Trivial torsion group, rank ≥ 31

Alpöge - Howell (2026)

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

	Independent points of infinite order: 

P1 = [1263245291274938020853531555751275, -30875792151319155023549502085318937060164541111118]
P2 = [1509015114241048221368018731012091, 45364228052610936059681767354495712041165503811650]
P3 = [662420427938855184766328802924635, 95054453411041626474786779983789153341766444962]
P4 = [-1274064978411953403919045775834965, 11361326404274821361376855837501975916280315657202]
P5 = [683190793048161716166372132552295, -1238229222668474322829449299990459843066115330978]
P6 = [635659300182584389829032012909835, -751939018017306841014926132214353636475342351198]
P7 = [541154563406718311726866801998739, -4858827251830739778680340725627199824356503109862]
P8 = [-248764213201171682042751874834695, -29403540252951684175535897243155341955795212105758]
P9 = [1772363559466530608913055731013355, 62055142509877943380345910348057784968467160623842]
P10 = [622049681035694365885489857478031, -1379616094770646093119416251171251161705912428410]
P11 = [-343746913367434053557936715792581, -31005994945795167281668477280091616990476695982462]
P12 = [821399009280840681555588652225649, -7699942068198872209289633789793873375706107955662]
P13 = [675242449916499793171390350166715, -862549312102926165170271654828059658606743329278]
P14 = [272992011693366534571309684756731, -15165504792829176871974957060801880450175014251710]
P15 = [752668954339285868300409092384315, 4448429560131722871335630396999897920170009189762]
P16 = [89431696367926552962484585958315, 21123167740148319234230516075010880044321068817762]
P17 = [598411815312243379807959031572491, -2421021411595123693318302691990458888164586763950]
P18 = [606309803233922015810498204451515, -2076484241323086126807851625127741561250109759678]
P19 = [233837808792187868132006939409965, -16514741167716491660784443791196642149254800680278]
P20 = [646611139369643576819246499294339, -31825885223542099587515180973775322358863263478]
P21 = [493614052227906856620900369900239, 6817631043387865785366293808335080293095315399622]
P22 = [371398150004535415769267101995970475/529, -25673671464070329706527254671267379841559771253806386/12167]
P23 = [6648342230547707157192362267504217695/10609, -1279406283632646884188989792501166077042020059186901946/1092727]
P24 = [-273924812714221436173405491394011989/225, 60373346051110255663733961669667262981505391909016846/3375]
P25 = [38368256409131941134523074897450035/49, 2014359005174955519289384851469923497847481729757766/343]
P26 = [120971002303599578436024196003831811/169, 6022976189817356741598683668048277209447327349873274/2197]
P27 = [1393711818959871147732895669252112453695/358801, -50025639931226124235079066472628859085108106242215846185882/214921799] 
P28 = [33880100347989327554317013369507240405/52441, -1625428840401701290033088199518465040555146308891387272/12008989]
P29 = [2414006859927475817114231297593193376132555/6576723409, -6271312273744705023066953347310446139544895013645275806533834414/533352538299673]
P30 = [1851239518361586157919581984143016220091115/3122686161, -464653340281388886077120757283748710955475003244028385364438798/174498825362841]
P31 = [-2702861639436925505489655192302807733068149/157176567025, 1503880072281146506820107479051421413520955451434212606442373245934/62313435879896375]

Li (2026)

y2 + xy = x3 - 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]
P29 = [67874840820182268597647907936582871/502681, 80581032075217681677822192102437784338628376179256742/356400829]
P30 = [6713014538784551199121920448085296653/237169, 17389121945217432551592433873627102231449587231623719598/115501303]
P31 = [-316408858372840476395533341767285708/4748041, 3621148460240821165722126549633182690560042357051383059/10345981339]

Li (2026)

y2 + xy = x3 - 68906458325887563494253912743860596601427440743924636890868535x  
              + 219726256542658179767501421714589587209898779390156657938983977041099527263435646417385213097 

	Independent points of infinite order: 

P1 = [4492189723065914919256867882934, 914724714768630901238259806376949142963232533]
P2 = [4567121567039911266825199317014, 534643646961232792435902595697216090643639413]
P3 = [4586421283438238075388908357654, 410821791963280157575133208491598963338280053]
P4 = [5037509468610356621679378871334, 666098792397671427124894554496634456754265733]
P5 = [2659630394938362489510492764438, 7434634969447538378141678087206140041567980661]
P6 = [3663881909579733757226379828044, 4055267373636724119245464691819620518768714863]
P7 = [3677812647428541656906378144054, 4006049353458969139310786882901444065827897493]
P8 = [4098341294835689182470629375254, 2482228482598568995384324738277746299302465653]
P9 = [4164965728556382418786791819254, 2232180041009927043930804781152182748288551093]
P10 = [4210076388396811999333015844534, 2060904208652281762820569092863917211521306133]
P11 = [4417431261416015121795299132054, 1239813367034742170365682325542949695429486453]
P12 = [4431929075218277083918138761814, 1178834426993205829246391686760443853789620213]
P13 = [5105676126910662784165003206038, 1003293868066583439325644633458295443785691381]
P14 = [5176341638430813588645978697934, 1319257072366563187890692056699670017335927533]
P15 = [5316559979035457788289474404022, 1912540767664534490942467385279679856808556053]
P16 = [5324224606886431739743202668022, 1944363137574878874350541396985641121979088053]
P17 = [5482287408296083204197995043734, 2594995395099938359808519242326412410251570933]
P18 = [5548697434495430663915305345814, 2866828288411478685823776455429234788876484213]
P19 = [5615878802954628697457810529302, 3141664450647791215948876141974015871913938805]
P20 = [5733477463417257660942569682134, 3623241806674604699060111526578454755278054133]
P21 = [5874686722234872651737487188438, 4203481938879821615647418800850411741692040501]
P22 = [7017936048521021994702580091414, 9043699426822951025499785852469862997523101813]
P23 = [757752158220185258826363604934, 12959449505271470233127111406483601080657066533]
P24 = [973557344004335969787009680534, 12392119236646578470308138231491128734565177333]
P25 = [1013068729907318835646725610934, 12286537312070329678298138251534690134629856533]
P26 = [1639444315586958180395107629614, 10543453748450346001731175965059489249608951373]
P27 = [226164351008217133335210359392886/49, 36401096114268784547188426613387846966845341059/343]
P28 = [123713595815943158865232342397366/49, 2697460873874820084336950933813275583929181950019/343]
P29 = [3708780681953243884687375900371494/841, 30996726003060644535638723469585545194108289295337/24389]
P30 = [-1732275322545229786628751579745114/1369, 884459401907288495674534376294947971656404061298689/50653]
P31 = [3228375666134932050157146796346896575014/650199001, 159093427395242901986547740315901913477350720665857393087/16579424326499]

Li (2026)

y2 + xy = x3 - 890388688605835123536218905979825104833945491596640055500x  
              + 10383942932786635402481776305650782666752061853846503229365337279722917845189552250000 

	Independent points of infinite order: 

P1 = [246903839132700267388401700, 3188121327502976252139143196605062933291300]
P2 = [13735221956016104405303357700, 863419438176752476255018850716626096547800]
P3 = [14204911884288920776530111160, 776086588809006874326947032147663239914740]
P4 = [15606692452157860482383562700, 537796084708345829764729957721489621155300]
P5 = [16076783298623407312658506860, 473930325000169644749358625919749076165040]
P6 = [16087957805670676396343433100, 472578999453593319458162790925371890592400]
P7 = [16296061295293784603481077320, 449135415897484574850330133995494621848900]
P8 = [16568411683186986164870625400, 424085854719631496430158661059312324384500]
P9 = [17723940962630179680284595520, 412930047009402888263979581650058341846900]
P10 = [17996478501037614020510572300, 434349874342215936644201522079134805974500]
P11 = [18122883512480209908403040680, 446981540313852638736066211072794799324900]
P12 = [18212493410544071231825107000, 456876662638324809692674773179921159372500]
P13 = [18271753949455002106834343800, 463824012366257374286569788260827195424500]
P14 = [19252025532291819863674540840, 614603357423984877890516236360887281353060]
P15 = [20238538484311092098154369130, 808363150514794040503421186767760903782720]
P16 = [20282353822643527985937346960, 817554297359845332625147738159621036797940]
P17 = [20332342845982824776529564070, 828088207220924971417804985763806553479530]
P18 = [20588330122500116062946060056, 882776529757234040057811783525664691035252]
P19 = [7532291938414538664354033250, 2025986957543122490709466896149116893598750]
P20 = [-6094929823737951738016840460, 3947706171837435510458690137450613266925860]
P21 = [-7096526865082153033891718600, 4042922725843764217265650579818226813734100]
P22 = [12444459332239272339840307000, 1109388537402127269190354396556098206572500]
P23 = [34580845142558597889966569800, 4576737852018557251334833321521881051474500]
P24 = [-32515929227128769161036698920, 2226461999214643031039151068815247843398900]
P25 = [296344480861340216491234297000, 160535073089701490619497843844349961107862500]
P26 = [207412012485989609655327802762000, 2987107509234258206509163890577324199320159552500]
P27 = [6046131034045634958520968103000/49, 14467779933557731713359418423192314932570187500/343]
P28 = [-153683116569304359031524893746280/4489, 258811059320796439530022906518656944919303232700/300763]
P29 = [-1517602753169735672452174701399680/54289, 46355405876566991674405005574607099727696684765140/12649337]
P30 = [-8785220863216662674597365083088208/358801, 899237389388525394440521996088244363745469013283244/214921799]
P31 = [721348849592570893354406666356723600/22667121, 407776444486702851074776093633925824126775779719684500/107918163081]

Li (2026)

y2 + xy = x3 - 11164664418280499372340781211856561720993310743403866094775x  
              + 452347706083137132420429396490299265309643996076020970449633416838423618553239978675625 

	Independent points of infinite order: 

P1 = [57218701078468290567741107950, 923568580088759083659433202655335451549325]
P2 = [57224022109916449505041926550, 919695112330442121607598807179887867716725]
P3 = [57523436975521346388241735000, 678064401069994130106748428957805381187275]
P4 = [57546213933925068818878625590, 657013650643863507167242685761715318808085]
P5 = [57897813501492133509238909270, 143518287370045689715077112918587166571125]
P6 = [64050214586358567076298676550, 100376293409116164998985858754099992056725]
P7 = [64684647195444638814916913950, 901438413696317978028864414422577054017725]
P8 = [37077068844755007482420269750, 9453298700320956202628601201697519860152725]
P9 = [43886164346933538622874233300, 6848208534013988304616349433001515275924725]
P10 = [49151359380403322417165601220, 4725662461876341018416726069831682053008405]
P11 = [50196540142499562904507577110, 4289525999248650701408874259320467549688565]
P12 = [51851457297898121916945494550, 3584696110694168466412919345766145544020725]
P13 = [53411427985328368084157120950, 2897951726089820534310832298101275428492725]
P14 = [54579236852281001612937072550, 2361049392070935111801315184210079721998725]
P15 = [56702668689178319939465772100, 1261523438320548650085592804546481696552725]
P16 = [65190430587342162428727232780, 1250684346759681660263665875324840010667845]
P17 = [70200200833874728928058290950, 3812794098509529395035321773275013483242725]
P18 = [78192176813052471731312592790, 7578030891390922630306934736457079466702325]
P19 = [93978290811800058996255822550, 15268277319273297996130082767294664260748725]
P20 = [94106417809268577531949553800, 15332620444951331877200659633365984813252475]
P21 = [-45233448582802899822412807850, 29407710041179746156943113088789995797252725]
P22 = [117668353964875603090021422550, 27709879157007787033767586753075023877988725]
P23 = [174532968457358867688433255830, 61808790399703501589994611900411883664171125]
P24 = [725866602420262089962623439830, 612204682638379303969788056474269643688011125]
P25 = [1605301980898502888351062320166/25, 55910741298205010320876892715259819597973921/125]
P26 = [162023228062974589165719289126/25, 2437540134807440647711389546592060377298387361/125]
P27 = [56476862152013802903720661787590/9, 424369987092196551252672984334701424261969514455/27]
P28 = [47482272603709913602994373432550/729, 23901043601710841191613139988555366932663306175/19683]
P29 = [809291070847677074646412602784774/5329, 18490842696643555933591026214888479550472543917309/389017]
P30 = [1506462024755672824838351784648550/38809, 67285612330260818405414543012565588649294818824225/7645373]
P31 = [3658645971983415072918087269882590935670/65089786129, 25631307625279300828780412245404217970386124985413556317875/16606161865733383]

Ritik Jain (2026)

y2 + xy = x3 - 1322551785398487139025836657336851201005130805938072030859332427256x  
              + 554523175636064067020051837648138504981185208833595836423403320180559080489913613598985487888822336 

	Independent points of infinite order: 

P1 = [508086172393280415011987766008800, -3703530544753798414431407391405047135020381607144]
P2 = [795745727792954162757736998196336, 2446098972285446177882732904894132542675541641992]
P3 = [858486927121752049722573328463920, -7199614884746283286836440582530735179525470649656]
P4 = [-1215040807695562653464902499979152, -19175073926708457161348059542366039834567079750520]
P5 = [-874637434698074038959112273174784, -32282921547347307854382528523530595401249730698488]
P6 = [-1304715260327514218958291162425264, -7686578339479440041395745473006473810051069665528]
P7 = [23602811212429599986052336224368, -22876196897220482749530643189431187580590856610680]
P8 = [1173428260601068695769314114237784, 24866376205049725653390210146628948261462879957448]
P9 = [349206995094052980139237102575256, 11630265315064029590510946180143100649485885086152]
P10 = [321495898505956772436643839881200, -12749819317974161526873427261516758092917832599544]
P11 = [58778542407188866808174411281635166/25, 12773150411412783535333891781915835158233138358898374/125]
P12 = [-377179172892290101027001833086914, 31618081818062612998988895000426790095270444445492]
P13 = [20783548009339876269525505206294070, 2991762477564244422491268338657618988222943459017844]
P14 = [-5260430241186803514045311142826301/4, -35174644498349224600775932948681428967348150600909/8]
P15 = [514584884589503614934603674900048, 3196695480180135181096448482705353676071589948072]
P16 =  [1354096863032183137945370766876593296/961, 1149384273433861814904921780271815248811849512496888472/29791]
P17 = [31686012779255653247226228834736, -22641744538074982729600642925763805079728451689208]
P18 = [22562571496558578651126885326762544/25, -1224842373610551864920176427402094428778115674492888/125]
P19 = [42282641930787647343807031522744816/25, -7021939849843218232479986370976265244877974960638104/125]
P20 = [-1317978150828833974149643840931144, -2863906842444062311896991394429754166838750910008]
P21 = [22137401862757209300373089213731440, -3289381825439734464023719098811527939751203491601656]
P22 = [579491758164737106605754443418822160/729, -46180215841476132461951837895766473870252400479875048/19683]
P23 = [360080616986562768729676108290450217648/383161, -2822244378071460171835242686116787858579374197438933786920/237176659]
P24 = [1714511020712679975936781513217991600064/1408969, -45710455851505201554678923692437978739828663241539349509144/1672446203]
P25 = [-1378142485874190468896920896616038776/2809, 4904577454361682195642617455261021828114538991111176744/148877]
P26 = [3431223488566182644439448188574564999/14884, -29385641456875197798690740016713320933960424706558997759/1815848]
P27 = [46790469265850874603609653475948194416/58081, 48301317403133428986789204668728236275395739303879316232/13997521]
P28 = [4576562091632019392302851552424146938947696/5588908081, 1896998408590013548875697859200353599627225652636487015589544248/417821179227479]
P29 = [-1630251154838329373316711548742103832011904/1899129241, -2691069732188862509036491474092305043201332153281176779666991512/82762153193539]
P30 = [-2239332072333088027707410218165592332200752/5573966281, 13297261630471447058802177274162474254604127451180242673073534168/416146748573179]
P31 = [33593165529343245030720681661276682512816/41925625, -825015281203855144006423897766508424581436838542808083258664/271468421875]

Li (2026)

y2 + xy = x3 - 46538582067849420177295619802498366364593586948210230000088332133930x  
              + 122020162104048395110524763896188225683232675891947197925364672457083229997908236521974767041655897252 

	Independent points of infinite order: 

P1 = [3813111821913591339221118086158694, 2304541619896838359661639817871742469305917672928]
P2 = [4061734988932134590156805408594164, 1422759204980708148426127019233925951856550754438]
P3 = [4062120606705616642686590969225294, 1779145896181161652554174654150813165535105685528]
P4 = [-198793882327254285322742659859836, 362303590065521028524541271515238708605681883472438]
P5 = [3804907057116939048517616307571472, 5480063989135491020233438113078917856333485309258]
P6 = [3807183538239637081669784948512052, 4797716777719126685639562275823057526990803503750]
P7 = [4124212274128430032627944037949684, 15310799408851111588600198260127771263845455248518]
P8 = [4214360330521585012463320580352344, 27209520608145533977638711822812355770814521555078]
P9 = [4240559954009384741465990285256814, 30426017597717658555408165281582879663883033856888]
P10 = [4300981218292284879000040053724724, 37683355793693730017026451238400994981963888927558]
P11 = [1620964254919769627442858031446644, 225481503429688628198187663197872673421915474985478]
P12 = [1731839010132566481824733596107604, 215909880926165547818280353485575646730073416169638]
P13 = [2753194005242422585496119115466764, 121490082445678642674396384536316784181794136457998]
P14 = [-1115944199017656090007521787205068, 415409319081496061901236318317990864357274623778630]
P15 = [3339007240568585662540603273372514, 62080504240948776592023090579751694993070098689988]
P16 = [3459437513886503067051189978419444, 49238025581171877287937182309682181388698613228678]
P17 = [3467588588397718087990792033189844, 48356828128256811298989968598010245860733363095078]
P18 = [5763660840903216975542609830587494, 212732556101042448233651175930907180862954803745978]
P19 = [27257422652061617041125155454064451/4, 2786451064803544366528024372945023602062259888799199/8]
P20 = [-6372388826434455597685445690219116, 399770367701986293616572252670118443189560786075398]
P21 = [-3080600880736832005005140470258816, 485954466065549476211184772588204943564366779445018]
P22 = [3751529834568065498902896992047646594, 7266275989803309705327716882522750786013160051246433828]
P23 = [-718869999431193588616974050526762919/1024, 12873498663504467930927483108106903280833062480596364179/32768]
P24 = [18418985922487046288868726281265193796/4489, 3625504917956793306525042921633767538998499011660971874/300763]
P25 = [1956330482225652397451000612340728893696/514089, 1964510883726948137902433156920682552500608359501958204194/368601813]
P26 = [462936914600579975837735889793563397736/113569, 263821914775195463286107703364303639143172997216646413234/38272753]
P27 = [2473672659856695191375852575556793850004/606841, 3264262213036544746906889098640949212734211267680038886642/472729139]
P28 = [-1091499676880372275526895751812755869564/619369, 217205714467593840491079395657899191911727738859319520786434/487443403]
P29 = [6812405553180606462722940067129401027476/946729, 368150522272826287710334144589601557868844806231826152022526/921167317]
P30 = [-28041953813744181363582514008537684558316/6661561, 8481329264432613713176388569366291430738819933375717379489998/17193488941]
P31 = [206754728967167350900233622792512496243818496796/7724158302169, 91216257627297446494570906984583597509900692751555169910693193861539494/21467266547245265053]

Li (2026)

y2 + xy + y = x3 - x2 - 9526711476845901608280382793633750919674617782992759276238362125085687x  
              + 463206907181874406177037011231363730963974319416722261144137263207860246343963112239868204998888573429711 

	Independent points of infinite order: 

P1 = [-5515667988066484253173413839555939, 22706503116551854837813714112857474195848110785354424]
P2 = [-8218642219266668233124346553712909, 23258297514478733987652196704312237777278940532264094]
P3 = [25289423452240850802869647797004931, 15442015483688527075171854476181793675300351449735134]
P4 = [38716273931629572162501164676956851, 12345116023878787787255119222969018673219084297796254]
P5 = [42305885614897402215272083264293931, 11657165425952370383680271086221100280508727133818934]
P6 = [43707992556214663983330458898915331, 11415459874925119600034525818234092722792043797482334]
P7 = [45731835230733101963308479945958841, 11098490991831278753475890900826956524388252726220314]
P8 = [51057693786559570515463430068252579, 10483161922073231750498313898043999653629464654222366]
P9 = [56393610268470496280017189627006981, 10261892003095398267299750661776959858763636985635984]
P10 = [70883114285985610976094371969812741, 12002923586050681529283280307784252814827471497329514]
P11 = [79891156588326866423153990405567831, 14560905098000309245783311110341940241685754351032334]
P12 = [-116184108806706709445017575498635309, 1311290815113076889895827869301995608939953847703694]
P13 = [-18429680328525187550268629638390919, 25149979242797099932782147356873591000215051843319834]
P14 = [-21361017094012830310135935587443709, 25631236221064481321591380134628305805807291771616414]
P15 = [-70689507279912309466046658317240189, 27989453806690024712241445732444460453347379873054174]
P16 = [-72741379357992255459135647064744941, 27772215945426442515455255237619193512793027560415646]
P17 = [-74975178358295287215864581270940269, 27495772873659102965063716520435295381300668479331534]
P18 = [-88074283347075888403588466824890749, 24881036659957400049042216561130353635347255392242654]
P19 = [123451906689225929652705988467655331, 34184343625879673495396836887986027256682724614882334]
P20 = [133563315004465937886789964877715331, 39666673285421871487429278798580404726900701889682334]
P21 = [-106708078214660288865811164181460339, 16270841648060735427936928851727363243342883588945864]
P22 = [1223123260506047969392822242017681809, 1348569827702526602804528576200080917897363703890697044]
P23 = [1359213992068263529420393404185503651, 1580699682650308637657589011690857742468360045693441054]
P24 = [1446574675512500253654447661620725731, 1736016379772457940064166353385117978463098798271169534]
P25 = [2603271632076432593054874316354280491/25, 3062622089319064085879178898779231707718566886203908774/125]
P26 = [76122783668470924595879384538262356211/361, 606951603391216789561714176241442416052643178223833471946/6859]
P27 = [4315294783266558879983089266710853965459/3364, 282685034989457238853037027057512404588714211201593896101433/195112]
P28 = [1606548540810930830673559226858238491251/19321, 42118674501487726875810506591489829579860196767751846426746/2685619]
P29 = [184343405324842213410210354878411287756639/1369, 79148245957611724094225582579945957360588240035299803007086102/50653]
P30 = [783331738456205104150160176465694385571/112225, 749130111890695641847680402379911134083609511065173337704514/37595375]
P31 = [217163153667942248938850322849706195948821859/2749219489, 2056606691694398307231045743043694088700397399517248253707095188158/144149825466737]

Ritik Jain (2026)

y2 + xy + y = x3 - x2 - 82836057443790943042116289417351290235862297467097252984667x  
              + 10696452582629970322564365905835258421863312229899500227353576404194467503993863279559691 

	Independent points of infinite order: 

P1 = [165234804870151506790963419771, -38992086202261674929266389293466703222640886]
P2 = [243308398617909525293298402511, -70323199247725912965036903090982117873347866]
P3 = [158221331525574946241167399131, -39381879934096409813207828463974703272848006]
P4 = [330620781576829635919698971, 103291167563908154648041101079307839255886714]
P5 = [619761708580679381591350378959/4, 318132691389640769517253128676524789807964287/8]
P6 = [64594843315307102125868023771, 74934584894645993800099459186918083324691114]
P7 = [-189716857445329880329100162621, 139940943789387452377062458304149969954745346]
P8 = [39574778423650743380954226737371, -248952384267116820976273035950295909008324209286]
P9 = [712479717417465197854037360991, -559778198082110583054234717348347889163754026]
P10 = [77226097688666446215476157391, -68992126639467532922662773031190170796134326]
P11 = [-85054355324616135151515579029, -130869080156181218251349063808745746413497286]
P12 = [133062653519970432245041971931, 45055858363553787937057134293412519314359674]
P13 = [1591103570573475695142465496849/9, 1072337663778251778264971531518715847496548368/27]
P14 = [112433130034171501510920686021, -52954943389474738791318847411089665598304636]
P15 = [270098779155909672359076371931, 89594338564247485857891116784545101631509674]
P16 = [2350966365422643061912089549019/9, -2239928168501651783919644641454234002360921802/27] 
P17 = [345413618731811898988673264171, 152627799784930726765933005842322032652272314]
P18 = [16972341721789408149062410458029/49, 52648530810261951898245349737359562033441296602/343]
P19 = [863260689259440708829457644079, -763220560086430427299614716860022387606104954]
P20 = [234686728867551611509188713771, -64668169845027773699387489446942556753638886]
P21 = [295836081576905034821319056771, 109917387891974908462916999467689329766125114]
P22 = [185048020807289920253146530371, 41283906277211151353709111508715387783225914]
P23 = [276181260091101167186459518171, -94258750644182416726955587965060673785252486]
P24 = [1895710089964672732364823145951, 2581919517919285365133515861961659868896085194]
P25 = [10202907364749349482346848370861/16, 29825579764356212857595473656641491826048080771/64]
P26 = [189733569677786909369284130151151/1156, 1533366601322224826270465145022992057587224904031/39304]
P27 = [36381296672256298635612998280491, 219433898299891109714586542351693411272135275034]
P28 = [469940945453767100257699312581108309/1329409, -245186774760058763841620036624861634556189975245826762/1532808577]
P29 = [22712460691359802597739016665017459/15129, 3364931851255065143248112009263220477671558257841838/1860867]
P30 = [429972068029770548194997547997300991/3279721, -271529043051607889904291227566579178359409975074214866/5939574731]
P31 = [3794017588942700854849330193540488429/34304449, 10797071760022488941662981969196614271475303395024230902/200921157793]

Li (2026)

y2 + xy = x3 - 421406774965967963069585603310038116733186236957137864114786695x  
              + 3329896781397420269474143720108844845817406288637117577630017209595524115341966347568817100937 

	Independent points of infinite order: 

P1 = [11721108169971897557180165765614, 916258153786013576391806358225846317233748893]
P2 = [11724361296396180193671930091414, 899887709675889404122136575047777153649649893]
P3 = [11736495625156902939668286241564, 839921637235502627218705699861481216514187093]
P4 = [11751443446372011980332018062814, 768897495600260992470405827029563055271902093]
P5 = [11789584053849865739721223334314, 609161335719944927116218287503173319636979593]
P6 = [11794882176231965944317128323414, 590450785884935841554652140874019473105550693]
P7 = [11871446231301025521081236488678, 496484529788031465664075343578096208032483141]
P8 = [11891491287472087250647961769454, 537229991713329248153075614491864354785399413]
P9 = [11101145775198925705618690400104, 4455704035089235666631553102226427020682490283]
P10 = [11627589724051087094950594639414, 1418317833087603378892761560467415275716713893]
P11 = [11648850407959393454237459260534, 1300535884982076647044115628011105201934682213]
P12 = [12006772150316662034806879147024, 1043490326440402510567270777055815693427126173]
P13 = [12034993845906313634336855297014, 1195954361893268533460325100089896623291832293]
P14 = [12095985748756153965391524904624, 1537804372211516241285371322655898220695522653]
P15 = [12122178338031255610899041037814, 1687886229541719592726529309550414640536877093]
P16 = [12146333722460112763886309299414, 1827470788669098400385290765513355847298606693]
P17 = [12219869304665770357612447245814, 2257336098888633132609383664369932358219565093]
P18 = [12566584313542982173921497766774, 4330815260156885951801852117685093331811579173]
P19 = [12617536691593087675812009474814, 4639110031825791742919437140707470466602219093]
P20 = [3184780305585646120588021247782, 44945650012463934414598061309309416550340097045]
P21 = [4191864746629010269794742892314, 40460782162208986189522298776996489413244361593]
P22 = [8202441391669623012777043283214, 20620209306754890379245658182915764221983074493]
P23 = [14201611308350273540138541320374, 14474266263950769272496412196920202700584159653]
P24 = [25302214844902688941718932433314, 94158921492726422542388279460997619666129440093]
P25 = [57887643490802402368855864048984, 415831630460896115677386929251758237154667475853]
P26 = [66655667318744454521573062146814, 520951446397071758695753388352120931342488811093]
P27 = [3237575778935520723502251063440806/289, 18995029770873913923141547474310375556333607890629/4913]
P28 = [61689941162148256583431907157228046/289, 15253831323836650965662544570666226904729297018860709/4913]
P29 = [9003150263276625625323713296431526/2809, 6678203819867059072564571882016568286208477078470561/148877]
P30 = [-10513624071449481622666510667514786/1681, 5212944369497093135994577978784455236578484636589293/68921]
P31 = [512892177060206991150371826466052014/39601, 52598909518053018098616083140659247717062880588029307/7880599]

Ritik Jain (2026)

y2 = x3 + x2 - 396137825789612441729444338863306159122947430920364402299145x  
              + 3512038234266144362533916241901734617982932157407905278925467439474016818118945979916226837 

	Independent points of infinite order: 

P1 = [343855485792034218308157681235, -12139094422521994628612438002974587647808250]
P2 = [-653696109716762637861154489265, 274449388721128485485988957376954755357519750]
P3 = [44911840726453909387010661235, -279292914042872168046435723282728997894488250]
P4 = [304894807543927274033674683775, 57167584129848972463903392137652640055793690]
P5 = [-611115986398524271197026990090, -331001997445485164781606670864333079143404825]
P6 = [-102769061000518795942947758465, 367872993439837951486676335357591939447873050]
P7 = [-610540392212921985956404433765, 331630199816921153636870585949858937856781750]
P8 = [-242926958081316047162577874515, -421428292309690580379249564500482149431793750]
P9 = [-683273761160022213042349583690, -217670719582821994722995130070528773602831175]
P10 = [1485865641925550118772630227685, 1669606811518632261672282487112392315151953950]
P11 = [-300871952821517801579910705245, 433192646898205584647270839877093588543767770]
P12 = [282254077161820223059087049110, -79875792145103242645127688329380453120296375]
P13 = [303160422874122490585264290063535, -5278467125126584489541880408543685495931009012550]
P14 = [874574293902800535026823824935, -646682161468296983776609184101654037392497450]
P15 = [7411531806772906498955361463315, -20106743050224849942339402487243023789524900730]
P16 = [-693113759154563106536978912705, 193125129904866213189065101218431814898958490]
P17 = [27924028693812713268378220054, 290972902650751250403662155722057226177749495]
P18 = [649021522982931583430900463666115/2209, 7101985746317029828656729219156036739523576775750/103823]
P19 = [114524326510471727682445088157415/289, -151788907979044298451334950338150235334576365750/4913]
P20 = [-603131103168054223322148272278490/841, 2552284740137189779247049151409302559119390183875/24389]
P21 = [2891036908793816938284927165235, -4807703772665151955374608843175034653214760250]
P22 = [14918864260450448989556351685910/81, 124125660547243062320309663742548079023471236625/729]
P23 = [2638438201811911702353626391184465/7744, -11564323367134065415113950645372082008901069989625/681472]
P24 = [14626731499021214914143566045485/484, -3081696198053973329219530642624820529207937105965/10648]
P25 = [-11637165977509914044399336404610/49, 144055885023086506740349530292185418212943105125/343]
P26 = [-35387748524537353380182334102410/1369, 16485808509153626301272130110137851766760646835875/50653]
P27 = [298403832088730175458551718572033699/912025, -29145921682051344490677293586282118867300394486374818/870983875]
P28 = [268120137064184617411285599114571285/707281, 2053127172036575264058314008196219317502410460411250/594823321]
P29 = [84315314139965237198638462833720777910/101989801, -594564697350787098960221091539846728603384334666889173925/1029995000299]
P30 = [307845445340624377493962241504220421/198025, 158540931990296897096637722358407486028667648213392606/88121125]
P31 = [868069270681186620871873341184283068915/2247423649, -1909788350206704399066393157070913755128269906123921721050/106543612928143]

Ritik Jain (2026)

y2 + xy + y = x3 + x2 - 4362362153060584207614231549318410819022525030515538724235860x  
              + 3512038234266144362533916241901734617982932157407905278925467439474016818118945979916226837 

	Independent points of infinite order: 

P1 = [1252565160636020787995732259197, 114320889646768499250683272210010413280684481]
P2 = [-1105040699356793544321582062083, 2642582873524362751658744352688958546192546241]
P3 = [1562706814525828859714065758717, 714952719559418410209729317590651493204274241]
P4 = [2858096440651670834961052106717, -3793542211798022563273355284733310674710384959]
P5 = [1247921995634005161452974017917, -107475992598035400940964656539913612709373759]
P6 = [1187818133970973863127359913367, 79070381304862624788294760616928650811424941]
P7 = [2116479317756852168740518081821, 1939052781071667121876189287776570333378213409]
P8 = [930285353747107125809139058037, 508816519095738357639560971361945935891780161]
P9 = [-2144871226008492060362161235203, 1732432405362263962990801771698756229562073921]
P10 = [1794946174368193453143835653245, -1210308879761174683909454735341825190291168319]
P11 = [1178971690284062093429965684257, -87618718766035512636353925491149407395363219]
P12 = [-1217959750685611541345236907683, 2649238881105356130297349928493981967373980641]
P13 = [324475475840607410142422453117, -1459698901613922402611943990781917759024938559]
P14 = [1324727195971485680090228520437, -240551548570531739367057318555755841664400439]
P15 = [-1531710121742373018367538199715, 2569107315454786277187210031705258302592133601]
P16 = [5559692612580794854538719822717, 12292670817171531814196544911816598396643746241]
P17 = [999693039445928871594130257917, -387420385746970638603297043508954332253124159]
P18 = [3669467012783653111466213177094453/2809, 30727701861341740145692122079142273811359476826957/148877]
P19 = [11035213000917341965768925688957, 36044357730620720974616554868086610960987628481]
P20 = [92743889693283303015074364278393/64, -247426937221850479744077264935136589528274967663/512]
P21 = [6096135555292099064294606348333/49, -591238716673918519348234402520287040737943826537/343]
P22 = [-1751918226549639267864229143043, 2403647638418837938493370537943745230223276481]
P23 = [6801549860857778180198394154365077/3481, -321332554702201308801686066657151745386619852385661/205379]
P24 = [-2295491784272909949470734369495, 1195913858181828776615261869253604756042109233]
P25 = [76107687862157750819104893711884157/83521, 13077458056378493772590447822020649496682185722365729/24137569]
P26 = [11015271729600687093927911987712252317/5997601, 19124103092412624713146863536282665323864608990717783409/14688124849]
P27 = [2754095559892560566080783366373625233/243049, 4497847668923473146105740082079126978795959771598213737/119823157]
P28 = [5326424983513750976122674968759927957/3916441, -2385150201944447177928460715736979797386652955347542781/7750636739]
P29 = [1571834114499728882597805333350901767157/4044121, -62316608260610737218840096797705945744389997196218282064029/8132727331]
P30 = [480765076499901708013371113419162830323/41126569, -10383576088298075172045985491863297186469482027204952546573/263744686997]
P31 = [37379134563208190086850878424641833872717/8714035801, 6493692921539545127974177234157663625082752125427726892331109/813446527987549]
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