Trivial torsion group, rank ≥ 30

Alpöge - Howell (2026)

y2 + xy = x3 - 201769035260418549083594900060734240952308696994802735114305555x  
              + 1151107939141058565733479426024323225135665982951300586808823640527729578307228357301072889377 

	Independent points of infinite order: 

P1 = [-4761204159891138283979053265906, 44764265461782973805868732003346421827415264953]
P2 = [-14158422539541566469588779426546, 34199834254251713784176619895082644508395077433]
P3 = [-11522667358396562420423130332066, -44115070023357103726405378140637465204943359607]
P4 = [-204839531927226269712122049566, -34531574232452693997231136031282772551453427107]
P5 = [3899324051227528532535432912094, 20582352852872417675268569815574934013539218953]
P6 = [149851368287976334870008075289384, -1826442728148288630645637436047625928557963231657]
P7 = [240440240734591134232325971191694, 3721941824016160691689265341458606456425791434553]
P8 = [58446054919170749975942104376446/9, -289145377197241504032247540119122580900747897469/27]
P9 = [25642661602146479458845459929344, -113306861325798987289137854129016658652160209297]
P10 = [25720885078613923889202869994094, -113918565504468051036791617945007239588074855047]
P11 = [4956414590296956229584100339596814, 348939117745197814060339374812186839746231405619513]
P12 = [725964821994104294477684670330094, -19556488133953913131900670560396205869420775943047]
P13 = [20802191136944676997135829374, 33866070189878993817062821320678356972094522793] 
P14 = [79052318332408565020526148386446/9, -202982221452031541387733280916787231176177841869/27]
P15 = [-11232245340662775388535509780886, -44725045659073489550941507272219743508825024527]
P16 = [8362456338772815315335239525614, -6972475614865802969141741730862843401376795527]
P17 = [2011658715643038193607509024534, 27447371869432010931671648582500375378543228993] 
P18 = [5027695440284894460797358334207726/529, -116678851641395817353208818767411586490893208148849/12167] 
P19 = [24649144267565165528439068441554, 105612540318783792474731275264940719325335867213]
P20 = [-12211389420609043025008816968566, 42356225616159991618318584560811156010370207173]
P21 = [7798692390172953821075781106768126/1369, -691870045568822811690292896396241871567834072004011/50653]
P22 = [87157992815740534253438806045216/9, 277139378791840529410298740802253693668472375061/27]
P23 = [30786757706172245427369935940751/4, 58841476683002984849182029306774218124047405249/8]
P24 = [245309280348041323814668746104926/25, 1346501028820415725958868015485008289981037919061/125]
P25 = [-343878076324392159036619356326, -34934957027779219869199839566344035316624147307]
P26 = [544211807917340289404451270094, -32271721754226832038590040491036826507284103047] 
P27 = [20286216384652039303944170492166526/9409, -24593234902246769413006506020777691495865223432164871/912673] 
P28 = [-25558163204018019740775243468600589874/1760929, 74707049582033426178338768659390679551818201954095350999/2336752783] 
P29 = [4546264873863829383537112534021848799606/398521369, -145386763829319577901520209264368135012688669041886277075149/7955682089347]
P30 = [1709164065046406773620054102684586/169, 26450264171408287955631955124255640794301463854841/2197]

Elkies - Klagsbrun (2026)

y2 + xy = x3 - 12892599774455576272301592959047823530919513428112484011550x  
              + 560755046348395412977088824999503890617558856687636981223935662848386484980312656296132 

	Independent points of infinite order: 

P1 = [255988718874926192236548018804, 118471064338162161016351388941923332809406898]
P2 = [1066205451814910034394295004, 23388247010770888011772107443289505784724498]
P3 = [113917476844419910281731337891/4, 117760659397165502491412133574198971289541859/8]
P4 = [522896578822997794950102133404, 369851424817590721242251565943695349358330898]
P5 = [752250152281037640723473915004, 645403618239626742448521759727984493932740498]
P6 = [61804854104619698852310920004, 120214803870231627655588231658347056480498]
P7 = [20899766145823995773278515804, 17332967103311289782942907345374643157361298]
P8 = [-93254972898947099650813769268, 30855520614764535640323270586626377097290850]
P9 = [69713966164712304592492810404, 879341993008471697868322219521408586366698]
P10 = [60167736519416976225848653404, 1689119177236660096430130798955079194250898]
P11 = [-14046491075437784120213373996, 27186014904486995598713981999311252231560498]
P12 = [-130496575431865945133637499386, 4574111880086616610004422937977208277497508]
P13 = [42793421047455475252311947484, 9348967782365268719193346216572636318501138]
P14 = [307730699417145077975860872144, 160420835618426919369879316139943794781559218]
P15 = [41901340123969117728348005179516/729, 61004412435011873236122293828203075348860441334/19683]
P16 = [58130399340913205751269159004, 2781011322605723291068350540205198693092498]
P17 = [70345925383762050501865122204, 1386888123108366471979074557284674228142098]
P18 = [47475994387288306853069249004, 7461603107396365118336363485188332292162498] 
P19 = [49620349750437331732008503004, 6572208525446476744123366284203963791284498]
P20 = [61805127635300558534355248604, 118573275438486609948049447535048685070098]
P21 = [59748726488260912696099828854, 1932931405414328200984836878686914954536598]
P22 = [59841577032822742951082534364, 1880183337056779424275360836669144786159378]
P23 = [-118760392546072421990801612496, 20417762309697775621233399179218700962920498]
P24 = [81538823240248716448164032604, 7185053709895791307497392456302587466433298]
P25 = [69804093051086399717838542304, 962825039084755225856791363805867260214898]
P26 = [60968339684582708695209678044, 1158621242686468864160533635604978093990418]
P27 = [84118386438376926037642559004, 8453631999329921882625980842970149886292498] 
P28 = [-100777109571234949083590688996, 28922984981811818231788164913656183379020498]
P29 = [52934781599204816902071863004, 5159086508060990188972690790046932481972498]
P30 = [52008008801558252504808621504, 5559638586120196928159633060055634855729998]

Jason Ma (2026)

y2 = x3 + x2 - 1403690649658805063822079426565364532330941688777465x  
              + 20330956262404043956056670578860396389806578729906895240267998011765438043400 

	Independent points of infinite order: 

P1 = [25643969080536974638027485, -34620339898516189944890934242028206415]
P2 = [16248317330407575130883445, -42579665642521512985563313263456501375]
P3 = [18236957900920008027959945, 28236101215350759604142653387122582375]
P4 = [19528972089838300990547795, -19138829172609567261476925549241890975]
P5 = [11780804662995131736641355, 73684310892975053809039413780161744385]
P6 = [20538984671513006639016945, -12843051715169473538638988726321280375]
P7 = [138028300106979766260016245, 1567249531925129448619330694998204034175]
P8 = [22125998767621488305505945, 10242005874749512234755155397264197625]
P9 = [20450597019232948656009645, 13327963769501301839570300835854254575]
P10 = [134415208723034261294536845/4, -843138834044680323736236008471742062445/8]
P11 = [54278510940067775318948820, 322573587507499515325109766741198507750]
P12 = [-43244151219096409733287560, 12781077749908063780973937942011286420]
P13 = [745543701465260297187151635, 20331586455631405397379923615746041574815]
P14 = [22204429034830927325102865, -10507357817899264156996162765039576905]
P15 = [459620814787170159712743705/64, -52764913032914496864752477053386921287325/512]
P16 = [357852459868528909768630545/16, -713569440441120874714246204774065902175/64]
P17 = [17965750211851668220731405, -30188958018730964708996570493650293935]
P18 = [971072121974849592393013695, -30238384065881314686074283115856000378625]
P19 = [79517361998368140003971248755/11449, 127997103663865496608335580716732189821316525/1225043]
P20 = [-280934597967344386683603675/169, -330717954340015338399202393270075077941115/2197]
P21 = [3201533480679137051057485705/144, -18336643622656585358933005015877013006125/1728]
P22 = [133292143003650647147992514855/5329, -11471980172608631478253510448091546597387975/389017]
P23 = [3933996691670231331123791175/169, -36060573927746648623484985062388645867435/2197]
P24 = [381510470620890012129566227005/24649, 186111720556679732975600637192790088887438725/3869893]
P25 = [11174174753726072457726314673720/502681, -3777012575826631765317923345772580714260467700/356400829]
P26 = [6321939816710140070382492507/289, -47318017383767384410560451750003490780943/4913]
P27 = [6236739309376440239548034109675/54289, 14833780466619229665701620422963350316990179385/12649337]
P28 = [-5033543361442477981379286020495/294849, -31746799804466507104792125113559683512019149225/160103007]
P29 = [2721526757238726331600574725719/674041, -67161290842777442333111128812340180108461643725/553387661]
P30 = [1302682039504667281187401507330252845/10416447721, 1426604658582102577485916066188237232918138281803311125/1063113070852981]

Ritik Jain (2026)

y2 + xy = x3 - 409814159690250367299180863217962628472973392587261030x  
              + 101046273555389793109464401287072761874361882945160895394109386177847129246684900 

	Independent points of infinite order: 

P1 = [290755813676664171600404610, 2543740024465776460109693764654751445870]
P2 = [3452201896799940898424660140/9, -14469763995505213060207128104279343856210/27]
P3 = [350825153088537746885129460, 672396822994732298302349094618540844770]
P4 = [377180045643668209776796020, 363302750731833619228739608923067504170]
P5 = [341921793961643708618503980, 946643166526607961731326568176066621770]
P6 = [278210328309170887422754110, -2926686793141946257563331112876257871730]
P7 = [436524422405407568104716510, 2309474372378672068594622316097735200270]
P8 = [358959815840453612839887660, 438408575653403889843715296986431242570]
P9 = [1283132212346232592390631995260/3481, -53926984601204491590120721387175570667043170/205379]
P10 = [289046000269170142816604460, 2596194646667711909347431758672011069770]
P11 = [2085290053138588257903913260, 91182302629110387823298518634212713277770]
P12 = [375158836832957356843814460, -319816766309849589592170456636488794230]
P13 = [646043744425849123842678692940/289, 499967073356192847207384941830831010662664010/4913]
P14 = [4129213506585635393607105636, -262323455594870979360246287652548056683798]
P15 = [372338670770650896292536270, 276025293766582229800259954950093497060]
P16 = [643355036575594020944208420, 10182268342916918951896735133813942887410]
P17 = [-459492238299581121377657429940/841, -310284076976613631933218164080676229909619470/24389]
P18 = [-184733162400526060600638840, 13055583951847301108274311593877346349770]
P19 = [-664843173357362837653479540, -8923945534033138933255704166125123538230]
P20 = [419369709278054501417415660, -1713936697567850099274358472843206325430]
P21 = [59964151938803704086371834190/169, 1217206970135899805674029961100520511571940/2197]
P22 = [2110662758615089801228964160, 92945427926131687014171722127944004146070]
P23 = [371851146673952065089362220, 271079913669923001594806621681619364170]
P24 = [84033941855257814662647960, 8197645824263509112986291955873646757770]
P25 = [-722121535347737095445734164, -4519376114696279609176330694994998337078]
P26 = [475475262048543352192879122240/1369, -39478337093911122835991171660428965471296190/50653]
P27 = [303041265044363576008141762924/2025, 598025904231533449527281445340387178691033778/91125]
P28 = [397838602149595015821550812, -987170717982894696974470194969901769958]
P29 = [811069925233297000336557636390/1369, -412138684931124873905682948356131614098717940/50653]
P30 = [2328300623327056569605760355860/3481, 2306910598910147357598810860471068803309880830/205379]
Rank records history Andrej Dujella home page