10383561539853147479038665421363]
P11 = [25934726315159755354, 145244295192942091588704117043]
P12 = [11390807473140285528634, 1215713015403696823110293448560083]

Eroshkin (2007)

y2 + xy = x3 - 2274412631042746230240014920326936789445x 
         + 41861448474250945269536760003663301866430401503884473151937

	Torsion points:

O, [22935589249135150514, 41970474374182395872085761543], 
[22935589249135150514, -41970474397117985121220912057]

	Independent points of infinite order:

P1 = [29092047839687511794, 17779258075082697894047718023]
P2 = [-40042222833789788686, 262166172995985021961841950343]
P3 = [12749333601877491314, 122215179045634694783184958343]
P4 = [16413671046029425094, 94614860436193229223216411323
P5 = [34231124382078831914, 64161882695108616251348629943]
P6 = [184459149322898168414, 2428699358856826709350842982643]
P7 = [-44298691225405686286, 235975315425044672490727505543]
P8 = [16889375335100119778, 90916176066253752082469675111]
P9 = [-20672826778209059836, 282922477635927184125760441943]
P10 = [19785262531937505914, 67872448967870615838814873943]
P11 = [28528319860348946564, 13945589112868893015324845093]
P12 = [29117718101778919154, 17970883254639240872733936263]

Eroshkin (2007)

y2 + xy = x3 - 3460212184869409462879787098424302220365x 
         + 76725332182289401244480579162479048671720653011227833006225

	Torsion points:

O, [46627887868145828930, 129458223785945094358421180735], 
[46627887868145828930, -129458223832572982226567009665]

	Independent points of infinite order:

P1 = [10105607718142501250, 206856963386201917943798521535]
P2 = [-13009857994099020670, 345745859768995019046066150335]
P3 = [28976934109135016690, 28104140780973505497462530975]
P4 = [22373995425270287330, 102503101634127369791658258335]
P5 = [-53363024164255686118, 330780156751016363506359596471]
P6 = [24999947745865277780, 76453321659288905279671764335]
P7 = [20906847293512979330, 116282024692195140609164550335]
P8 = [-67729462416577636990, 19739037236318087978172159935]
P9 = [38091918964547592830, 13797353499110074433329467335]
P10 = [38299291670039711150, 19504522661314717583193861035]
P11 = [29117728961170881002, 25671954825563261656393399247]
P12 = [-27346770928406869870, 388458224524004677675308091535]

Eroshkin (2007)

y2 + xy = x3 - 5915232167158349511943570504558263178160x 
         + 189989838940979212731233721328707449324240816328756166793472

	Torsion points:

O, [23628255529510549984, 251822846302087015215164745808], 
[23628255529510549984, -251822846325715270744675295792]

	Independent points of infinite order:

P1 = [-89519732833201272536, 46132069718947335099772316488]
P2 = [84691875391497876064, 544507404023764849497002816848]
P3 = [45970901032972411744, 123339705593822530812696363088]
P4 = [-57127814943564887456, 584356500482526458711993489488]
P5 = [100781099570779355794, 785787971074967111000920421218]
P6 = [403846687831081059904, 7979057989786159612197872334928]
P7 = [412126705956115260904, 8231113852951392740039735265928]
P8 = [59425667905190826184, 219839975510776823289040042408]
P9 = [93986866492572171184, 681374792998977476568890161408]
P10 = [-46025836157323353776, 603939769734558083592818509648]
P11 = [52315483791428293684, 153994704354157120312282501408]
P12 = [221794131787439765464, 3128684107727828196905846489488]

Eroshkin (2007)

y2 + xy = x3 - 2930856110646876366568351217006462280065x 
         + 61162147490056987962447877881747110669389916249870892297225

	Torsion points:

O, [26933945442536577130, 41970474372183217775385048235], 
[26933945442536577130, -41970474399117163217921625365]

	Independent points of infinite order:

P1 = [-40235999407913384390, 337562702986720130719858556155]
P2 = [44707073688835602730, 139603648537041562948391294635]
P3 = [32087052019039392970, 12480663538868816838967321195]
P4 = [31528035511767762730, 9869952437945552272618814635]
P5 = [-2651803878244516310, 262517724044421919452063662635]
P6 = [50985119243826724642, 210397287358064083790394170947]
P7 = [78131376617041716730, 555989435091996521706297986635]
P8 = [-21386746977088898720, 337729889632422785641062096835]
P9 = [128227595750239630330, 1339292426618763632469963887035]
P10 = [445902862569267147730, 9349484855879169740480431667635]
P11 = [33615911520963816730, 25014390445822677658407626635]
P12 = [304589704490274688170, 5237053493530840603650300449835]

Eroshkin (2007)

y2 + xy = x3 - 1604115720432703747645854436115825419710x 
         + 24854471963630464651375308856010948374447644530671142636100

	Torsion points:

O, [18399347760464240220, 39605940608244872081398493490], 
[18399347760464240220, -39605940626644219841862733710]

	Independent points of infinite order:

P1 = [33959885299314421020, 97693051842804472247186693490]
P2 = [-41545117463335066980, 140680164997460409038803589490]
P3 = [-9174361359411383940, 196974651376491328835587458450]
P4 = [-46183272832717250580, 20826151013849617443427335090]
P5 = [30899189560960660716, 69209131284197626008455306082]
P6 = [180624464887584204060, 2372344986496528817720837550450]
P7 = [27006994203855343020, 35077461214397158037804099490]
P8 = [24660095404027993560, 17121039832417463073774918450]
P9 = [27325131129439513020, 37741995642991621184505617490]
P10 = [60354618219869892660, 384566297479682544533572034490]
P11 = [37276788414939323670, 129832525945243340323635807990]
P12 = [26994959517410041500, 34977300506696000086569912690]

Eroshkin (2007)

y2 + xy = x3 - 435143747768370112106161383366802254050x 
         - 1718988230824332939656053494953243446622485861836978983868 

	Torsion points:

O, [33624478289064400804, 147192227053989133160632318798], 
[33624478289064400804, -147192227087613611449696719602]

	Independent points of infinite order:

P1 = [-16628901140393603996, 30310798695492477365141969998]
P2 = [-18281748975241043996, 11226375646978704018619073998]
P3 = [-18120188830565924036, 14706913979321483142093261358]
P4 = [-11720178835264088396, 42083965709751428011846582798]
P5 = [-14795543899109216996, 38475128378435732798601362998]
P6 = [-4273399721615984036, 7906601566687286327464882678]
P7 = [-5536028259149890436, 22810388529597166759825074478]
P8 = [194437378280745414244, 2695283237214064642263975036238]
P9 = [75692504667782440504, 631674870120181974483487621498]
P10 = [22627717884668796004, 4516799350135623773217233998]
P11 = [26892917340098748004, 77643188526643345552629713998]
P12 = [113711726980224511204, 1191273802614516606072894037198]

Eroshkin (2007)

y2 + xy = x3 - 9961897895877072436363696377038852218385x 
         + 386641226009676505641094591224658839288977045345666728365225

	Torsion points:

O, [42985521501155122570, 194552215642803468217376764315], 
[42985521501155122570, -194552215685788989718531886885]

	Independent points of infinite order:

P1 = [86006937761814759370, 407503179290727132395383756315]
P2 = [257537050800611271370, 3860351532566427118874999500315]
P3 = [-47636031804866054660, 867808291817112587252415481915]
P4 = [126186608735304687250, 1067173877208511473292263447235]
P5 = [7987084975889469130, 554602763521880669996015811355]
P6 = [570049510686960737050, 13414511860509253690265185331515]
P7 = [-84492898325929220150, 790665439693907320113017087515]
P8 = [57759465938964442270, 62784609544384827153200068915]
P9 = [71086828408411911370, 194183449779612402337287340315]
P10 = [69187816955840072170, 169109893321039996359466787515]
P11 = [-74967801385490523830, 843878502711079387679916143515]
P12 = [1002185439564440266, 613725153108493160747144097307]

Eroshkin (2007)

y2 + xy = x3 - 4296561733833813622211765271899645505485x 
         + 110491051080034418321885623194222767606021500012097359146897

	Torsion points:

O, [26223340349492762434, 125911423143838901315213629183], 
[26223340349492762434, -125911423170062241664706391617]

	Independent points of infinite order:

P1 = [-66335161885466989466, 321878849982494645592135800983]
P2 = [43130574497326462594, 73562038494881453388816313663]
P3 = [62500645297143948034, 293430327832772473627293425983]
P4 = [39724630583973479284, 49990286460186219613552775983]
P5 = [52832646183411684394, 175964436870295185202888378063]
P6 = [40152375753476762734, 52039247228172469251933608683]
P7 = [-43021119583048421246, 464445625850749479637376423743]
P8 = [153607000090204783234, 1753534496181588293632680229183]
P9 = [118284677678737657474, 1121260852277368266767215651903]
P10 = [154293008929717511014, 1766553792553501566858185182723]
P11 = [-36517228798129793756, 467646834983113396146797102863]
P12 = [65891296272938921134, 336842436424133652476023884583]

Eroshkin (2007)

y2 + xy = x3 - 12639274175693629669246799362435207748655x 
         + 546025924766214300252759666836267029730941947556294567508025

	Torsion points:

O, [74922305077099245510, 140098625728723286161546726845], 
[74922305077099245510, -140098625803645591238645972355]

	Independent points of infinite order:

P1 = [62404916143288979550, 17353244355641300882904733005]
P2 = [56276436815756921970, 113851600823870891463914866485]
P3 = [62531566218939297840, 13529345076882993325316878965]
P4 = [51182707218235510662, 182196139172293920440128655421]
P5 = [-123867936409446284290, 459443226430915765703642501645]
P6 = [60618127466215160586, 51005678040053921720440874097]
P7 = [-127187469539387764290, 310025383890038138131817502645]
P8 = [25521611479838189460, 489974335234694949980396822895]
P9 = [67239902885857680870, 12950322288230406443644134045]
P10 = [62116378179347057310, 24343049461826315636280381645]
P11 = [357795446003224168320, 6467446075259235162330804622635]
P12 = [55800806143121197110, 120386373477442685338038375645]

Eroshkin (2007)

y2 + xy = x3 - 6822056808999168202154285328075085085610x 
         + 220557528518736836412923854565947623298384352585736677636772

	Torsion points:

O, [33597116630822085684, 171112189663847522584115050758], 
[33597116630822085684, -171112189697444639214937136442]

	Independent points of infinite order:

P1 = [108627574395471512724, 872521622980212548899508902278]
P2 = [114514187165976087924, 970060268305152231750877519878]
P3 = [216357056754008293284, 2978642740267652747017153454358]
P4 = [-89798247197031297036, 330239130732818562816044808198]
P5 = [-58658584952057101788, 647221393889295451414622620998]
P6 = [254956008985391713524, 3879952072334244308419153212678]
P7 = [48497740467581546484, 61411470632628481940457111558]
P8 = [83529709333896481284, 483236613887851803796032674358]
P9 = [48488203694843207484, 61393403651322347995436444958]
P10 = [135873991883978443284, 1342418703213815188287580052358]
P11 = [-19427775789555083916, 588015415298039058142649281158
P12 = [41805113563993625034, 91772043806874993968808258108]

Eroshkin (2007)

y2 + xy = x3 - 12330133292485949436592445464751904238410x 
         + 526068270853525264701322899592369976900342938683261805191972

	Torsion points:

O, [74184882325187335284, 140098625729091997537502681958], 
[74184882325187335284, -140098625803276879862690017242]

	Independent points of infinite order:

P1 = [4905857288241651324, 682419568870385061776935663998]
P2 = [34359554411510493132, 378119759804359472985568201158]
P3 = [-123551365829723864268, 404312866407648058578378398310]
P4 = [10573197557735714484, 629985190214737097356430163558]
P5 = [50496972003071345334, 179438945385893124021529037808]
P6 = [240637491252319647684, 3390195758311167639456901381158]
P7 = [-54066540977993417676, 1017186860794534459636769104998]
P8 = [-127405192710395498316, 170116430525758365494958061158]
P9 = [153470113559116816044, 1499484753154575992628934619718]
P10 = [133467655369957502484, 1121577336373929494033258787558]
P11 = [133352576988517939764, 1119468654460961408048670372198]
P12 = [-35142395903241602316, 957067531429956634361875381158]

Eroshkin (2007)

y2 + xy = x3 + 1211947908078559188213163163805214393450x 
         + 12487523185136437111960490242590614939802936818380403902500

	Torsion points:

O, [7218786168825388300, 147011881271311887688419583450], 
[7218786168825388300, -147011881278530673857244971750]

	Independent points of infinite order:

P1 = [174094593215162577100, 2345229516490102683769073767450]
P2 = [-8035801177852010420, 47219128802467687802481165850]
P3 = [26074175133613931740, 248626068460093118029171627210]
P4 = [140989496423786047180, 1727991328121643130655269581850]
P5 = [106953989353422468400, 1168577570592871806712912288150]
P6 = [51717261544505514100, 462053053176382853062507651450]
P7 = [24625984926364779100, 239305443017407344954033679450]
P8 = [159741400003774698700, 2069361587765377286656166983450]
P9 = [835832012784956824780, 24185747504567921047082803917850]
P10 = [79712777204662517740, 784602035812828926688901021050]
P11 = [-8823809056205042900, 33264214399934792981206807450]
P12 = [113723458726001881900, 1273225342616024250739608343450]

Eroshkin (2007)

y2 + xy = x3 - 10550832125564514944256818543885357615825x 
         + 419625737407708712632765113547497120508448337766481231555625

	Torsion points:

O, [46568389338256803850, 171112189657361886230397691675], 
[46568389338256803850, -171112189703930275568654495525]

	Independent points of infinite order:

P1 = [79926686962542884170, 294832766376023415742867978075]
P2 = [68394704416818080770, 133954751548629569050308352075]
P3 = [58781576061642482650, 50374886492965431536600359675]
P4 = [57089477649503099710, 57885672002021166531197824015]
P5 = [87439631048272000330, 406941151935030924215555061595]
P6 = [57055238293350232810, 58115624990308030371508693915]
P7 = [-52895681251992268550, 910889430960719041550892841675]
P8 = [-41657627788383789350, 887049874192718065508231143675]
P9 = [146766331280154663910, 1425663999120670449199218350275]
P10 = [-93353275141765100630, 768779667753179092858150774555]
P11 = [-28714328596516645250, 836008654504751183851122218575]
P12 = [52626044843740954990, 100623234782195569640335616275]

Eroshkin (2007)

y2 + xy = x3 - 347224739173438611349143152475095606405720x 
         + 131243904079630053502971198622754870300832280107569949516942912

	Torsion points:

O, [86298823828501546864, 10095619191114969361157788748968], 
[86298823828501546864, -10095619191201268184986290295832]

	Independent points of infinite order:

P1 = [-614749561874797341776, 10600746933282861402536629692328]
P2 = [475572648547118323264, 8583313415804625610117065113368]
P3 = [1140115197226494169744, 34890699024068910929778689551048]
P4 = [2241986579258382073264, 103063692963121940550568634963368]
P5 = [1311751861331224953514, 43964683331285560396293198649618]
P6 = [-474782825444657910656, 13750465615490121007791640210648]
P7 = [5382285515975580852514, 392659898899681462964182663836868]
P8 = [8111735896673083257184, 728744793151651671448361112233128]
P9 = [514662210343360953904, 9426703177126795738830590218408]
P10 = [-223741136935838459336, 14061716080321850957746462801768]
P11 = [600560387177739981664, 11803400521273993098007178520568]
P12 = [1318878437198782868464, 44355462228166611148551290598568]

Eroshkin (2007)

y2 + xy = x3 - 17843714757132874055011263134593557643365x 
         + 918347652275294697563872916161846113738275813342846617692417

	Torsion points:

O, [67742206905192793074, 142984158466394799764815818663], 
[67742206905192793074, -142984158534137006670008611737]

	Independent points of infinite order:

P1 = [91031845232245022514, 219917666655100106137993308903]
P2 = [75850129567446638274, 35829302244905203784126713863]
P3 = [-137804054554449291726, 872003877048826363953721477863]
P4 = [91062395491766900274, 220405073187266395835724037863]
P5 = [75682870370568788274, 37252885795807094056920997863]
P6 = [-43461201205976781126, 1269552720180201640215358539663]
P7 = [78298623351617992242, 35110091367943702503987927783]
P8 = [81041969416392215514, 67270935567918458208126201903]
P9 = [-77551713161158629726, 1354895382504045581515597267863]
P10 = [79109995916346260274, 42810898873578978681203077863]
P11 = [-150651515023150645326, 432853048054021516088940124263]
P12 = [239250925801094695974, 3216237854232421576479347077563]

Eroshkin (2007)

y2 + xy = x3 - 52511008932588540734880878160900060971565x 
         + 97380634788478317255437122176207105241892655674522315598342225

	Torsion points:

O, [2361149176057356930, 9861879213742511369857994107935], 
[2361149176057356930, -9861879213744872519034051464865]

	Independent points of infinite order:

P1 = [-426102705225606443790, 6510838092680261462121208566735]
P2 = [3343745689432367507610, 193150195569462104709098100245655]
P3 = [427098414979613366850, 12363720848581623589396527790815]
P4 = [1202309048647612481730, 42097986533251394009715414903135]
P5 = [524692568212638206190, 14638222734209170743127650297675]
P6 = [712538705984535824730, 20536030233502363544972515629135]
P7 = [873036429752929199730, 26776081191861791519634591171135]
P8 = [-496070042618737410348, 1163673112647665178974098749597]
P9 = [2788649868203438524530, 147095449923948584602844068726335]
P10 = [557822760544244141730, 15545559026343680710275887283135]
P11 = [-305618370360821388990, 9213219683852819728026254496735]
P12 = [-410787872703756098670, 7045028902450906395063671211135]

Eroshkin (2007)

y2 + xy = x3 + 98978303519591143407307037145293552090x 
         + 794337445180275349451540964168292632664086352466344240100

	Torsion points:

O, [968989078863982220, 29852238823111767715229889290], 
[968989078863982220, -29852238824080756794093871510]

	Independent points of infinite order:

P1 = [19182000926554729580, 98746850443195292840030404490]
P2 = [14782697749109983100, 74080627834707871017232012490]
P3 = [53052008495695613630, 394158649573849157853291834740]
P4 = [40489689825849640420, 266798317776257444240802050290]
P5 = [-3958480825750074580, 18452805598072851636103785290
P6 = [16484365638831906860, 83098133074569811029689169290]
P7 = [6324915932142029420, 40907118970742667385127697290]
P8 = [13405385824790184500, 67306657869643798869823900010]
P9 = [7149486591756870020, 43213755209129620767690827690]
P10 = [8023893831368328650, 45881697604716023179804884110]
P11 = [9508937492380105670, 50944215071843473861548398540]
P12 = [201069928400514303020, 2854781143660789370924472657290]

Eroshkin (2007)

y2 + xy = x3 + 638371365356923079870386876943741844375x 
         + 20030368902810195212582937188953841745810795518138076948057

	Torsion points:

O, [1651844044492906554, 145221782263137413170399129923], 
[1651844044492906554, -145221782264789257214892036477]

	Independent points of infinite order:

P1 = [42441636600850842414, 351530506252767713137151371803]
P2 = [41203862702740337274, 341010168386418516658793009283]
P3 = [52766998714318953354, 447925694124039097546025914323]
P4 = [144504256692915183354, 1769107267404576858141747832323]
P5 = [102586174216930268154, 1079410699236925910547920094723]
P6 = [270139188022135996854, 4461606841207189046650698889323]
P7 = [22100440326298398354, 211974440337439716831646219323]
P8 = [517512923870299020954, 11787727249501276437227068473123]
P9 = [260910836779080595254, 4236502559399149599355238096223]
P10 = [-14421048913218808746, 88460652055944670240153796523]
P11 = [12383609688219652634, 172727485338785469085408307363]
P12 = [40786313305102460154, 337514693114546744286354270723]

Eroshkin - Siddiq - Voznyy (2023)

y2 + xy = x3 - 175743527201228400282498060739701250x 
      + 20803849458555294950424936016610860362724265150562500

	Torsion points:

O, [470182460476840100, 205222534910914932215048750], 
[470182460476840100, -205222535381097392691888850]

	Independent points of infinite order:

P1 = [-108212670932578300, -196352635505106128499591250]
P2 = [-16542899935774300, -153969574222170994538391250]
P3 = [124581701919311780, 29034540708690231081935150]
P4 = [130318772010699140, 10694400725859072288686510]
P5 = [360836588756154500, 66114875395366478228792750]
P6 = [339527957979473252, 16568993223134663229659246]
P7 = [127894161601844900, 20475168349275158351586350]
P8 = [385292962859755940, 101430284116503616047618350]
P9 = [430067471555210660, 157374417710882686291458350]
P10 = [339634887877876580, 17109393234921867137903150]
P11 = [381428248756958756, 96246295335682811109175982]
P12 = [-340810441652884060, -202764264265230561863534290]

Eroshkin - Siddiq - Voznyy (2023)

y2 + xy = x3 - 4896051339671660713324400998980395642445x 
      + 131569862667060107081557431308017075692434207520531963899025

	Torsion points:

O, [47288293462684487490, 76086883185519293794807593855], 
[47288293462684487490, -76086883232807587257492081345]

	Independent points of infinite order:

P1 = [6894385196043598530, 313276725084327181402020649215]
P2 = [41954266674484370826, 2421113338004867714016790287]
P3 = [41939807561956860690, 574092209922003091699587855]
P4 = [43852374468985711290, 34581068704747807814833215855]
P5 = [42404804031524863920, 14307393272647109480425644855]
P6 = [45010326469096598370, 48833811580717506146856575775]
P7 = [44565521142886557090, 43424950194656001383104185855]
P8 = [34818672031289264010, 57514545603831871909650512655]
P9 = [30074187792134150370, 107358735211719575870908463775]
P10 = [61854634216557155250, 255698977411322583082082872095]
P11 = [51709793575731304290, 129085312405828752514049001855]
P12 = [12634565613164163330, 267819463223746304141640884415]

Eroshkin - Voznyy (2024)

y2 + xy = x3 - 6125558623193180362615556286047464602185x 
      + 184575351713773195262092962368163579626649810199900583211225

	Torsion points:

O, [41689079568070657870, 40757950130763395574099689515], 
[41689079568070657870, -40757950172452475142170347385]

	Independent points of infinite order:

P1 = [-45940662863764169900, -607476552397160344866499227815]
P2 = [45746956390502745070, 9390833953506819865021413415]
P3 = [44201817593782292110, 13269061783021966171243075015]
P4 = [47464872099446608870, 27582130906785219581025691615]
P5 = [52291101703672476220, 85122287985117144763162652515]
P6 = [32321988686146056070, 142661314956444667731279851515]
P7 = [35300886142274926320, 111031499660778377531733825915]
P8 = [40812463095703152504070, 8244957301415994663255417171094015]
P9 = [49060121235968396470, 46230810509567026759955711515]
P10 = [-2471051038465216130, -446874513976419830571735862985]
P11 = [19129576763319063370, 272756848962770288370148619515]
P12 = [9868127662085793520, 353678547454820768995395919765]

Eroshkin - Voznyy (2024)

y2 + xy = x3 - 28869472695885471661757372297624036951x 
      + 64622389720504351007385122791720566287900055948843115881

	Torsion points:

O, [1662764468660354710, 4606131096094012565783485909], 
[1662764468660354710, -4606131097756777034443840619]

	Independent points of infinite order:

P1 = [-1659604692679081274, -10390537676796644701248417083]
P2 = [-1914852673719792422, -10624597004918268380011532963]
P3 = [4109357091269126998, 3921909384565816504004501077]
P4 = [-3144281441798937770, -11149447531284652239735768491]
P5 = [5007892717428628720, 6755742471452922425467651459]
P6 = [1038004217878312726, 5981150039388949374785096917]
P7 = [1556385582455515126, 4843598619699546936644237557]
P8 = [4872117816146473858, 6294360179098678803863858545]
P9 = [2755114884223377228, 2448838642318405312754881935]
P10 = [3006346183195489426, 2236620292287813887263246957]
P11 = [52247998345704563941/16, 145531464673032264371820149623/64]
P12 = [1202117295854335722694/169, 32525186995722548118433094742649/2197]

Eroshkin - Voznyy (2024)

y2 + xy + y = x3 - 306712529628966886031813979241588447553x 
      + 2087309681193098988004390954784199077210072693310176887448

	Torsion points:

O, [7598112544108917624, 13982931922127741966436943975], 
[7598112544108917624, -13982931929725854510545861600]

	Independent points of infinite order:

P1 = [-4446536026891982181, -57993124546496639102801694680]
P2 = [-2430669404763685131, -53089223711779074610181993240]
P3 = [3898647955206715707, 30835090922196705482279804140]
P4 = [499244428132403649, 43980786644766673247690131300]
P5 = [10721671774176143637, 5598240808578970543993554082]
P6 = [11791329512931275394, 10496380482688500977432738335]
P7 = [10754986577292401349, 5713755590743574868440289250]
P8 = [-8428157947608512901, -63825127088623618174504642100]
P9 = [12467700335762277357, 14189199893989650597618049840]
P10 = [12095532095845591074, 12126735822842445351681116575]
P11 = [11664080433220143984, 9833474326630569732385597735]
P12 = [10215486011542797849, 4487868222298119078624839500]

Torsion group Z/3Z, rank = 11

Elkies - Rogers (2004)

y2 + y = x3 + 44182596082121121317135170025680399046545625711306

	Torsion points:

O, [0, 6646999028292476087078617], [0, -6646999028292476087078618]

	Independent points of infinite order:

P1 = [-30156002278649820, 4093799681127459731025817]
P2 = [11364087102067560, 6756491872572362690626342]
P3 = [-20835788771691894, 5927660006237675713476241]
P4 = [1134264920569989390, 1208031685828825118221478017]
P5 = [8907565209691176834, 26585114133655761890666064910]
P6 = [111849199886121334, 37992674604901443769570910]
P7 = [11724873521668020, 6767159346634715672034457]
P8 = [-138658831412368575/4, 12719819443574268333325811/8]
P9 = [165971060901522240, 67941788876402816577138982]
P10 = [994768217796990, 6647073075327662243966017]
P11 = [532896351059436225/16, 576457310785324883248677823/64]

Eroshkin (2006)

y2 + xy = x3 - x2 - 4536184139377606759385840064x 
         + 117756938515259070124029037981449282901220 

	Torsion points:

O, [33618885611941, 57031010525639772307], 
[33618885611941, -57031044144525384248]

	Independent points of infinite order:

P1 = [-19490912550326, -445832687088207682886]
P2 = [99694509539254, -810177524574020786498]
P3 = [44209842630586, 60179204272303054492]
P4 = [19042989810446, -195653002235648166468]
P5 = [39937171925371, 17130985069181971582]
P6 = [-5180338033241, -375655273818610196726]
P7 = [40648708695664, 23053923614463599554]
P8 = [264049839228931, 4162948942598699706502]
P9 = [-11554767109984, -410644344382289546298]
P10 = [8407403487266, 283220194078161953452]
P11 = [37535093077111, 19323726664997391322]

Eroshkin (2006)

y2 = x3 - 2902707625457545077500107175592x 
    + 1904597097766869546326844055645258034709879849 

	Torsion points:

O, [883309903897008, 5458684044929332492275], 
[883309903897008, -5458684044929332492275]

	Independent points of infinite order:

P1 = [1063302229460058, 4508156633116678166625]
P2 = [-825654663146817, 61142274721662403678500]
P3 = [-68584647656817, 45862355645427412806000]
P4 = [1030369575281634, 2763900126117107202225]
P5 = [1042423080209358, 3390046797825888711225]
P6 = [-760622971612917, 60600386362297642810950]
P7 = [164575959418458, 37833038284168124572575]
P8 = [-75434443283612, 46077456339616112696285]
P9 = [696962978964658, 14834832809096935264175]
P10 = [979980554131023, 1065853725738960152280]
P11 = [973568051517558, 1181237636592991511625]

Eroshkin (2023)

y2 + xy = x3 - 19381405000232295858279660x 
    + 32840137866325074034091731864481219472 

	Torsion points:

O, [2659787424984, 326042595719178708], [2659787424984, -326045255506603692]

	Independent points of infinite order:

P1 = [2055128982024, 1299537851617235028]
P2 = [2558537333184, 23764901654317908]
P3 = [2560149274944, 31604788042325868]
P4 = [724785670584, 4378756710176644308]
P5 = [2153616502584, 1043375878842321108]
P6 = [2560641704024, 33755458168724948]
P7 = [2582011626312, 104107876468451940]
P8 = [384922167384, 5043494376631511508]
P9 = [2582080359834, 104312168121256608]
P10 = [3706952400984, 3454453825643044308]
P11 = [2563692341592, 45752868317464788]

Torsion group Z/3Z, rank = 10

Elkies - Rogers (2004)

y2 = x3 + 4692726937524378378756566939402025

	Torsion points:

O, [0, 68503481207339955], [0, -68503481207339955]

	Independent points of infinite order:

P1 = [-135797482140, 46781315964225555]
P2 = [-150436201545, 35891470127810220]
P3 = [-42200591214, 67952721291406041]
P4 = [2327642247924, 3551854243978575507]
P5 = [5504535148140, 12914782107290941395]
P6 = [140506152430, 86409469562070095]
P7 = [397507563420, 259814927561209005]
P8 = [7162660587075, 19169656506442936830]
P9 = [73148794740, 71303068454026605]
P10 = [-102758626586, 60063833881519937]

Eroshkin - Siddiq - Voznyy (2023)

y2 + xy = x3 - 38690898580374797692832240x 
      + 71645473412331804273266091664555065600

	Torsion points:

O, [6779986730080, 10999342793991442960], [6779986730080, -10999349573978173040]

	Independent points of infinite order:

P1 = [2020834202080, 1307747476141906960]
P2 = [-1413936357920, -11114187406056493040]
P3 = [2042874944830, 1063208190423500710]
P4 = [34040558580, 8386206101214516960]
P5 = [-774541703840, -10057262105626179440]
P6 = [-204110181920, -8918193455655853040]
P7 = [4977572386272, 1544079016781181456]
P8 = [1754121626080, 3028903739399506960]
P9 = [4911027872730, 280209611489120460]
P10 = [2054214947920, 913467905280053920]

Eroshkin - Siddiq - Voznyy (2023)

y2 + xy + y = x3 - 12837942817555687537193x 
      + 559879351290579624180331147258556

	Torsion points:

O, [63825564660, 704792348148232], [63825564660, -704856173712893]

	Independent points of infinite order:

P1 = [66952731255, 686185706907982]
P2 = [-17480766810, -27909751417746218]
P3 = [57434291655, 3464011425623482]
P4 = [52042038360, 5719758035573107]
P5 = [65744596086, 158922899506819]
P6 = [65283584535, 86943389703982]
P7 = [12312387035, 20091786331163982]
P8 = [-10106969340, -26241176521229768]
P9 = [65527769535, 80802462168982]
P10 = [86318162235, 9740247026507782]

Torsion group Z/3Z, rank = 9

Elkies - Rogers (2004)

y2 + y = x3 + 87299817093221362429969788356520

	Torsion points:

O, [0, 9343437113462120], [0, -9343437113462121]

	Independent points of infinite order:

P1 = [-34739896854, 6735993625205487]
P2 = [59816792760, 17358779174601879]
P3 = [-44207258970, 952038792981504]
P4 = [6576595440, 9358646559113879]
P5 = [55473407808, 16062629859888488]
P6 = [-19255568904, 8953228261141352]
P7 = [101583478425/4, 81458264395412407/8]
P8 = [11986370039488/81, 42053694778844172287/729]
P9 = [-2123513196948/49, 833762857240311704/343]

Elkies - Rogers (2004)

y2 = x3 + 30457819633596695100179965225

	Torsion points:

O, [0, 174521688146765], [0, -174521688146765]

	Independent points of infinite order:

P1 = [-734843410, 173381106196815]
P2 = [5130038900, 406775844709485]
P3 = [-2676929565, 106184137901590]
P4 = [690947990, 175464197227935]
P5 = [291207945620, 157146639625792365]
P6 = [25120488440, 3985280944128435]
P7 = [-872639080, 172607374924365]
P8 = [2918890200, 235215923409485]
P9 = [-102315705, 174518619468760]

Eroshkin - Siddiq - Voznyy (2023)

y2 + xy = x3 - 681179687329862935396x 
      + 6877345661595357821090874031376

	Torsion points:

O, [11999853320, 656671976043356], [11999853320, -656683975896676]

	Independent points of infinite order:

P1 = [2458097096, 2284247502195932]
P2 = [-3345689464, -3019751781976228]
P3 = [18104595656, 692177157469532]
P4 = [16699524296, 398782230906332]
P5 = [-5852418754, -3265490050886518]
P6 = [8383466546, 1325103209028872]
P7 = [13533349856, 370609954059332]
P8 = [15986794850, 270791853120776]
P9 = [22090509896, 1615445801268572]

Eroshkin - Voznyy (2023)

y2 + xy + y = x3 - 18922511049224630228x 
      + 31686836332788381734601386006

	Torsion points:

O, [2348333430, 14167620266347], [2348333430, -14169968599778]

	Independent points of infinite order:

P1 = [1378698225, 90658263047977]
P2 = [-1935504645, -247103663927903]
P3 = [2557613655, 4549101125347]
P4 = [2577866580, 6168963821722]
P5 = [2592946680, 7423021430722]
P6 = [1921401771, 49217855606929]
P7 = [2625372060, 10186298114617]
P8 = [916162305, 122961848790097]
P9 = [2498593155, 2410440755347]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 3016379149728660400142350912959846785025

	Torsion points:

O, [0, 54921572717181546495], [0, -54921572717181546495]

	Independent points of infinite order:

P1 = [-13580508502085, 22621297461076022030]
P2 = [-10281458610626, 43926560316060880843]
P3 = [-8656352901329, 48659402557655022644]
P4 = [-6807472203630, 51970274832964203555]
P5 = [12548957682066, 70657929934228138461]
P6 = [13712674495500, 74798899606655172255]
P7 = [57608146068540, 440681797579709656095]
P8 = [68386202831625/4, 716148138308019950085/8]
P9 = [-199209456983031/16, 2109402682852246586853/64]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 147030510084918957720971419130560628730225

	Torsion points:

O, [0, 383445576431543877735], [0, -383445576431543877735]

	Independent points of infinite order:

P1 = [-50154557991956, 144456700181089923847]
P2 = [-52727805577425, 20869412354078043360]
P3 = [104496923780715, 1134943108177955543310]
P4 = [35732870795130, 438925505456087126085]
P5 = [-10022944407720, 382130358591343277415]
P6 = [27499517203680, 409666071038349426585]
P7 = [57526981904095, 580868003465329308740]
P8 = [327485861645514, 5938762775261107748187]
P9 = [3375317195236690, 196097798509380387638315]

Voznyy (2023)

y2 = x3 + 60671430811931370152283076408801110225

	Torsion points:

O, [0, 7789186787587737015], [0, -7789186787587737015]

	Independent points of infinite order:

P1 = [-2909261474130, 6004000065254229885]
P2 = [7344217497970, 21372889071530629115]
P3 = [365538418165734/49, 7482016308063596893077/343]
P4 = [258113994926040, 4146847791478672762185]
P5 = [2811324076330, 9104441317888334165]
P6 = [470629092629169/4, 10210011798426888899703/8]
P7 = [34411950129915114/2401, 6449014913552113264743387/117649]
P8 = [-1205481387407917044/332929, 697950472351067426941176471/192100033]
P9 = [129263092444431027390/13344409, 1517900726905104766710224224695/48747126077]

Voznyy (2023)

y2 = x3 + 19988905798337416857887750048967131889

	Torsion points:

O, [0, 4470895413486812583], [0, -4470895413486812583]

	Independent points of infinite order:

P1 = [-2640021124104, 1260444411760220505]
P2 = [-1493609987870, 4081281036143667083]
P3 = [495076126904128/81, 11487657144818347102351/729]
P4 = [-5029068819772305/10609, 4872433274311350166204716/1092727]
P5 = [133138704688704/25, 1634726880503373945483/125]
P6 = [569664793333145419/112225, 461648166803843299551365822/37595375]
P7 = [6128715173559771793/391876, 15211985511221581947227379289/245314376]
P8 = [-1212444784307181698/514089, 966187173700541562163401007/368601813]
P9 = [338768307272049511667230/2363612689, 197176718832242653058548030024322629/114911758101113]

Torsion group Z/3Z, rank = 8

Kulesz - Stahlke (2001)

y2 + xy = x3 - 101802374223975950878048695685575x
	   + 423441144264551283172186925767924238963073455625

	Torsion points:

O, [3228164584980150, 358395281084675043437925],
[3228164584980150, -358395284312839628418075] 

	Independent points of infinite order:

P1 = [7954744215743400, 342036905663438696325675]
P2 = [6842957560456950, 217349059459739953181925]
P3 = [-4372846691530578, -885996990679464617293923]
P4 = [7252229378427006, 258024710976748981568877]
P5 = [6542261582905110, 193494030879178504719525]
P6 = [6844236205549350, 217462870102024754638725]
P7 = [7420355659278390, 276782501766541023783525]
P8 = [7319642360196150, 265423911498353437421925]

Dujella (2001)

y2 + xy + y = x3 - 4213613959455x + 3328416232976793662

	Torsion points: 

O, [1275964, 170749461], [1275964, -172025426]

	Independent points of infinite order: 

P1 = [137177, 1659144070]
P2 = [-563615, -2350086962]
P3 = [960817, 408055152]
P4 = [1245950, 111922021]
P5 = [-1065128, -2570083998]
P6 = [1154661, 50052976]
P7 = [1168285, 16589026]
P8 = [-537260, -2331499659]

Dujella (2001)

 
y2 + xy = x3 - 31931594312439500340x + 69606680279441891519909634192 

	Torsion points:

O, [2747698824, 51116009544588], [2747698824, -51118757243412]

	Independent points of infinite order:

P1 = [3748352324, 50798802385088]
P2 = [-2164600176, -358584463952412]
P3 = [-2468360706, -365219430020952]
P4 = [2411566104, 81401231483868]
P5 = [3553831074, 31795437908838]
P6 = [-694085676, -302382792881412]
P7 = [3026475564, 26221716181308]
P8 = [88647240552, 26341149984421260]

Dujella (2001)

y2 + xy = x3 - 286699610291764606928838x 
	   + 59110806824657235777329702047745892

	Torsion points:

O, [280891170252, 27233824468946634], [280891170252, -27234105360116886]

	Independent points of infinite order: 

P1 = [318396699432, 10223679281236974]
P2 = [-49377625704, -270456927185157570]
P3 = [330898542492, 21763968085963194]
P4 = [137818873836, 149050012878717930]
P5 = [-50858768268, -271220261144527926]
P6 = [257135268852, 48904223792122314]
P7 = [833100033612, 631251633372206154]
P8 = [242226726588, 62264299617796122]

Elkies - Rogers (2003)

y2 = x3 + 708291392738196762720225

	Torsion points:

O, [0, 841600494735], [0, -841600494735] 

	Independent points of infinite order:

P1 = [-88860785, 81396479060]
P2 = [-87348261, 204569627262]
P3 = [-63256830, 674665720365]
P4 = [-40588401, 800890328532]
P5 = [126513660, 1653248880465]
P6 = [101707060, 1326794024465]
P7 = [-29810690, 825711396965]
P8 = [-44793980, 786391914385]

Elkies - Rogers (2004)

y2 = x3 + 136677874368461861091875625

	Torsion points:

O, [0, 11690931287475], [0, -11690931287475] 

	Independent points of infinite order:

P1 = [-505980450, 2671829766225]
P2 = [-241990650, 11068289345025]
P3 = [-351203600, 9662247592525]
P4 = [259337650, 12414503280025]
P5 = [-464466761, 6039764206388]
P6 = [73448775, 11707865310600]
P7 = [-96861050, 11652000596225]
P8 = [-101098316, 11646654527123]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 169668019404529696518779025

	Torsion points:

O, [0, 13025667714345], [0, -13025667714345]

	Independent points of infinite order:

P1 = [-428102730, 9550329752205]
P2 = [763845580, 24806076171895]
P3 = [-553322286, 509702045787]
P4 = [315018990, 14174961996195]
P5 = [-508590650, 6173629323155]
P6 = [-394073550, 10414930445595]
P7 = [656902246, 21286964203219]
P8 = [362575915, 14742208735930]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 1445887233929012173476363837608705921792025

	Torsion points:

O, [0, 1202450512049877324795], [0, -1202450512049877324795]

	Independent points of infinite order:

P1 = [-90454951997465, 840104708514152065820]
P2 = [-111740734847060, 225151798133797564645]
P3 = [178706657475175, 2674523406877491109580]
P4 = [144596524874835, 2114032589465798340570]
P5 = [542848290518094, 12704909978273363426103]
P6 = [108077862367750, 1645699170152010956455]
P7 = [-113062834860920, 24122193456850425595]
P8 = [20650256798038891980/139129, 112693729343637600569697181665/51895117]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 301359421157274208903983634665127715652225

	Torsion points:

O, [0, 548962130895450937665], [0, -548962130895450937665]

	Independent points of infinite order:

P1 = [-64069202651090, 195867657258517492885]
P2 = [-49985711740965, 420079221671105486490]
P3 = [-49254382194654, 426460522597100886531]
P4 = [2085001298840460, 95206525109820416907615]
P5 = [78719427340554, 888348970871574078867]
P6 = [2364341993823481/4, 115048798645689631188179/8]
P7 = [397919024013966, 7956612249540681975339]
P8 = [572089347821979, 13694463138586879339158]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 69138874026594049497777557929883961

	Torsion points:

O, [0, 262942720048671531], [0, -262942720048671531]

	Independent points of infinite order:

P1 = [-304609226108, 202176073227773557]
P2 = [375428706690, 349362729127383969]
P3 = [354590738835, 337228699734978594]
P4 = [797317246704, 758950091630963925]
P5 = [1511044782060, 1875964136837945781]
P6 = [859648171704, 839294082381381675]
P7 = [-171885073998, 253101993523007937]
P8 = [13217754991122, 48055494138786071553]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 9838490519720684837781572747809222948716234225

	Torsion points:

O, [0, 99189165334328148271065], [0, -99189165334328148271065]

	Independent points of infinite order:

P1 = [-2140322687580180, 5806247588882652187335]
P2 = [-2055766256829270, 33918472242343862275035]
P3 = [-749020644033540, 97047751266027171020985]
P4 = [-905731190025570, 95370198982979990422035]
P5 = [-8140878237198735/4, 300226007350806375297495/8]
P6 = [248410287230746975, 123809652558963159260357060]
P7 = [18908428758713025/4, 2718449729672018324247105/8]
P8 = [8048973258698335686/2209, 25050184671426738954614475891/103823]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 1849496130345279660785418100862767969

	Torsion points:

O, [0, 1359961812090795313], [0, -1359961812090795313]


	Independent points of infinite order:

P1 = [-1166129563540, 513541944692568687]
P2 = [14246581505820, 53790409152078382063]
P3 = [1676770339428, 2561998143135108689]
P4 = [24944596292585/4, 125058849541657867371/8]
P5 = [46551077787515, 317613235578643392438]
P6 = [840548308843380, 24369354111123441513937]
P7 = [2113241318423435932/10131489, 43964229112463276244213508927/32248529487]
P8 = [17926277208084730531651208/280085451361, 75899111823497266436757408870070864449/148229903509233391]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 152723545143455519945088904235486565815049

	Torsion points:

O, [0, 390798599208665946243], [0, -390798599208665946243]


	Independent points of infinite order:

P1 = [53641992005448, 554144754388817745021]
P2 = [-44701660004540, 251791523404798078493]
P3 = [105282653263260, 1148791752521290337757]
P4 = [-398264256963716/9, 6940101651401307267505/27]
P5 = [1245181114525830, 43940598166126668293007]
P6 = [29440566815220, 422186057746120614243]
P7 = [47025614644503, 506671872044318995476]
P8 = [-1843581955873445436/47089, 3111351479974110229009702725/10218313]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 690692889755727249719717688458280431156686225

	Torsion points:

O, [0, 26281036694843817551785], [0, -26281036694843817551785]


	Independent points of infinite order:

P1 = [-601302165408670, 21755079792172998703365]
P2 = [2943215862263490, 161821994854246571900485]
P3 = [3512870545282230, 209858111341263388155635]
P4 = [15008197011196424, 1838811219128481440382807]
P5 = [19760722030019240874/841, 87844742662657162963377997207/24389]
P6 = [-285038733249419573885/335241, 1692446697876374794572815387710/194104539]
P7 = [376811853224714647267770/246144721, 252592375868037034548050450790530765/3861764527769]
P8 = [28966585984891785381618575482476/13354393950031369, 
      161089297748451300175476234099478847731706598899/1543249963745486389690597]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 33369130174946127176878019982225

	Torsion points:

O, [0, 5776601957461335], [0, -5776601957461335]


	Independent points of infinite order:

P1 = [-13357648590, 5566486145140635]
P2 = [-5282495790, 5763828882641115]
P3 = [-8533262430, 5722566495333765]
P4 = [37407372640, 9258169254007465]
P5 = [-32169640230, 277898969014485]
P6 = [-22119117020, 4748392791957815]
P7 = [601791107346, 466876370496290469]
P8 = [17813519579171316/11449, 2377532853891025170551211/1225043]

Choudhry - Shamsi Zargar - Voznyy (2023)

y2 = x3 + 38658753177868039008353847582858239376760572225

	Torsion points:

O, [0, 196618293090617687338335], [0, -196618293090617687338335]


	Independent points of infinite order:

P1 = [-1481358798920286, 188170199654910959792163]
P2 = [5887451636734470, 492676478729511734037765]
P3 = [-3000701156787246, 107887985983983086907267]
P4 = [3954831836726980, 317041119014600830992065]
P5 = [109806228333915, 196621659936894912330090]
P6 = [321829001972335, 196703040812364393189740]
P7 = [3433278743237200, 281297339289170658261665]
P8 = [-349586594412175350/289, 943612775167625800976547645/4913]