Trivial torsion group, rank ≥ 31

Alpöge - Howell (2026)

y2 + xy + y = x3 + x2 - 1284727764113567728281797636015784768866707681415849262157224232063x  
              + 560368321454261339256859338901915312332769858684945406858043869199456710681989058863306170127006181 

	Independent points of infinite order: 

P1 = [-343746913367434053557936715792581, -31005994945795167281668477280091616990476695982462]
P2 = [739387066521976456185309925500635, 3830813927243125109605210052773411116315836476962]
P3 = [794918160232596023344129483097739, 6435806665987309972375508567418781686192466975138]
P4 = [552229821725543064455253103966115, -4394327660511778936811414681478594663009127495638]
P5 = [621111741565922967463663441261795, 1421683191996488160880565213880200774834445145522]
P6 = [672709190481033101277453068263195, 739179814158401859498464392317452223287910224482]
P7 = [821399009280840681555588652225649, -7699942068198872209289633789793873375706107955662]
P8 = [765203485098820658379256355481245, -5034345308760298854369394902951465889284652680168]
P9 = [5399595603851507511243908479925915, 388654067273650918227195921742660550010310029109122]
P10 = [752668954339285868300409092384315, 4448429560131722871335630396999897920170009189762]
P11 = [665267573831105472345147950403835, -339632380726246052348178275348181915170728600638]
P12 = [2348702476455858806523697264250795, 102466166054736815527265002023727452468648322685682]
P13 = [-485301239340016399925702578908055, -32703997667874232639739344802658052943027850955518]
P14 = [1671562150318197588290891477490065, -55528472789639333688258829014227823359584879796238]
P15 = [533792265728561313885832952129951, -5165872676030535509137532850965718653070997396662]
P16 = [8050101189939018648004785213625795/9, -304090504548899861983493167851611965128494136836666/27]
P17 = [641126276756517817380587191947425, 475393761225519880788251703870437551199359754892]
P18 = [-28607312564278696457116932283279229/25, 2883492798944948226520935391601885431994599305099066/125]
P19 = [56549444812087071231733196839837255/4, 13405577698133557868624072930089297336164276021592851/8]
P20 = [1436747523773762064481823443661979, 40991896134294415817647888149334972652525806317538]
P21 = [-5064387018792502839844249474841045/9, 897570094862437631568619794896775730629375504169894/27]
P22 = [487104273264503086273608258117432335/484, -180314887995146178717423515157548125006198119570993099/10648]
P23 = [25599342024506626201702677512314955, 4091893764954118703319829997327215311497327967339042]
P24 = [980266989498097787981788009573917995/961, -525575897314714907955968902277850024344706090034709538/29791]
P25 = [5818507410725525932023407879789875/9, 1789182708478717961552943758166081786352797832214/27]
P26 = [19197683320190396803412613672365855/49, -3712724452088495818457900090458774587078999890353974/343]
P27 = [887913128901643530184527456121547975/841, -475948997371817062287059043149845391545859899478008302/24389] 
P28 = [1816718417495680977851120075801270195/1849, 1248545591609034704022269662335921779441905481482388694/79507]
P29 = [4528903098191425769774358894938128668457595/3630906049, -6558125371552387528524400913253424783234119422079260125534495614/218787505794593]
P30 = [33880100347989327554317013369507240405/52441, -1625428840401701290033088199518465040555146308891387272/12008989]
P31 = [-11477003798992551861310315487329257523535/8922169, -244939776343182008186576362205394985355490153552116110249914/26650518803]

Li (2026)

y2 + xy = x3 + x2 - 500155818938812672789174015511502418542914025471083342999327356190x  
              + 135726649501359436415916160439700483690089900096812327554427167585750287637491352376425892437884100 

	Independent points of infinite order: 

P1 = [441117661625954589143045653904380, -966370721861153920696174088384167004962423898990]
P2 = [439295274846162564121600666466080, 886517003187467241176966098503914165737886604910]
P3 = [985302836880853196886977840812720, -24484179749136029625161725223684880698407793384930]
P4 = [583217395396577612275801524269620, -6511844290875339902772109696621341165318036514790]
P5 = [-786673812301098841241104416525140, -6507578029130630825844645880529285982978609065390]
P6 = [428960563669956826847322586045660, -333683657402002655519064685269202950416520545390]
P7 = [-509393320283524104563833602425570, -16072476848437556337820465759511194005689896470740]
P8 = [54106696280894100129460921352440, 10431839230807707831353056689103726780782929949010]
P9 = [-612495682259503757874089381822420, 14570229421017750644567008945310403641493938139010]
P10 = [5762722002947756207237356817205820, -434312675770200955443264086964278478067914152995790]
P11 = [437941634092629422214927504568252, 825637864323025269813779859221308476693774287026]
P12 = [1465405043856260306354133575801980, -50493802407602787104439680821856002850633766494990]
P13 = [389657805224680231788014068077580, -4930624689948230620781086641734028555503360990]
P14 = [455853265068379115817245008669180, -1567257931027448110179857333796988943288432782190]
P15 = [-519284336048836633758550355662820, 15981911019363792332260400729811107339914317933010]
P16 = [38795885481082215046407714677453980, 7640237118387974568130268540309778087253055594901010]
P17 = [373398452551161154823544125027230, 1015305033368293887710450991503516808921610265260]
P18 = [-10619823959707010297074018411221380/49, -5246281119771603625159629945619549144220276353133570/343]
P19 = [9060332593397113526347187394213304/25, 179588213108921848155701032939531374832048533541982/125]
P20 = [-61928766271365829538965992877938140/81, -6151736300985366340415964434083637075462290792654310/729]
P21 = [-1056489198334749821805630139429728020/17161, 28989445503569025024405855214939551446091957215101573910/2248091]
P22 = [31198901717404287105419478628666332220/2809, 173919531799939402604858942235119986268513717213500550570/148877]
P23 = [831098095331436820600938629452730146780/2193361, 2544124336349274485995869268733330561375939049622929897410/3248367641]
P24 = [991544486307150542901055136032411468/4489, -1805264699651082891515753358330051695299705128081230042/300763]
P25 = [-5497536451147064550636543200675996612/6889, -2952213908678812992657440509144645098923380152634930618/571787]
P26 = [6352866942420018916479041917313786407420/16589329, -40083767594112919488376731292066214280047671870477742264830/67568337017]
P27 = [149343976658259180417615426346383784036/326041, 307828302500053303287169552634293576106442453989008392022/186169411] 
P28 = [325102715125918025247878485334263194448/66049, 5804347035658363760935293992666775160668385265744746078718/16974593]
P29 = [-4029490102946808240961260471329184540887780/6357351289, 7134459487365995435260585003812646443399219011553365170341943370/506890690325837]
P30 = [9384020793859891045482811091862200975775280/21936868321, -738127137838057445573045146849912304327355944610805755155540690/3249091503891631]
P31 = [54226577442542070628258988594404494173587180/87633168841, -206187080609125630018990227991760274111839699010826583731539537110/25941959338832389]

Alpöge - Howell (2026)

y2 + xy = x3 - x2 - 31345484910176375196540418777187943071895696555975520575563754x  
              + 66974244086367930138572439137439687631297215201959427754054664641394923903641564905062093028 

	Independent points of infinite order: 

P1 = [3529742779969213731891497041717, 556872661776590737280086339127299575176284954]
P2 = [2720856310035824225122027309267, -1352901930656683663359757961025247060685136971]
P3 = [1898069470179602410153376367967, -3783708465094207800903723501566186682943528796]
P4 = [2654495402239810431020248076752, -1572352106736555163783329144722138650427976806]
P5 = [3510655448291638247411649354952, -445901106421781126419510904599713310372776506]
P6 = [17985826888476467471785315368592, -72948162134981389554959728090403389230973195046]
P7 = [-5182903369779258913481015168783, -9497851720105733498120897524665119126207999546]
P8 = [-1759012691035891400816930210783, -10801330638810812254177787610110756820858473546]
P9 = [5436880849835847504978059030467, -7567364108528584813547333458360162350504464171]
P10 = [2221259091128225688858508582597, 2882269112718112073426082172989503462140404049]
P11 = [45587304550949432341400159348152, -305577585724134866699149169276685388403380483706]
P12 = [7035427487516001986958023472892, -13952763178461678674306212961154344561829391046]
P13 = [4918121921505544395801033924967, 5636708534786846929636449657391717057957230829]
P14 = [56074847011380476462553352927864, -417888003206882852332068965183258003286852329594]
P15 = [374161484326278583014818939848048/169, -6378251404810393858705838972385429140216791704062/2197]
P16 = [636380437609344131161644751408/9, -217275681415591179964887173602999744467822009442/27]
P17 = [82984041258841058749044940551448, -754269874341471472933419751590609238741824754334]
P18 = [-4045996706092748484290578229968, -11294451966015046854216725585108321848503942086]
P19 = [12211462080437666231445555833428381/144, 1346563639917235353508012932377886590490149994694685/1728]
P20 = [-6103455857683629338449031332223, 5560848099237512602998048291089204608516828054]
P21 = [8691943680580894697204697976497727/3481, -424583381995136951885780234923410500343971075455084/205379]
P22 = [2381312271890685012304620296802972268/24649, -3668683099618008097853256735906564130428244608155663798/3869893]
P23 = [-16407275549207377511941517357233463/3136, -1645161558026801477245229085676350209861437973889961/175616]
P24 = [-50929713331070474648111205595763/16, -742230311081736290153353988689374811745166974669/64]
P25 = [9821310054870402393719948712066928/2809, 51696898235207614025436339537558557311273817472158/148877]
P26 = [89545960015902622710515055296321/25, 101027215288220029692844808859945593417286579366/125]
P27 = [53843302322314695516933963789443776/529, -12475367615733889408509845028237300894493180687255450/12167]
P28 = [77786370007087035006721640203463053/2209, -21435660363088187707980744134491249564893107341553233/103823]
P29 = [11440560695784386417958268358176263692/1442401, 30859436400376679630849705017581384722041087543530498354/1732323601]
P30 = [4755522497199061887281632526285930012/1692601, -2293308202905213200329042090407572905578461588405012046/2202073901]
P31 = [312198561204292314758925860004535822147/106523041, 583026386401115722574088755488616369513145839776819686469/1099424306161]

Li (2026)

y2 + xy + y = x3 - 367647057132661536131605961085494446124192297751927792319003x  
              + 98299706024123968245526077311124299338038232423912535744363101436934755800391181044000522 

	Independent points of infinite order: 

P1 = [71105554675963492526000845558, 269290675255312661174603378631020782861321040]
P2 = [-29704042557861724610106945056, 330445307249640665572931018435703810679069958]
P3 = [175101426740732876260582241005, 198224319312459426980672090698178741258477618]
P4 = [248378798255456892042084921013, 149355158680850764630077956833029679160249216]
P5 = [313900322947403427672774580018, 117579105889397601198987345355815654946667156]
P6 = [333644300759445321399900785433, 113036057509898777135980865422754832845580526]
P7 = [337344195504174583822768511518, 112544426509569291513494634176313983515956200]
P8 = [391217840641677214394952534118, 119775022330853746321023953719517878370351456]
P9 = [401963771980612812461862639853, 124362992830515375551949254364908031216155186]
P10 = [464097733011782059104565490608, 166240855799271737505324188103123394017945590]
P11 = [475134242728882677950838093241, 175729241082718683415630197857841349279480910]
P12 = [529405211752483370638248246438, 228126658076586731082043695240804220559709241]
P13 = [579189497967497021932404809974, 282236736482276291601963258814603820401596857]
P14 = [625994455340224943038994586453, 336842016882366329700990861656802279403168810]
P15 = [749358867473339221965452104495, 493552567740834272838256207863808118503287806]
P16 = [797654601729686318207740399225, 559065616589325930326866108761173159611508216]
P17 = [837492018316060116985328636506, 614661833404508719417509841535129164548977708]
P18 = [-107722374782128106191710128387, 369666738848101167311314042281532540472062370]
P19 = [-233000608579077662642766453614, 413898849914606078624628944133861443516843060]
P20 = [-660994466322383342336243257631, 229161184819056810524321051792070900954383558]
P21 = [-688879569869957765431042713827, 157012629314028111691246563988770194336931680]
P22 = [12634572579830709787101640264696, 44859120442512732892230119991041047323065577593]
P23 = [1703911033642285521501475481995/4, 1102343813623627845916169407198261090774939219/8]
P24 = [3652077712422208611593320763671/4, 5789523681738628554391589071533545434744586569/8]
P25 = [352026727500770729632317877760971/4, 6604701297779016008758035713310404474080024860031/8]
P26 = [1776743818924411871583988405023175/121, 74829595482338766924870171133618492376860560105646/1331]
P27 = [-5599939107511436010602903625091823/7921, 49159336464846881946892694393203461251199881716504/704969]
P28 = [32940950888329124567379444930867961/5041, 5953929118344485676637543868016722944517491049985298/357911]
P28 = [67874840820182268597647907936582871/502681, 80581032075217681677822192102437784338628376179256742/356400829]
P30 = [6713014538784551199121920448085296653/237169, 17389121945217432551592433873627102231449587231623719598/115501303]
P31 = [-316408858372840476395533341767285708/4748041, 3621148460240821165722126549633182690560042357051383059/10345981339]
Previous records with rank ≥ 30, rank ≥ 29 and rank ≥ 28.
Examples with rank ≥ 27, rank ≥ 26 and rank ≥ 25.
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