Torsion group Z/8Z, rank = 5


Dujella - Lecacheux (2002)

y2 = x3 + x2 - 11849634571550798667743047864720x  
    + 15613761915399875450490670165233536220551598068

	Torsion points:

O, [2106658439922331, 0], [3316780242030356, 113133244154085840132450], 
[3316780242030356, -113133244154085840132450], 
[1709475188895326, 18780687692370615107460], 
[-1580291531282794, 174336293807843204446500], 
[-1580291531282794, -174336293807843204446500], 
[1709475188895326, -18780687692370615107460]

	Independent points of infinite order:

P1 = [1408739633544836, 41429730450937797313410]
P2 = [-218498788069669, 134879432087724391611000]
P3 = [740396620160686, 85124664142477399663860]
P4 = [1278614340522956, 50527198607648165081250]
P5 = [1650944116469206, 23463118324372658662500]

Dujella - Lecacheux (2003)

y2 + xy = x3 - 304241169811532712979315990x  
         + 2065986446448965089594679105215890328100
 
        Torsion points:

O, [18327955577780, -51443996747795938990], 
[3624507249380, -31794301049998014790], [3624507249380, 31794297425490765410], 
[18327955577780, 51443978419840361210], [10096048933580, 4842002440093973210], 
[10096048933580, -4842012536142906790], [-80665991468641/4, 80665991468641/8]

	Independent points of infinite order:

P1 = [10070073195332, 4839954434963286146]
P2 = [9371130643052, 6152839124497660826]
P3 = [9726832852580, 5191480617042218210]
P4 = [12667127971346, 15641060126117318462]
P5 = [63419650125532, 487696851536311240522]

Dujella - Lecacheux (2005)

y2 + xy = x3 - 476388667785943389435012002002520620x + 
         + 126543502804781110382893072580356730547061880243200400
 
        Torsion points:

O, [493667139543209240, -108057069556113274863099820], 
[493667139543209240, 108057069062446135319890580], [1289643093590974640, 1287276137791469192822348180], 
[411460148407645400, 13730593880654560995960980], [411460148407645400, -13730594292114709403606380], 
[1289643093590974640, -1287276139081112286413322820], [401992113591463640, -200996056795731820]

	Independent points of infinite order:

P1 = [402532692356724440, 2212637731387190951391380]
P2 = [324782305006667480, 77974882314020296026529940]
P3 = [1390354700866905935/4, 434594151457905802575483565/8]
P4 = [882656384433521059082840/2337841, 79957957553437414478707133108477620/3574558889]
P5 = [323969654355843125370440/829921, 6039142065890653356117619477905580/756058031]

MacLeod (2012)

y2 + xy = x3 - 3561865423528009610395543810x + 
         + 81454964632284157241813402084313871228100
 
        Torsion points:

O, [36321858637220, -18160929318610], 
[39035821525220, 43557132138191321390], [182221724405240, 2341587779512438706990], 
[56220773709620, -242704319349797055010], [56220773709620, 242704263129023345390], 
[39035821525220, -43557171174012846610], [182221724405240, -2341587961734163112230]

	Independent points of infinite order:

P1 = [5949581008982, 245914531223693387432]
P2 = [-1195249703128, -292763685098693066518]
P3 = [71998581355220, 445232088662833121390]
P4 = [982331112925220, 30732830735127701753390]
P5 = [56005666625120, 240082528969829449490]

Dujella - Peral (2012)

y2 + xy = x3 - 480510101420343678511903884x + 
         + 2444567252166483991593076470098546378640
 
        Torsion points:

O, [-17323159448172, -74632298902171904220], 
[2911609363932, -32713800070251167040], [-17323159448172, 74632316225331352392], 
[2911609363932, 32713797158641803108], [74812439159615/4, -74812439159615/8], 
[42554919910296, 243022828193040306756], [42554919910296, -243022870747960217052]

	Independent points of infinite order:

P1 = [152142052528410, 1857687175181176518630]
P2 = [-17446157816898, -74280290007379818960]
P3 = [-13924080691168, -80222356449268800940]
P4 = [135431528216497152/49729, 377069596220028001934128980/11089567]
P5 = [255085146694101509910/6661561, 3446938201295351045754826491120/17193488941]

Dujella - Peral (2012)

y2 + xy = x3 - 4271691045320482126621379967109933780x + 
         + 3405752246986521064914214812675778553292700806723712400
 
        Torsion points:

O, [664292095093001960, -928031851917201703610528980], 
[1815336257724085460, -1278099655521678025514386480], [664292095093001960, 928031851252909608517527020], 
[1815336257724085460, 1278099653706341767790301020], [-2387132903893102040, 1193566451946551020], 
[1193861295659147960, 86959002218509523776551020], [1193861295659147960, -86959003412370819435698980]

	Independent points of infinite order:

P1 = [228562615660897960, 1562479976749216983222551020]
P2 = [2628972178002762280/9, 39891362539299541650001338340/27]
P3 = [13771595863068107064988/12769, 332774380360754013808140834100424/1442897]
P4 = [1812321581372356897960, 77152963252173815252239446551020]
P5 = [1985568691873065786080874868043200384143612539858960/953193451415560177888969926861601, 
     55419375596823882813941489864115564434831875747904523564094764380938876625980/29428733696013954912586729933306891107344982747599]

Dujella - Peral (2012)

y2 + xy = x3 - 15431325141995640971648011388750x + 
         + 28075861328873003447771017200972297203151562500
 
        Torsion points:

O, [6702248138813500, 475096986293999335425250], 
[-310791823591340, 181222987504824626760490], [-4635516169602500, 2317758084801250], 
[6702248138813500, -475096992996247474238750], [-310791823591340, -181222987194032803169150], 
[2366819535999100, 69363039116735853921250], [2366819535999100, -69363041483555389920350]

	Independent points of infinite order:

P1 = [-3933616698398660, -167064625300798315877150]
P2 = [-304174496034500, -180946218395393828126750]
P3 = [30125238846195629500/1089, 163781914453907721439422841250/35937]
P4 = [4704373642915683100/841, 8334771827186805028417366250/24389]
P5 = [3873097779952529057020/3129361, 577240587069060000283078461520850/5535839609]

Dujella - Peral (2012)

y2 + xy = x3 + 36206619016320507513893296274455378004840x + 
         + 4164017008409107952186593775986362717428213475209847776897600
 
        Torsion points:

O, [156291840727092030320, -3693315412233862929036112831960], 
[156291840727092030320, 3693315412077571088309020801640], 
[-23669649321485915080, 1814871361218248759825904571040], 
[804202125944347841840, 23524285150866281051240886076520], 
[804202125944347841840, -23524285151670483177185233918360], 
[-23669649321485915080, -1814871361194579110504418655960], 
[-92878370948454608080, 46439185474227304040]

	Independent points of infinite order:

P1 = [-67064195467452854080, -1197589614369326142568015345960]
P2 = [-65897305576033403080, -1221450897422362864342649380960]
P3 = [-33136093288424303020, -1711107141606955142785711926220]
P4 = [-1199450455321992953655760/14641, -1426062696163063335753522082921519960/1771561]
P5 = [768239157666096330051444908236512975536/1218306864093433321, 
     22410342584682979788472958396439186031027846303718850846088/1344729822994914935632410869]

Dujella - Peral (2012)

y2 + xy = x3 - 65757608422928824814414058034068422990x + 
         + 205242095780348539295058638592647749404597811660012388100
 
        Torsion points:

O, [5054226848205421180, -1414039306655551068866231390], 
[5054226848205421180, 1414039301601324220660810210], [10061040755942099140, -23708149465911097936036953350], 
[4686603439366827580, -2343301719683413790], [4711749694270443580, -110927181243547634151829790], 
[10061040755942099140, 23708149455850057180094854210], [4711749694270443580, 110927176531797939881386210]

	Independent points of infinite order:

P1 = [4690074293979129580, 25268425953526489155394210]
P2 = [3287154859686661180, 4960406666627157383270928610]
P3 = [4691951723573631580, 33542898997926248710906210]
P4 = [610225905796845476925033352/129937201, 76089646070271390795437793082053265274/1481154154199]
P5 = [15080919944248669590143074202285722365730/2033873413244877232681, 
     1026866614433066076575338431243510157145935662081773735756840/91724611061139147195523546255829]

Dujella - Peral (2012)

y2 + xy = x3 - 19743860929325035502863459210x + 
         + 4096735464261663217563807627752085945979172
 
        Torsion points:

O, [-55832256650896, -2241659442641343641722], 
[496544119254584, -10803657048771739616542], [-55832256650896, 2241659498473600292618], 
[496544119254584, 10803656552227620361958], [-801810746064289/4, 801810746064289/8], 
[117037435637684, -1840955102182875899242], [117037435637684, 1840954985145440261558]

	Independent points of infinite order:

P1 = [64786343948084, 1757705489176705699958]
P2 = [23596343445404, 1908923949002165628518]
P3 = [-36402206521336, -2183396018286616851502]
P4 = [161860995539084, 2267502458684815282958]
P5 = [7071203454338, 1989340636071804829946]

Dujella - Peral (2012)

y2 + xy = x3 - 3311208879480661613210248501333720x + 
         + 90685812410781204721505439347438829261545665934912
 
        Torsion points:

O, [34897371938719664, -4199079831539765277150232], 
[34897371938719664, 4199079796642393338430568], [101521894343920424, 28299837729476791710467048], 
[-272534946734685569/4, 272534946734685569/8], [-5563403593857736, 10437203092192784495142368], 
[101521894343920424, -28299837830998686054387472], [-5563403593857736, -10437203086629380901284632]

	Independent points of infinite order:

P1 = [-38545427066733136, -12690504561982447160267032]
P2 = [938042469674528, 9358449831722316580767416]
P3 = [10694702454349364, 7516424878414658923610168]
P4 = [-67602742432698736, -2362041271545037246700632]
P5 = [54226730963098031073802/2241009, 16683461082499484751259653062651108/3354790473]

Dujella - MacLeod - Peral (2013)

y2 + xy = x3 - 14308480358642264809261681298200x  
             + 20832355619626831304182830954139201872779240000

	Torsion points:

O, [2183978504356400, -1091989252178200], 
[2212368006468800, 2307895092736193421800], [2212368006468800, -2307897305104199890600], 
[3106181689024880, -79731987895753504301080], [2184852459080900, -75561605897010200200], 
[2184852459080900, 75559421044551119300], [3106181689024880, 79731984789571815276200]

	Independent points of infinite order:

P1 = [1211191369036400, 72655671639587522661800]
P2 = [2184005996951920, 5205662428129388200]
P3 = [1791360394830560, 30808050954630647143400]
P4 = [2718620200653680, 45011891361086352564200]
P5 = [1332000953825150, 64317318766862744365550]

Dujella - MacLeod - Peral (2013)

y2 + xy = x3 - 18517021882744014923118961383526900x  
             + 19572370331271742985622541294683515971125406951610000

	Torsion points:

O, [-292346212988066200, 146173106494033100], 
[195385170934116200, -153014114040205657487029300], 
[195385170934116200, 153014113844820486552913100], 
[-97108866171554200, 143020236083801361459793100], 
[926077960160596280, 892551120908963174263623500], 
[-97108866171554200, -143020235986692495288238900], 
[926077960160596280, -892551121835041134424219780]

	Independent points of infinite order:

P1 = [-118336470086786200, -141797327820264979482286900]
P2 = [210989399689331300, 158297157992907011362903100]
P3 = [-40713731034438040, -142333341041312150111908660]
P4 = [142209347416659560, 140765930666241784687041740]
P5 = [-282818931441843136473040/1885129, -356522449865003833741985045328757100/2588282117]

Izadi - Khoshnam - MacLeod - Zargar (2016)

y2 + xy + y = x3 + x2 - 102395458352937356410639152133170x  
             + 856158496702096425441260926362420647519951478207

	Torsion points:

O, [-2879723444279123, -1061672315068051126304749], 
[26913300116021317, 4194562462418425206704051], 
[-2879723444279123, 1061672317947774570583871], 
[26913300116021317, -4194562489331725322725369], 
[-51903875988466381/4, 51903875988466377/8], 
[7092211186683225, -697626458730384925525437], 
[7092211186683225, 697626451638173738842211]

	Independent points of infinite order:

P1 = [-7896575981427135, -1082743647577851862986069]
P2 = [137070045628542207361569/7102225, 46626711890018575051636134176228867/18927429625]
P3 = [391116510751635260703649927/6650239401, 7636134805651687303025726440017358353779/542320372912149]
P4 = [3897894206123854765273077373/476703250969, 247527251076001399279606156324559408601703/329133562489283453]
P5 = [225725576468973542959010209417/1368429700401, 107051808263784530981136586223519810630198149/1600787695099389399]

Izadi - Khoshnam - MacLeod - Zargar (2016)

y2 + xy = x3 - 78282589422876534310543926547680x  
             + 242679292548703638017812138671209901736667366400

	Torsion points:

O, [2988208040525760, 188248494673544728569120], 
[-6089783557389480, 702538878072933585669960], 
[6311074531469760, -3155537265734880], 
[12730295341071360, 1144200942810374295993120], 
[12730295341071360, -1144200955540669637064480], 
[2988208040525760, -188248497661752769094880], 
[-6089783557389480, -702538871983150028280480]

	Independent points of infinite order:

P1 = [2066566626072000, 299547636274067229154080]
P2 = [-1550787758431860, -600291110493998418028560]
P3 = [3263855129292810, 148139443657335755220570]
P4 = [97385236225472, 484826409788603375426336]
P5 = [523951619159330520210/1849, 11987463546918439737167965101030/79507]

High rank curves with prescribed torsion Andrej Dujella home page