Torsion group Z/2Z × Z/6Z, rank = 6


Elkies (2006)

y2 + xy = x3 - 37680956700999226080263982005713090640x 
         - 36992898397926078743894505902555362159162611772488902400      

 	Torsion points: 

O, [-22285870802215159041/4, 22285870802215159041/8], 
[6580469341357086880, -3290234670678543440], [-1957686771918842720, -5410333087492938317922831440], 
[-1009001640803297120, 504500820401648560], [35490678423593760640, -208156974649394363225766417200], 
[-4221250881727100480, 6844675077153789599452504720], [9770580256649272480, -22969181409893916968242729040], 
[9770580256649272480, 22969181400123336711593456560], [-1957686771918842720, 5410333089450625089841674160], 
[35490678423593760640, 208156974613903684802172656560], [-4221250881727100480, -6844675072932538717725404240]

	Independent points of infinite order:

P1 = [-2989863404519469920, 6995768683639734630607549360]
P2 = [10655036526409965352, 27770115828849915693054478576]
P3 = [-1612366285436148320, 4423900631065189129485603760]
P4 = [540264925906143942595/4, 12544631527256833935404497716455/8]
P5 = [20269619893966105186720/169, 2882001444306457588145290707326320/2197]
P6 = [-77923188462673412909706080/23746129, 828968027924538661366491705732719337520/115714886617]

Dujella - Peral - Tadic (2015)

y2 + xy + y = x3 - 5012222351518888614250804048874855041913x 
	    + 136464417579052941096027626504118630642626009794008307407656

	Torsion points: 

O, [101385619622182196325, -818806680703570032251204452538], 
[101385619622182196325, 818806680602184412629022256212], [45916779191860753200, 55928347748290363259007318712], 
[45916779191860753200, -55928347794207142450868071913], [-81741568644889637425, 40870784322444818712], 
[159565345823179599399/4, -159565345823179599403/8], [41850232189094737575, -20925116094547368788], 
[36016729475079153825, -51591040962907792944544972538], [36016729475079153825, 51591040926891063469465818712], 
[347988478320909450, -367042591656711869191839946913], [347988478320909450, 367042591656363880713519037462]

	Independent points of infinite order:

P1 = [28779233056263365700, 126699395587412255986907128087]
P2 = [2624446826249701950, 351181125621986823809112974962]
P3 = [678256361971058208084885/4624, 502815865789791524906312336273492159/314432]
P4 = [1630915216944315465870675/484, 2082341605198145016507753918275661001/10648]
P5 = [3470132949068123274359401800/88491649, 12231036918653937148600392720595412654816/832440942143]
P6 = [179857395173334923582146263/208849, 2404178034465750506657754080186704856764/95443993]

Some curves with torsion group Z/2Z × Z/6Z and rank = 5
High rank curves with prescribed torsion Andrej Dujella home page