Torsion group Z/8Z, rank = 6


Elkies (2006)

y2 + xy + y = x3 + x2 + 690560257169377937125059126423812540x  
             - 33492188518937203020097127431557287129957999127407035

	Torsion points:

O, [48336483394805853, -24168241697402927], [883542747769245853, 1125336842235356652764997073], 
[883542747769245853, -1125336843118899400534242927], [4005178026762805853, 8184203792297326700054597073], 
[224631009320286053, 364641927591846714613884873], [4005178026762805853, -8184203796302504726817402927], 
[224631009320286053, -364641927816477723934170927]

	Independent points of infinite order:

P1 = [286126088836986533, 433035480275757031966872273]
P2 = [121390141610245853, 228306310388654208874997073]
P3 = [377567526557995573, 530156507956449179371813753]
P4 = [210538669890805853, 348180841544471134046597073]
P5 = [20853081120020405853, 95301363213306571952478597073]
P6 = [326728698269569003, 476458277653995656563456173]

Dujella - MacLeod - Peral (2013)

y2 + xy = x3 - 756828185715435657801375588494450478688880x  
             + 253401847133358152315350001760510755926602193561212158650118400

	Torsion points:

O, [488270373699396020980, -522119461339460945407102274900], 
[-118081345724039567840, -18469507450545438959675271900080], 
[505910156890389864160, -252955078445194932080], 
[610824764162880776560, -4360662646017169114212121556480], 
[610824764162880776560, 4360662645406344350049240779920], 
[488270373699396020980, 522119460851190571707706253920], 
[-118081345724039567840, 18469507450663520305399311467920]

	Independent points of infinite order:

P1 = [620360330455938117460, 4758091883195699130662218969420]
P2 = [497190314206408214800, 136962231152035964282861848720]
P3 = [4355219292628560802420, 282075815982149392450508425131820]
P4 = [490711380442247094160, 424022366175406313485509707920]
P5 = [-448543293347798650040, -22419392091058428164924864493080]
P6 = [1562969108844553330473522536893440/2684692081009, 
     13987784198174075611499325865842096055521336633840/4398876028809489527]

Some curves with torsion group Z/8Z and rank = 5
High rank curves with prescribed torsion Andrej Dujella home page