Rank ≥ 26

Elkies (2006)

y2 + xy = x3 - 271568164801421919805097494520335727505515190x
             + 1673523352045742769296938739782713519216640490554763586630258973092 

Independent points of infinite order: 

P1 = [8171422342464806465484, -6968834778825385043474309573382]
P2 = [15506445321751452148524, -1091324356491763240297753031369862]
P3 = [11039908252424365794924, -144827641060783314905542709878662]
P4 = [-1124601528095626940916, -1406238603313519698489557881370502]
P5 = [15979026151674771866124, -1189134868428022585423319170045062]
P6 = [11497067226620401106124, -266453792604568993030594792645062]
P7 = [7964508116120761886124, -125826249902394562842770754145062]
P8 = [2057879788890780254844, -1059897903660876637259757866892102]
P9 = [11013564232941423844134, -136087530861067930848004367296452]
P10 = [10829312030749636504524, -51221935741228061266466491121862]
P11 = [5848751988213230056404, -534099270092912134067240836612782]
P12 = [11638411535858522295324, -298922500066105927047772771275462]
P13 = [6868334166980311706124, -363743509637318702264451883645062]
P14 = [7381103602441056808764, -266792452337957421936180167447622]
P15 = [-18419992393365020549076, -652667554475023102928617867030662]
P16 = [5493669297710229516324, -589422439885808591350951805382462]
P17 = [10874362643126096194524, -79405280481739553235221538851862]
P18 = [12784040435352607621644, -539538954823537014427509667959942]
P19 = [5007446790086282822724, -662736416997155792108337796316262]
P20 = [36311702775006993614724, -6300067037234134629601408866834462]
P21 = [7959119841967332599004, -127557175274976008347318287354822]
P22 = [-12920905145755852965276, -1739335347799715861327567809614462]
P23 = [-17861114371778492133876, -908840632427321863500516490045062]
P24 = [19507248497293445799354, -1949133775462458912736164210505272]
P25 = [-17316278094075829267956, -1087992116843026234002459666650502]
P26 = [33532502811165086659374, -5502006283885945731489116619434562]