Rank ≥ 25

Elkies (2006)

y2 + xy = x3 - 1222583105876029916237789137035775062690200x
             + 523967447200209449943328506898413682821590945806099349816040000 

Independent points of infinite order: 

P1 = [641919671491882025200, 1917622583611100541324074905000]
P2 = [594994690844383471600, 2678847964182260740553494589800]
P3 = [208196841608040479152, 16686940720261011217265226309352]
P4 = [-740361154329309511040, -31989105845069848078376802650600]
P5 = [804899774308127940400, 7834276239548635107854922077800]
P6 = [77301598762979192560, 20734554180656765176390619976040]
P7 = [702911299694560038040, -3449260992056554312601103761600]
P8 = [915485567326094606800, -13114547973847689345711009896600]
P9 = [-26293910519288470100, -23581682704464936496514064216200]
P10 = [709908819594746168740, -3717262868674905037217938483100]
P11 = [875280814115630686150, 11154863867288165850338319192550]
P12 = [1643784276034119529840, 54367762877007597564101217327400]
P13 = [2310209163245804946160, 100146314980252246302935050670440]
P14 = [1909368489723249677080, -71767455269710117770413293033280]
P15 = [673388810420663671600, -2458352439913512016092058261400]
P16 = [-1225141077322557861200, 13524179089651555542230916989800]
P17 = [641884454793213323650, 1917498326574315724474707751300]
P18 = [1779583257516825231520, 63119495622551336426454649548280]
P19 = [1336811684405040135250, 35757154248294043850600009373700]
P20 = [591774410894432534320, -2776985009258775118824482266520]
P21 = [4566516049692282828100, 300278030461559541447358362839800]
P22 = [-499965890434284752520, -31784318126596227394232274064800]
P23 = [1239030190644321769050, -30187812602932900034742078815100]
P24 = [3417029330608564213375/4, -81177702701038029224605519995775/8]
P25 = [9868611464482118831434/25, 1267809878380229048466900915382388/125]