Trivial torsion group, rank ≥ 32

Elkies - Klagsbrun - Li (2026)

y2 + xy = x3 - 28623928991398919165500467001947502452240040593445344054331428726899413x  
              + 1636638097022027963025086119463360482368875730889253334875693440181143871885452358966522018914326964558817 

	Independent points of infinite order: 

P1 = [-17285086047619198200275927219245378, 46111179066568373276607383102482144371741815219099939]
P2 = [68215181559115791286912139413731752, 1215745994536280739541523550709421744253343778830649]
P3 = [-1387344455679348370056438549185082782/9, 1318887523853880583666480399346070475437416669116157773/27]
P4 = [-15173111738907417143928794817757678, 45469318950895008100110363225734405329931508848152439]
P5 = [63927798059157219113420682804111698, 8248098441085605177795173740047151713081679803826295]
P6 = [-175920256516773927216987811704653998, 35039949672608908854097547246063300466459786974561399]
P7 = [-93349800702048191841766577329428814, 59120290116614451959806566061007034385548422315557655]
P8 = [573754124312442257185948307163664322, 417240744005751235417966893171909121015622127878014439]
P9 = [180006952777798089015773645901232682, 48133228635307144812860154882618375388961803603653359]
P10 = [160321396413246682065277325922567122, 34181028216507334116572719306351091991947961215512439]
P11 = [187998867721967795592534802279222802, 53850940566565683134619753906882886861610126637377399]
P12 = [66617012700760176909764459672180402, 5043034032676679080907370401847501570294251282527799]
P13 = [20473830338934141513040567654006703/4, 308831275829136886260134068291679983315164149145219097/8]
P14 = [-144242496519268397216179549818488068, 52576900190391597126212347647253606266517688605410359]
P15 = [127127961818398385902253516853523682, 7233068110664577625624691797727005745748746428983959]
P16 = [128297156333802580224247307136843998, 8721305327064650980334193697546159894082981290426307]
P17 = [58654760853591405256596003080713082, 12629454082754085813424972634967232920799950887988359]
P18 = [264668488801898558937265106101400402, 112252649320378317400537269027978355118572889895583799]
P19 = [48875938435594063723701413074480706, 18824833359492620836802938656259778030946627675230039]
P20 = [-46751488921809892782619440189415918, 53597244206965961195885993921166172011561612233629559]
P21 = [63212355736975402066975330539562322, 8935108690519176550367540044314374317678553998365239]
P22 = [149965180266399526372714179597820802, 26771171797667975348875052454770449653345087625477799]
P23 = [-182854634263257276680050224594247678, 27509307978199386953099786312866803703357383022035239]
P24 = [27279643140022388956147032953143322, 29598791605568521450032539172535594675573347618078239]
P25 = [152646461785987818202823388078235682, 28707194425824194229749493385297886276143005584430359]
P26 = [-155412007773607516887022658957572372, 48285514471575161902643629074973045338520277417757335]
P27 = [-136511375279649473687107598119871134, 54774008346958134705142692377896672040680545719542679]
P28 = [-22380805593714240644839255686446314, 47603089514353981823409086886326475736479031451520155]
P29 = [60968847620036686567594611958700402, 10867545979063818686920688106980677205378178408655799]
P30 = [7632386621288771404751045484484178, 37664490123215897851101937789570480710930615542924087]
P31 = [23257921216948228220425920362648882, 31360579392942811665262265279771839971273135748081559]
P32 = [-151470070563153081195648444471171118, 49971003620032674130056543677631999602019036919525559]

Butler - Combes - Levav-Porat (2026)

y2 + xy = x3 - 2683271419743643911571415248437680644411114074492848524217257830790710x  
              + 55749809625327992708980272827639998709495346060890650113804084739761273223461464537986168161839288075172 

	Independent points of infinite order: 

P1 = [31054910854901946139918571632119564, 1539689403651685346330789656185100198921467263416698]
P2 = [20952306801909168595782926761245834, -2954171406485712079130329634080185644332588379247042]
P3 = [12361387083983954671656103675053084, 4946687881262216020303730277311598480696592520956458]
P4 = [33504051868357454167780070506794504, 1859667464679456973126770462123852296472486748717818]
P5 = [-38389350043541337258477183948726516, 10108553326471929292549752944133604199513584762836858]
P6 = [-43166271072938544515041939716012036, -9546923824506415798701246324283422160761022579799862]
P7 = [777094770402319651265429240805312444, -683549335804666840717134807822356278795139554323753462]
P8 = [66278467767637107113648057618019084, 13002195332789902688125634300309606935003918343278458]
P9 = [-58688191259814055184398397717636868, 3329593988505965147249900431977328072464209445798858]
P10 = [121132565109092386903327602649437888, -38834370066574032023752585741798025861050993421509914]
P11 = [3186484026988584742061538341755561644/25, -5280317737465265674794612824355930122622481195037379438/125]
P12 = [17521245565803419906996015763420684, -3756922973738973981664687547664638138272068126404742]
P13 = [-126292584116092845138387215417173089/4, -83520540824252738731123204396240488295279052719016511/8]
P14 = [268746408013312598521711713030911484, 136911433509290961237468661129543448757677215620563658]
P15 = [79431661143500398808243047525133784, 18541260367251618656234004436903706137123597578082158]
P16 = [43493303271843225945930082231122834, -4617395067168173573725521461897954897757040583379042]
P17 = [-1173049837614126255234588451051156, 7674360335434354879654680037302528947618481604423418]
P18 = [162796915001203721825600240986466388, 62717632003168392937188206524875604270241364328356186]
P19 = [220040835783228531259797400101878751/4, -69101734414311129583131444381896546975609866254978471/8]
P20 = [30587943443325349741163594931275874, 1514204974645515240692859135325060640323033542740928]
P21 = [1898828962027536132611131147672096116/49, 1082572579258515083110380110605262621608720813937372094/343]
P22 = [18754467558750246388369981403130709356/169, -73677126860868954200889878862266522802521612602564622334/2197]
P23 = [52084204881606293081216641165125213516/49, -375450637663477861875368981074430152335426759437283226506/343]
P24 = [291705290333348540940701876857956866604/19321, -11607246764771587077733960028404496167939984064775659682738/2685619]
P25 = [4820350595242300239324985537432110275916/18769, 328354650712128467140073719578533252934985057358449757575914/2571353]
P26 = [11568520176683788246770787564897169004/361, -11210037093029122705284774864269354707915311113774343618/6859]
P27 = [-37843037892509901974920803020429873556/841, 225357442787458100715148571765866738281491338063601697762/24389]
P28 = [3141337155766669840578453288811227862561896/16507969, 5380751028283531894324944153640098477283881838494106370472229026/67071878047]
P29 = [2152104672056579469632588626088682594356543436/59571629329, 35502088786718441044239122309194307716781998171350086816873599234986/14539826285217017]
P30 = [-203638427529714146842724424062289353802622203636/3755886636121, 47088537999748428133009000026670074957448833861736738910459501476458798/7278949615555495331]
P31 = [-2360551738460929437576444970789311904350828/306915361, -46853458215677255185448962761722711514651367835719041470627677938/5376850209359]
P32 = [1743885406146398385426868478560246502197732627836/242749705502329, 22957411843082952727598636198204269897681482061670229656224487170576992414/3782144065850535314483]

Elkies - Klagsbrun - Li (2026)

y2 = x3 + x2 - 36300261496678491655403474595672911062118111090068419850096188425x  
              + 2681570521104550428683375705711866392405455703802435221668128544559353811070192017100657661506875 

	Independent points of infinite order: 

P1 = [-217737259215735501722954156451125, 512515675895862413593611951417625523258740226850]
P2 = [668150890406211088561600057320925, 16634517993387136840613136490762274323455344835150]
P3 = [116813430887944439128034400634075, 187552155276556166771618756677678557564141959550]
P4 = [449728649852048501487403879042150, 8792983401314113149444847925209206757020155187375]
P5 = [-219608228113661995238520957706475, 249380565082687474418586718401091641211559397750]
P6 = [183244352172882810036784534109110, 1477437559921123492824176147085062209639962629385]
P7 = [2604161316096319780774317664240045/9, 109354107670543533401443811747947322476279132010110/27]
P8 = [4966814256926269628804693798423725/49, 71549776449844175523700094401467059352526601862750/343]
P9 = [85467144329502865658986087998175, 450995493246330022001774029895651914050821293650]
P10 = [165705717076075735841335348563775, 1102914766209323492052464298825895036981496880750]
P11 = [125536790188551730892118296468575, 320854263580030606037601011932424436721063781550]
P12 = [120259694994630458102974593983965, 235276997403922850701673336858709237669189990990]
P13 = [1844376851987684027832307919265025/16, 10873307488177873078866486433621271852028111707625/64]
P14 = [148504878861319645582809498076975, 752254272250530029183898346750132723650838457550]
P15 = [106971285278198652343631130734275, 150139578301567696486328883254207503357599312750]
P16 = [16558236667467718297326306836725, 1443967471249599745501707645952215927930066820950]
P17 = [113957635835277631684913190346075, 157389518093054391587635670518583812412141965950]
P18 = [4554452880990451484155920745685950/49, 114040221524282385220555559725198081916370590618325/343]
P19 = [111004337703058574485441983912643, 140980055333262477802087075687215347647935226734]
P20 = [-10364712789501990496668445583037503945/47089, 899555833989613875274870846451042972453177729153940210/10218313]
P21 = [102984275693856412610299976707525, 188257808011878568812709221690326072988901268250]
P22 = [-57547260517332652632009383818391891/841, 53684061402178371847066815541703902661630966278142850/24389]
P23 = [1739664614918281071037537172714596375/16129, 297153893174501427656737614803486757889717766021554050/2048383]
P24 = [318431034812169127154213807419724541813/2114116, 2441965030673034608419119458908818223802742882096210100401/3073924664]
P25 = [-2087501578700468206743854441900061962/22201, 7589186883381895262319091941454276131661215554998840221/3307949]
P26 = [59122280139467037951903339518866935/841, 16845925123924944506804849536125696273654569253127590/24389]
P27 = [-10876163542763051503053293843645385485/73441, 43647126741280520909428014166444526227955281160650852590/19902511]
P28 = [112802282734952061977301288658975178797525/799928089, 13702546755443931093264550767303684522241191406902952430712150/22624366141187]
P29 = [9397155528237503833607187345975937150/83521, 3550776431873725641389226699487109989656090758353984125/24137569]
P30 = [4575135750415992557625547258399124182975/50027329, 125943731285412412583427430318356261398082437646391072392750/353843298017]
P31 = [38782612995915587861002824789433322350/69169, 229094939017000693018045144198834922228649306785355655875/18191447]
P32 = [39735786218786611005216242414579187503005/730566841, 18398073134506615179799286813174076909136917914584541209015470/19746491145389]
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