Torsion group Z/2Z × Z/6Z, rank = 5


Dujella - Lecacheux (2002)

y2 + xy = x3 - 990429690240960203866170x 
           + 343844266388187332499781887080604900     

 	Torsion points: 

O, [-1137029249260, 568514624630], [424281982740, -212140991370], 
[943719301140, 499638233430874230], [943719301140, -499639177150175370], 
[2850989066079/4, -2850989066079/8], [250969723620, 333292433878461750], 
[250969723620, -333292684848185370], [-442779492540, 834013215008711430], 
[-442779492540, -834012772229218890], [3022967672340, 4997467270386528630], 
[3022967672340, -4997470293354200970]

	Independent points of infinite order:

P1 = [39632200740, 551954334123294630]
P2 = [376196308530, 156489276347919690]
P3 = [-387729217260, -818274779331071370]
P4 = [233546798100, 353936579560301430]
P5 = [276487413090, 301892365554581130]

Dujella - Lecacheux (2002)

y2 + xy = x3 - 2353799432200918732090882185x 
	   + 39989567111692230080439457690563434811225

	Torsion points:

O, [34624316332970, -17312158166485], [-21382917634260, 283802358972118214865], 
[-221809019191761/4, 221809019191761/8], [146226404851800, -1680011543007889672035], 
[20827938464970, -10413969232485], [45759685113870, -167627237761782692985], 
[45759685113870, 167627192002097579115], [146226404851800, 1680011396781484820235], 
[12435568124070, -112435818541217438085], [12435568124070, 112435806105649314015], 
[-21382917634260, -283802337589200580605]      

	Independent points of infinite order:

P1 = [-13272789126870, -262474418962008447285]
P2 = [62997061649340, 376455986997155384865]
P3 = [235292735902330015/49284, 1858717131514493882047582075/10941048]
P4 = [26587804667725663180/471969, 95178457006708979892110235295/324242703]
P5 = [642213269652960633874197/709476496,  
     513932348728130104651370222396511867/18897615947456]     

Dujella (2005)

y2 + xy = x3 - 59964332656283235183736182568175070x 
	   + 5262192586204697880378286484023061659919913201000900

	Torsion points:

O, [-100706799382144800, 101388630581288134821084270], [-100706799382144800, -101388630480581335438939470], 
[699032192471396940, -552199676783521388201772270], [699032192471396940, 552199676084489195730375330], 
[-280570274119751460, 140285137059875730], [109878334916952540, -54939167458476270], 
[682767756811195679/4, -682767756811195679/8], [222633621670444140, 54287327183221293445091730], 
[222633621670444140, -54287327405854915115535870], [69575471922080280, 37774899712984645725722130], 
[69575471922080280, -37774899782560117647802410]

	Independent points of infinite order:

P1 = [-80523452065536324, -97819279597823790397110798]
P2 = [206354965015177680, 40930900421205070714160190]
P3 = [719688550990256835/4, 138035757763448396051529615/8]
P4 = [472440225903704060, 287021215160097630172260530]
P5 = [2425719338252784270722090/841, 3777990174973980213385863394320563470/24389]  

Dujella - Lecacheux (2005)

y2 + xy = x3 - 3150412655044310138959718911736x 
	   + 2017307529136250088009457662898185285042999360

	Torsion points:

O, [1593887607912064, -32328622377009159286664], [1593887607912064, 32328620783121551374600], 
[4972732153199968, 330631504589067259261336], [4972732153199968, -330631509561799412461304], 
[-711649981802372, -62441064393262230727784], [-711649981802372, 62441065104912212530156], 
[520580652338596, 22767151633951757574040], [520580652338596, -22767152154532409912636], 
[4910819220603007/4, -4910819220603007/8], [807403499295088, -403701749647544], 
[-2035108304445840, 1017554152222920]
    
	Independent points of infinite order:

P1 = [751615498434496, 8603230476282383128840]
P2 = [-104225584039806191/64, -27231631312774403459151205/512]
P3 = [18351330710909180554744/13225, 2485999215821573343630941982730648/1520875]
P4 = [3164329883365981282447/4, 178001076569511351346622945349293/8]
P5 = [6012159494875581605596/1352569, 433193404879008808392299881834820/1573037747]         

Dujella - Lecacheux (2005)

y2 + xy = x3 - 1574953367944818401647093988620060x 
	   + 22354378546192575469417478920309874871779369315600

	Torsion points:

O, [-16436081600420560, -6618180935008016054759260], [-16436081600420560, 6618180951444097655179820], 
[113889505087841600, 36334885624328686241602340], [11171372531379320, 2480759994323992163309540], 
[113889505087841600, -36334885738218191329443940], [36208265339600120, 3577496063891096259039140], 
[36208265339600120, -3577496100099361598639260], [11171372531379320, -2480760005495364694688860], 
[-181843109471757761/4, 181843109471757761/8], [17736226214465720, -8868113107232860], 
[27724551153473720, -13862275576736860]
   
	Independent points of infinite order:

P1 = [-715874645805280, -4845769377325125124865860]
P2 = [13341553880886920, 1927903179156513170395940]
P3 = [-1286533431046780, -4937456761403462286962860]
P4 = [4465692559968843235743251576/28561, 298423645416988262182359757871185770972548/4826809]
P5 = [3066243125273656053875457039039671824535840/3120685480327594971834289, 
     5364885633228551570655712209620348921025054651409294263574657820/5512835079841624786126134349007271913]        

Dujella - Lecacheux (2008)

y2 + xy = x3 - 14381792753033634870375842403810854602660x 
	   + 662806499738374270959359777990475800630625702118528528547600

	Torsion points:

O, [59208052914026106680, -137287729378657545002575294420], 
[59208052914026106680, 137287729319449492088549187740], [66988426022129214920, -33494213011064607460], 
[79726693539644536520, 151539232681722156018778592540], [285855606678826504639/4, -285855606678826504639/8], 
[-138452327691835841080, 69226163845917920540], [79726693539644536520, -151539232761448849558423129060], 
[-5191333823748374440, -858677634929515911888813327460], [-5191333823748374440, 858677634934707245712561701900], 
[185163361528655877320, -2085243000169909266116860959460], [185163361528655877320, 2085242999984745904588205082140]
   
	Independent points of infinite order:

P1 = [28773623469583250120, 522314569080532537050235328540]
P2 = [81110098194846295892, 172941860066979775872332866976]
P3 = [99266531033757329720, 461879586573455081296428074540]
P4 = [-365244658089452594680/121, -1118508262713108993277475594769260/1331]
P5 = [6643475559305148645228758169122/99218070121, 164960503768175723311599079042881495649743134/31252600689343669]

Dujella - Lecacheux (2008)

y2 + xy = x3 - 131004709010049976906075974805380776438618525256030x 
	   + 532804231132851068847796904272978574164851588764408546154301817681097384900

	Torsion points:

O, [33389045949132825834906060, 182706700209073226375795426180018347170], 
[33389045949132825834906060, -182706700209106615421744559005853253230], 
[3127620701446987822919880, 12396193453631576739116597007002914770], 
[3127620701446987822919880, -12396193453634704359818043994825834650], 
[10558029488395025352538860, 18071446338362979133433696833680970770], 
[5050331710548974320586460, -2525165855274487160293230], 
[32207788920270924494380319/4, -32207788920270924494380319/8], 
[-13102278940616705444181540, 6551139470308352722090770], 
[10558029488395025352538860, -18071446338373537162922091859033509630], 
[-4842578185394737600209360, -32459876617406316825543566607834908430], 
[-4842578185394737600209360, 32459876617411159403728961345435117790]
   
	Independent points of infinite order:

P1 = [4378088878468599339759660, 6570517514971845768410563030167274770]
P2 = [-7979836683805661968982050116/4225, -7637823045463085681864384316252420666014478/274625]
P3 = [-5231043492938682868912140, -32786494932889596996791154709889209230]
P4 = [713612332777702004897920158469380/59151481, 12106759422063621824787844255195559625208869141750/454934040371]
P5 = [-61885908913425738474318485559213679090785215652/4788303286054431104809, 
      -2713566141931187976838998756110252588426581005608756309845220986212678/331338937706851736182789648257877]

Dujella - Lecacheux (2009)

y2 + xy = x3 - 83451301835836505339622094504595112570x 
	   + 293100037604964283613171289154269227976286656300109500900

	Torsion points:

O, [-137997199399934460, 17453178954992624957144122230], 
[5416960199902475540, -2708480099951237770], 
[20520494531516195679/4, -20520494531516195679/8], 
[-10547083832781524460, 5273541916390762230], 
[4592638561230514440, 2589944805914873981042467830], 
[4592638561230514440, -2589944810507512542272982270], 
[5983710046920885540, 2827914715000468862695402230], 
[5983710046920885540, -2827914720984178909616287770], 
[13496488778932076640, -40314466615391592460655073270], 
[13496488778932076640, 40314466601895103681722996630], 
[-137997199399934460, -17453178954854627757744187770]
   
	Independent points of infinite order:

P1 = [2227320680915289990, 10875517008467119620485618730]
P2 = [5130102277196175540, 9799955405563966460062230]
P3 = [915177317355011795860/9, 27577777338096544291625464472210/27]
P4 = [77307991022750019285927135/145924, 679629834938989962662288789445044730015/55742968]
P5 = [1881847753085976389016547277940/243967832761, 1254262607371269085596579330198284678872884930/120503275603473491]

Dujella (2009)

y2 + xy = x3 - x2 - 1013844706869085241884030494287504370x 
	   + 383312079581955140879990492812070932757128820095066500

	Torsion points:

O, [372381816545586408, -239608573897394989830314706], 
[2365196974526377335, -3349127083733781065594610540], 
[505435806420992700, -252717903210496350], 
[-1159495544674905108, 579747772337452554], 
[2616238953015649635/4, -2616238953015649635/8], 
[-302840664555601920, 813984778618660611482816670], 
[811579375505453941, -308305653494139285606496711], 
[811579375505453941, 308305652682559910101042770], 
[-302840664555601920, -813984778315819946927214750], 
[372381816545586408, 239608573525013173284728298], 
[2365196974526377335, 3349127081368584091068233205]
   
	Independent points of infinite order:

P1 = [138816488023051483, 495225922218480420902721944]
P2 = [-20805761910217395, -635922074440891126374828150]
P3 = [29173783901510224748640/27889, 3184252619282355301660145403115890/4657463]
P4 = [210239585179609966845856728/688485121, 5775024067059628021053518856356152671366/18065161089919]
P5 = [4734982045611269991267678213864193365/3852366745601621809, 
                 7538553348381114164657847937875659347956414918234933330/7561212400828731280214296873]

Dujella - Peral (2012)

y2 + xy = x3 - 9327366518423676643700621544483761x 
	   + 346725314827357591512921699486961685298392990114985

	Torsion points:

O, [55686165040391098, -27843082520195549], 
[-111518973825543590, 55759486912771795], 
[223331235140609967/4, -223331235140609967/8], 
[56728903041825214, -396496424071568625479489], 
[56728903041825214, 396496367342665583654275], 
[32171545190604448, -8941327563408817550451149], 
[54795624443899396, 391940364333221642439115], 
[32171545190604448, 8941327531237272359846701], 
[83219539490971798, 12117854675346295286376151], 
[54795624443899396, -391940419128846086338511], 
[83219539490971798, -12117854758565834777347949]
   
	Independent points of infinite order:

P1 = [54774717293086510, 400465728977361126606115]
P2 = [55675847163928048, 16455177652709572813651]
P3 = [80484750953001166, 10834125295358201848752355]
P4 = [-67602933383033156, -25852014585896411209576709]
P5 = [13073367425784981906862/237169, 29830231763553211898593122469753/115501303]


Dujella - Peral (2012)

y2 + xy = x3 - 141033921768261303722896459969632383196x 
	   + 642982736099115694423636613434152191277519936678545325840

	Torsion points:

O, [7140429854737564728, -3570214927368782364], 
[26274212552414019135/4, -26274212552414019135/8], 
[-13708982992841069512, 6854491496420534756], 
[8094887370939953784, 5635986171296721751204699620], 
[8094887370939953784, -5635986179391609122144653404], 
[19632650368084087248, 73765443624285117558990866916], 
[19632650368084087248, -73765443643917767927074954164], 
[-1028898674559648264, -28053575987518304973062301660], 
[-1028898674559648264, 28053575988547203647621949924], 
[5680910419295422368, 5012007929188345287733592676], 
[5680910419295422368, -5012007934869255707029015044]
   
	Independent points of infinite order:

P1 = [1624079315785374648, 20450334645410411496771746916]
P2 = [785210169637118546516648/109561, 20843012447119584444887913630674396/36264691]
P3 = [407002614541703170176282/54289, 33513228082843323119532320867368422/12649337]
P4 = [18873260401553352497272/1089, 2096758764023109763611133734437732/35937]
P5 = [31441397639117126712542905992/273529, 5575101292757919669044833844001015801588972/143055667]


Dujella - Peral (2012)

y2 + xy = x3 - 119375272592212606342695780292063782980x 
	   + 486730768842864202768007431828034011912604553878407184400

	Torsion points:

O, [7186702817601604360, -3593351408800802180], 
[-50293038049986181185/4, 50293038049986181185/8], 
[5386556694894940936, -2693278347447470468], 
[9009928973919570760, 11940830995394642537562717820], 
[9009928973919570760, -11940831004404571511482288580], 
[26696525803682351680, -127775426176178396551892925500], 
[26696525803682351680, 127775426149481870748210573820], 
[3869412613398909640, 9096856524414413314306898620], 
[3869412613398909640, -9096856528283825927705808260], 
[-3536151495322549040, -29404803688314714302694299780], 
[-3536151495322549040, 29404803691850865798016848820]
   
	Independent points of infinite order:

P1 = [2343446129014487560, 14827368095283138154659050620]
P2 = [1091725459764636760, 18913141813665214635383815420]
P3 = [545146403760595948024/9, 12533662930050056318388641080724/27]
P4 = [-1634630167078604856440/361, -209668865591953312468410008199020/6859]
P5 = [95832549059437402660, 932288089154737905966824615020]


Dujella - Peral (2012)

y2 + xy = x3 - 376654149758617929440196007339872300x 
	   + 88973290232367064730638426983006536029213915195290000

	Torsion points:

O, [557289341150487000, 228355169571059129333265300], 
[1420158389689723199/4, -1420158389689723199/8], 
[353624544151632600, -176812272075816300], 
[-708664141574063400, 354332070787031700], 
[362481868682122200, 8402883729749385334487700], 
[362481868682122200, -8402884092231254016609900], 
[557289341150487000, -228355170128348470483752300], 
[183912451736636040, -161004714997174783171070700], 
[183912451736636040, 161004714813262331434434660], 
[346243813159243560, 8275500081305727894561300], 
[346243813159243560, -8275500427549541053804860, 6]
   
	Independent points of infinite order:

P1 = [358347257963395800, 4082637225896721677476500]
P2 = [7025820502981588440, 18554037310801316873241201300]
P3 = [681661166805871800, 385959566883520032256900500]
P4 = [1112209998150190090440/2401, 13868637324189582326781004463700/117649]
P5 = [1299474683278195497790291456200/3632733076729, 
     23636020401140152349402528689524861972586500/6923891160452402317]


Dujella - Peral (2012)

y2 + xy = x3 - 12384379556305685688479674960590x 
	   - 2940026837163614801093525753452171000804443900

	Torsion points:

O, [19364467648309500, -2649260590238047898444190], 
[-2419685932643340, 113398904890013270211210], 
[-2419685932643340, -113398902470327337567870], 
[-3393831784606404, 1696915892303202], 
[14529300522366495/4, -14529300522366495/8], 
[-238493345985220, 119246672992610], 
[-861549338986620, -84203487104504958321990], 
[5361077458893180, -291118354281752440359390], 
[5361077458893180, 291118348920674981466210], 
[-861549338986620, 84203487966054297308610], 
[19364467648309500, 2649260570873580250134690]
   
	Independent points of infinite order:

P1 = [-238517610907860, -544394123414838588750]
P2 = [-3099342798468420, -75308390132878912748190]
P3 = [-244292692685220, -8414717216670269967390]
P4 = [-3064407798519120, -78956628508393608989490]
P5 = [2896703004612913836/25, 4927823471207915850562375194/125]


Dujella - Peral (2012)

y2 + xy = x3 - x2 - 465940285636697633201648611060890324x 
	   + 122417499952052899686605546724393820791691045238548368

	Torsion points:

O, [386768677089269472, 7944946865395561284150444], 
[394098211784164632, -197049105892082316], 
[-788196494652356232, 394098247326178116], 
[1576393131472766403/4, -1576393131472766403/8], 
[394109678033018857, -12429190695194022132541], 
[394109678033018857, 12428796585515989113684], 
[386768677089269472, -7944947252164238373419916], 
[394086816729850032, 12428556267497078425284], 
[394086816729850032, -12428950354313808275316], 
[401473540279937067, 8044379047495419296074359], 
[401473540279937067, -8044379448968959576011426]
   
	Independent points of infinite order:

P1 = [391646593074566904, 2663000925505702929711780]
P2 = [394075536109341567, 24694208712949089407199]
P3 = [394103796080989552, 6033041857857292520964]
P4 = [50872872039051807, 314396947978510314020534559]
P5 = [745247009732968074875473/1646089, 137957837725051088935246101835182033/2111932187]


Dujella - MacLeod - Peral (2013)

y2 + xy = x3 - 881063736059316881644156022557131x  
             + 10065931953809263718033274148403504846943847748145

	Torsion points:

O, [17185313444501358, -8592656722250679], 
[68357167785661455/4, -68357167785661455/8], [-34274605390916722, 17137302695458361], 
[17632006488294354, -112176451833709824317835], [17632006488294354, 112176434201703336023481], 
[28247179230462348, 2777927187230844394544721], [8001570889982826, -1878387525584867155671891], 
[28247179230462348, -2777927215478023625007069], [16647269344437888, 110048299065372250221561], 
[8001570889982826, 1878387517583296265689065], [16647269344437888, -110048315712641594659449]

	Independent points of infinite order:

P1 = [13498024849181028, 795365943647276665302561]
P2 = [4539421538415920, 2481925289338669423782455]
P3 = [11869989276795498, 1131440016786966052603821]
P4 = [4511386748981706337497/262144, 1645147139917406191237288916253/134217728]
P5 = [12072799671624900858/361, 29103076160457501209030460099/6859]

Dujella - Peral (2015)

y2 + xy = x3 - 2782943443885745159100949025619723531275020246x  
             + 55855240810754614011637810490868813280940664269627083967668818447140

	Torsion points:

O, [109044001755668684633888, -32388114159646556159099305470467674], 
[-11859255132686138171464, 9337610431701331808095791331289774], 
[-11859255132686138171464, -9337610431689472552963105193118310], 
[21920741782073992669628, -2320428269964301346440204723992934], 
[21920741782073992669628, 2320428269942380604658130731323306], 
[33091046042849616095948, -16545523021424808047974], 
[27745338877258652001740, -13872669438629326000870], 
[-243345539680433072390753/4, 243345539680433072390753/8], 
[39701829468352179710684, -2818992444225403465094308378033654], 
[39701829468352179710684, 2818992444185701635625956198322970], 
[109044001755668684633888, 32388114159537512157343636785833786]

	Independent points of infinite order:

P1 = [23750284577699813908448, 1776656408644779962971766512279526]
P2 = [691511353654463965517684/25, 25247144434411739376289791577276138/125]
P3 = [43882187782816701847822692716/1585081, 340503355387749636940932181921830291036910/1995616979]
P4 = [258228789379105828496987398005752939756/4798947067471489, 
     2615761019520217898452643310301670673767287754132891480410/332444337096100144140737]
P5 = [248639842573471033744096758586198560139616/7327510688458875121, 
     13931299141484207325603352816906406436354981512505798207492214/19835113544842763845899563831]

High rank curves with prescribed torsion Andrej Dujella home page