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 = [36942058536678896625, 41448919589550865715463314662]
P2 = [32759641624440479850, 86158367801450315742417467062]
P3 = [7799455463851058031, 312803876557218100682449398064]
P4 = [67536753983723643825, 325583659454350025193050287462]
P5 = [88968136411056293415, 628290086040065957453320659772]
P6 = [660750980183544551325, 16890918612236252777190840817462]

Dujella - Peral (2020)

y2 + xy + y = x3 - x2 - 106400353230449963598965706612473537209115813x 
	    + 422128950019755721902266808708794939178991483786699178861367557981

	Torsion points: 

O, [181726597026665480895, -634664618003129716542918251588768], 
[-47639326761252544702965/4, 47639326761252544702961/8], [181726597026665480895, 634664618002947989945891586107872], 
[5284984104250797096255, 86139228435516169417871532810592], [5284984104250797096255, -86139228440801153522122329906848], 
[6649943724390833913151, 92977136437621147651013368258912], [6649943724390833913151, -92977136444271091375404202172064], 
[6086388510603317132607, -3043194255301658566304], [5823443179709819043135, -2911721589854909521568], 
[14483120471895039162303, -1385319617540636827629234574010144], [14483120471895039162303, 1385319617526153707157339534847840]

	Independent points of infinite order:

P1 = [3381582657678094595391, 317798837174699492653979100463456]
P2 = [1405307609816280252915, 524765710721840000489508602178592]
P3 = [5182458350371657727295, 99515663037413274888605365739872]
P4 = [5809169908678191267135, 8373221198553485598701485793632]
P5 = [6300275580642266589471, 43095447699485136548183136831232]
P6 = [25698851534528457561883767/4489, 7540653954712106685208830187241252320/300763]

Dujella - Peral (2020)

y2 + xy + y = x3 - x2 - 11595592774044457871284428924350937512443357522730417x 
	    + 480542453538711190965783397398215616559186760599243038811874075581447924343009

	Torsion points: 

O, [66184635185156602798631767, -54844733355433883513630132786104647084], 
[66184635185156602798631767, 54844733355367698878444976183306015316], [125329443847710046928771607, -997938713633575943329029867050697157484], 
[125329443847710046928771607, 997938713633450613885182157003768385876], [58237881495355569255996567, -52557163652534920636364611375166869484], 
[14986580212404192211127127, 556893277087404466248278753159318098516], [58237881495355569255996567, 52557163652476682754869255805910872916], 
[14986580212404192211127127, -556893277087419452828491157351529225644], [62745125095719522700942167, -31372562547859761350471084], 
[-124339652055848226661803177, 62169826027924113330901588], [246378107840514815843444043/4, -246378107840514815843444047/8]

	Independent points of infinite order:

P1 = [20970067814964567466010967, 496591927893119661744866949211476716116]
P2 = [62860942214552355866740567, 5239961022805830002556384174024786516]
P3 = [57975572744384610409241967, 56097282454580990820241436028971360116]
P4 = [177047020645511504721795927, 1994299577244738077567579471279411794516]
P5 = [52258598034561837208087617, 131488510966568126706718335430338736966]
P6 = [42271752946059488735003414583/289, 6796720613937993899562259363016269627899668/4913]
Some curves with torsion group Z/2Z × Z/6Z and rank = 5
High rank curves with prescribed torsion Andrej Dujella home page