Torsion group Z/4Z, rank = 12


Elkies (2006)

y2 + xy = x3 - 1086750289534747727483801311783198679020964x
         + 435973165715323898311705750809969552813996511661336976727731984

	Torsion points: 

O, [608652855284830069192, -304326427642415034596], 
[765575145621101788360, 7258912294053670714797672847132], 
[765575145621101788360, -7258912294819245860418774635492]

	Independent points of infinite order: 

P1 = [667212090873280716424, 2811468741600154868659805081692]
P2 = [588300445738899279592, 496738531576486238655704034844]
P3 = [640878480652690741030, 1650227478438617514410537429218]
P4 = [594914072761535795656, 61252759091705020914823536412]
P5 = [638462258521963853092, 1543723151978346502319214760144]
P6 = [611110705367872899304, 267524019061985430016982751484]
P7 = [594950966257051623880, 53222697163458916334270893852]
P8 = [716616869926847885656, 5020041299716878281470542342412]
P9 = [627762722675286133960, 1069751374718071593281800264732]
P10 = [510340803979604535780, 3778525680405340851280110580752]
P11 = [617601639147843518920, 606053623188294616739131272988]
P12 = [610092172393399908808, 198061390862962920112854267676]

Dujella - Peral (2014)

y2 + xy + y = x3 + x2 - 204436185783252342785934887233210383183371840x
         + 1165977772244037776901948451219544033496456897571880326934385121505

	Torsion points: 

O, [8321256469951156635413, 202492403819802069202802590388693], 
[8321256469951156635413, -202492403828123325672753747024107], 
[-16576367231130988793387, 8288183615565494396693]

	Independent points of infinite order: 

P1 = [978748421021154077677013, 968189872097011478431309250996630293]
P2 = [40938350455374318726499411487/734449, 8034798036372727657292256069809358560313049/629422793]
P3 = [-1833858234229021806425707334957/129823236, -1643957467127684520633723817507798298370057463/1479205950984]
P4 = [466652543424944830571020577/104329, 19679459405477726600184522257021115789031/33698267]
P5 = [1855357476468407776750099333857/9048064, 2521230557158903421620671041761650894499866191/27216576512]
P6 = [-3057825789598843317991399927/474721, -486854952876376400114202289959672893443583/327082769]
P7 = [1316165334856113211967439783713/19351201, 1479078000627128774872361508989894420231052407/85125933199]
P8 = [1813152158641149066379817305273/226532601, 702396508269043497895380089462203516064428283/3409542177651]
P9 = [14504830996189006898486130877/4313929, 6440066697804847094677254799511744033478569/8960030533]
P10 = [5609636083108794972298867953144693/459099749761, 218252275733512453971818858150694480104231578525237/311071758345811009]
P11 = [-4174731052256571796429970002572787/492999175321, -523817741681937590070885889786485498002950586908993/346153947960711619]
P12 = [6380415804693738957573514663666885797/274098255263569, 13629281585957393563736821264699724609670040970930713421/4537946867595440433497]

Some curves with torsion group Z/4Z and rank = 9, 10 or 11
High rank curves with prescribed torsion Andrej Dujella home page