Torsion group Z/6Z, rank = 8


Eroshkin (2008)

y2 + xy = x3 - 3969894586069811345797915011354270x 
	    + 96060267726898136055233656334812211512766915208900

	Torsion points: 

O, [42608376638232540, 2064919764454782297078930], 
[42608376638232540, -2064919807063158935311470], 
[102000243167270700, -27428919904947941620316670], 
[102000243167270700, 27428919802947698453045970], 
[37773199222238940, -18886599611119470]

	Independent points of infinite order:

P1 = [19009234576838940, 5240669432779505183920530]
P2 = [37871089128607740, 177444177960827044499730]
P3 = [37969586449003500, 255667665375551443275330]
P4 = [39249365168545680, 841765753894805349859890]
P5 = [40358796111683340, 1256111928005948324372130]
P6 = [34516084242736860, 395140905689330572017810]
P7 = [38420722546570140, 498871396814474291589330]
P8 = [528262098613980, 9693465339791103915219090]

Dujella - Eroshkin (2008)

y2 + xy = x3 - 684648272485680709402900211994868050x 
	    + 78106637528224228839785060686283103205949536929802500

	Torsion points: 

O, [1102021082445716100, -813609135503177007756478050], 
[1102021082445716100, 813609134401155925310761950], 
[-537011663296475700, 539357490149052705460413750], 
[-537011663296475700, -539357489612041042163938050], 
[116385524298476100, -58192762149238050]

	Independent points of infinite order:

P1 = [91693232663164272, 126885573188412316276382226]
P2 = [113959322895490740, 39554299596194279632619550]
P3 = [-70343499630811740, -354850840132403770586907810]
P4 = [3332477014980584292, 5899579002826558525488247902]
P5 = [6257334030337361729700/1369, 487003540372174589875827818653350/50653]
P6 = [-1415059215147241461900/1849, -31183838105048001079416025456950/79507]
P7 = [-925432706614164340860/10609, -404700658151149195487201671059150/1092727]
P8 = [64651459180134662025/64, 330760852295513048563777169775/512]

Dujella - Eroshkin (2008)

y2 + xy = x3 - 475580051023836530278979780324370x 
	    + 3991178865068293690072267520912681524547928912900

	Torsion points: 

O, [13518316590105540, -180389476580872603967970], 
[13518316590105540, 180389463062556013862430], 
[11684356686817860, -171840809940319288581090], 
[11684356686817860, 171840798255962601763230], 
[-25180945414732860, 12590472707366430]

	Independent points of infinite order:

P1 = [12313149261235140, 46175189994956930342430]
P2 = [-1936872157800896604/121, -3645664495594016601432279558/1331]
P3 = [1152584970568784484/25, 1118447402666672077575856038/125]
P4 = [7461431333415354756/625, 1916495731899417237309797646/15625]
P5 = [1297092940611240900, 1477051543090251325360627230]
P6 = [-5126598523468860, -2508894081763080979161570]
P7 = [34032996272194500, 5217691706556788509171230]
P8 = [10040597866828740, 477807165098859594124830]

Elkies (2008)

y2 = x3 + x2 - 196348925423783757393146158660311426285x 
    + 1051757327090520025478687684152067952012419669968483731900

	Torsion points: 

O, [7537968017313745980, 0], 
[10132363338719401605, -10124998556114464265515325625], 
[10132363338719401605, 10124998556114464265515325625], 
[-2439709524759448545, -38939312707414052613432763725], 
[-2439709524759448545, 38939312707414052613432763725]

	Independent points of infinite order:

P1 = [6023881995371818689, 9357569440591044907223158455]
P2 = [7270559978599909245, 2918915446078886619276240615]
P3 = [3629951324598415980, 19668515653655328302780242500]
P4 = [5221270916602449105, 12996420964925856481700015625]
P5 = [-12379625066269681395, 39815104178698903249192193625]
P6 = [-6788411736470472855, 45517331456067424746735963735]
P7 = [8967887941144144365, 3485218083173721934181508135]
P8 = [-1643227814742177267, 37013055843255638108712796761]

Elkies (2008)

y2 + xy + y = x3 - 913217994232980248076710723083809326628x 
             + 11134659509213630442423739721043373303068542539029002813998

	Torsion points: 

O, [-35080242650405792251, 17540121325202896125], 
[10457916253930998974, 52230869394111038538388255350], 
[10457916253930998974, -52230869404568954792319254325], 
[25937864929179419519, 69985894996077621947272533495], 
[25937864929179419519, -69985895022015486876451953015]

	Independent points of infinite order:

P1 = [-649578426066748876, -108294007800484741498750710375]
P2 = [6720075382423660274, 72809617547796961950943751775]
P3 = [-12979543526370783241, -144226097960871109655672768265]
P4 = [13227370736142192374, 37006640679945108356282215875]
P5 = [15179996327376133699, 27748423439323339154524885325]
P6 = [-3105396365976348817, -118070387910224620788423175497]
P7 = [8377788197190378269, 63811654127849456017897604745]
P8 = [15677080881840697349, 25904379534934585054470613725]

Elkies (2008)

y2 = x3 + x2 - 2988953265941486315671511392675613660213547766700x 
    + 1988561710795654748570775576651431494771732919207799253936953545372685348

	Torsion points: 

O, [1009717521070207317070191, 0], 
[1073282221112051638300816, 130086018425624273993885089962026250], 
[1073282221112051638300816, -130086018425624273993885089962026250], 
[2105217870810973618012516, 2241956283721027904726941711993376850], 
[2105217870810973618012516, -2241956283721027904726941711993376850]

	Independent points of infinite order:

P1 = [-604068163732154016647549, 1890415231517780160670737258638609970]
P2 = [2857576690277708440973316, 4096539977629565042954566540469868750]
P3 = [-563231808943581685114004, 1869053557044598493055541403937978390]
P4 = [3013566251458771097202316, 4510997749515905302688211682319565750]
P5 = [-12342722470026536474684, 1423183632251359984976720720715995250]
P6 = [879646162190920779802636, 199976838486383464625330430583774410]
P7 = [1349260407411465907296492, 641886985312754258304009460566854550]
P8 = [20528665690170346229179576, 92692746035914225383396312639611897970]

Dujella (2008)

y2 + xy = x3 - 7215616324653838384913625350620220143981486228910x 
	    + 7312288533664556036160260732587179429322821986829787336941560686885443972

	Torsion points: 

O, [5475588829411348697262111/4, -5475588829411348697262111/8], 
[2121120609105544610860684, 1245127071749416083147406497213054658], 
[2121120609105544610860684, -1245127071751537203756512041823915342], 
[-750124386557461278266516, -3507536652081801573285227520396800942], 
[-750124386557461278266516, 3507536652082551697671784981675067458]

	Independent points of infinite order:

P1 = [818923650391563376777253524, 23434882858174236476471778528071875889698]
P2 = [87287885261768610231400407076/66049, 4935657245815827376883241873783387394597754/16974593]
P3 = [-1862584687967396306344316, -3780246785548224769247117633935370342]
P4 = [529053314630044378903008922/1369, 108420037538188871478601725317044976454448/50653]
P5 = [412069227782656191576833307856/1803649, 5770824731403908590585032401500281718089029726/2422300607]
P6 = [5901985286008909093081116649938816/35510764249, 
      16551274481098399703805024782604484858481003880964506/6691754947374307]
P7 = [-1913162851939056956970782278034/628849, -524339855620450523166533308418605458408772394/498677257]
P8 = [3824597950756972583237684803681626904/297364086721, 
      7327790129214131650483629221910897568193778338529651838/162155907493915231]

Dujella (2008)

y2 + xy + y = x3 - 360225153137805377563861499671657822554529441873x 
             + 83216446291933254309928055003080883290004599027629309383724863943212006

	Torsion points: 

O, [1386073885849902527145255/4, -1386073885849902527145259/8], 
[346518562747739275398970, -93100933945378027529949880423], 
[346518562747739275398970, 93100587426815279790674481452], 
[347134849579162392965920, 628636750636938998383579509354077], 
[347134849579162392965920, -628636750984073847962741902319998]

	Independent points of infinite order:

P1 = [-201350765871070825072280, -384167759776924072704852498720199798]
P2 = [3465453741425681714259167951170/13184161, 3916020594579611161495588016200914654954857132/47871688591]
P3 = [-1530891906851027420988621020/2209, -375226532559615842514877321739813505254/103823]
P4 = [208329125332353622742620998117865/635846656, -305793216111890946837183045973963735496750650933/16033509277696]
P5 = [143700762204124840536068555097557883894670/414695968638028561, 
      623123690513904427915920801326096141042018288195689409168/267051314637489154718295559]
P6 = [38665199748292222396169549408788065050829520/111581923925922542521, 
      21859453055797108234579056052243102483020191167988153101063/1178666032742375478596523332819]
P7 = [2595865259686900521076205534620, 4182377403483061900627454349269280339739736702]
P8 = [134167494836388112037821368497653010676658625761570/387187137927973517540875681, 
      7175269377858450105973312787736488735603282558693036044163524585292/7618708948796969539862652307413564038671]

Dujella (2008)

y2 + xy = x3 - 164709440877242365084184292915926531855737195x 
	    + 813625350097041580941646921583190206129348507310784165707920095025

	Torsion points: 

O, [-59277305281049993422321/4, 59277305281049993422321/8], 
[7547598942322316688230, -20565870659408465943020870326135], 
[7547598942322316688230, 20565870651860867000698553637905], 
[7272572815673036201750, 20312990487148414788069681298745], 
[7272572815673036201750, -20312990494420987603742717500495]

	Independent points of infinite order:

P1 = [7204921519339507272476, -30342594986928428224771428237583]
P2 = [553028774526498558335227670/78961, 1329781793949257140660840569253346992225/22188041]
P3 = [7823258661841924440650, 62214405062270243908610496702845]
P4 = [7304081951789605918670, 15621853339457769804240056773025]
P5 = [25406528236662964478578622/3481, 3375388464199383880408878427323611923/205379]
P6 = [19892700252670441957070, -2325734222986633537381401327065695]
P7 = [50684735214742788000780155/7396, -52115209593338433659753650190151348695/636056]
P8 = [68112130364806350968358710/7921, -128294785685905133481525886871050467335/704969]

Dujella - Peral (2012)

y2 + xy + y = x3 - 1335985556213327566393188804097819664x 
	    + 594359932384372250781816622950011820594941728829430686

	Torsion points: 

O, [666411001163892393, -333205500581946197], 
[381052561931994273, 374977929121253630729947603], 
[679288450334257194, -16921733012834847802357562], 
[679288450334257194, 16921732333546397468100367], 
[381052561931994273, -374977929502306192661941877]

	Independent points of infinite order:

P1 = [-140507948167530447, -882781142581362400909478117]
P2 = [970750778144885553, 460700310200518512519240883]
P3 = [655818982441251378, 16187331635074900618288183]
P4 = [663790984763164161, 4830472387434653237227987]
P5 = [653009534357684748, 20146964675065541058778498]
P6 = [816035821190266398, 218077775456715021304543123]
P7 = [694665952143846729, 38920446123657493602545947]
P8 = [-510779650064624895, -1069342730510619994725877973]

Dujella - Peral - Tadic (2014)

y2 + xy = x3 - 6425764677860152725987889739367599975652685x 
	    + 6269378239936665797804421444079658936297007509538479980249111697

	Torsion points: 

O, [1517955063240217792634, -3606855504159781197488412843817], 
[1517955063240217792634, 3606855502641826134248195051183], 
[2554304868262237681034, -80755868564113968693239261885617], 
[2554304868262237681034, 80755868561559663824977024204583], 
[5878015362945634044911/4, -5878015362945634044911/8]

	Independent points of infinite order:

P1 = [1479198619437225203114, 961674873976968940724553185423]
P2 = [983878248659722617554, 29993639837709698275453651811903]
P3 = [-195270002998284904246, -86698855872338945168351261518897]
P4 = [1905768976745927479129172/1369, 236977986013326425218546342851879443/50653]
P5 = [-6311475309396756863458144/2809, -14406515547556997158113573289642138709/148877]
P6 = [247988092428272416664, 68491721471136821252796551194823]
P7 = [-9690047463126775012507114/4489, -30199010299318291598491430799264039151/300763]
P8 = [64598145397069207714673134373185088874536/212324772722617129, 
      16417803290702261388641963692832568226039135864150739507379661/97836521164483623902429867]

Dujella - Peral - Tadic (2014)

y2 + xy + y = x3 - 196088112850250619096616465973482541791329679334818x 
	    + 1056769769874550553807438390237468946455487112972817882032821970832478722306

	Torsion points: 

O, [8568770960311046386464160, 2384884601829486253161022032872853857], 
[8568770960311046386464160, -2384884601838055024121333079259318018], 
[32606185327713118408341015/4, -32606185327713118408341019/8], 
[15953123067909853191678910, 44594347893083770521585821338568986982], 
[15953123067909853191678910, -44594347893099723644653731191760665893]

	Independent points of infinite order:

P1 = [1057973462598959467952560, 29163297964171432456943011018660933457]
P2 = [6390444722296280203020060, 8040592243914015726243050965950513982]
P3 = [10382194565777549006869345384/1369, 122534813782624874765286026024762963502662/50653]
P4 = [270353769673133729567976642040/26569, 46453901406064491867231675645914400643500554/4330747]
P5 = [43203697102467548760345377151565/1507984, 254953468763255199423326052551783700114691703789/1851804352]
P6 = [4871309206909977593031642361996240/779689, 339991027601031870439903062701394163947516029697659/688465387]
P7 = [-12563289877496241311102966112331660427615/950665009116736, 
      -1073052947602698733419285331654325649505466143175326000545413/29311717330335008278016]
P8 = [-175623332716620572051126606963232368719550/15374892203954089, 
      -81021788245304035137144982461959794292535587758641904563056146/1906417953646831973084437]

Dujella - Peral - Tadic (2014)

y2 + xy = x3 - 1601825625756180889302684290822230x 
	    + 24675947726759132681554784670325851665637575876900

	Torsion points: 

O, [-46214398621289140, 23107199310644570], 
[22563957778149260, 143035619776855373588570], 
[22563957778149260, -143035642340813151737830], 
[23654765830935140, 145304129651912056789370], 
[23654765830935140, -145304153306677887724510]

	Independent points of infinite order:

P1 = [22311083413962860, 208788897012877753988570]
P2 = [21662349223926860, 376641377102266218260570]
P3 = [-18959179128730940, -6944808179987330872541630]
P4 = [-18026270162169940, -6906034839364874838747430]
P5 = [23277178428047720, 46598373994440978778970]
P6 = [22597546137330830, 134296855095359975095070]
P7 = [-19676169383912140, -6969654126829489724751430]
P8 = [24054284208284660, 251384777499127214019770]

Dujella - Peral - Tadic (2014)

y2 + xy = x3 - 40261759047198447746757434668761470x 
	    + 3060380746085656600349631533426865337350014313816900

	Torsion points: 

O, [103749338665821340, -51874669332910670], 
[155127013688094940, 23403660202067945040817330], 
[155127013688094940, -23403660357194958728912270], 
[-51376077635273900, -70663036833673849226285870], 
[-51376077635273900, 70663036885049926861559770]

	Independent points of infinite order:

P1 = [57915247207650940, 30378758041630841142337330]
P2 = [55925258489949340, 31363118982496469145553330]
P3 = [43427614290454522, 37333781548767800284291312]
P4 = [128656237705564060, 3166599180935284108069810]
P5 = [-31117058267646860, -65445232762546245363178670]
P6 = [143976949354409740, 15753278881617607486963330]
P7 = [69595528562908840, 24401452861042487623969330]
P8 = [603422305010816860, 445514027200451219573049010]

Dujella - Peral - Tadic (2015)

y2 + xy = x3 - 18131452558471483129875821564394494454484900x 
	    + 27258515593219545269929733144764982359147216805507430448480010000

	Torsion points: 

O, [-4871082049418599238200, 2435541024709299619100], 
[1130860723404503757560, 90556969481739681683345838981740], 
[3967867345142114803400, 133361563224381251999394767721500], 
[3967867345142114803400, -133361563228349119344536882524900], 
[1130860723404503757560, -90556969482870542406750342739300]

	Independent points of infinite order:

P1 = [3677101340806938192380, 101516931635836622115160987431500]
P2 = [456697255017527735299328/49, 280320687988858352967786568660804868/343]
P3 = [1105017658807656237512, 92586379385650335920764736151836]
P4 = [37285131903192444003873230/10201, 102079480195952015031266992878925028770/1030301]
P5 = [-21153632112275493108374430700/4661281, -1276111158392901590687686440495353546165600/10063705679]
P6 = [-1031610275069089952242196650792/2047291009, -17641041671674171027159334240799001843934703772/92633776284223]
P7 = [22425927162893873857854908102140/49857721, 106195524618376134691950508628230065359986624760/352045367981]
P8 = [20567114201971284897580030787843900/13451531822689, 2751437681922221518913081493704773640024112414126200/49335282013444325137]

Dujella - Peral - Tadic (2015)

y2 + xy = x3 - x2 - 51900732008393603998885951715125368699x 
	    + 46417971051490003806207921373628830870755520996382489505

	Torsion points: 

O, [9705823536209070331, 21377474622103797304470834122], 
[9705823536209070331, -21377474631809620840679904453], 
[-30462022090411145901/4, 30462022090411145901/8], 
[-567141007878794379, 8698884079181711280939722937], 
[-567141007878794379, -8698884078614570273060928558]

	Independent points of infinite order:

P1 = [-6154572799863548636, -11520294242319441803593622511]
P2 = [7220627323148293479, 6937383364814481651691155279]
P3 = [455126116795783006069341/44944, 225331925195437391204566751648308041/9528128]
P4 = [2041470057744234449240730/69169, 2831337416979550828604821387230408105/18191447]
P5 = [5043342585840457843441936/57121, 11288647158812882569260229688399437663/13651919]
P6 = [48860035682311664588755/56169, 18488986295180666844578659313339090/13312053]
P7 = [2716009269149768367840805/394384, 978461322828567779158262455968895125/247673152]
P8 = [254967578145149321436294966592101/10741114788496, 3886625915284820478563009806677673568390692044693/35202542927684404544]

Dujella - Peral - Tadic (2015)

y2 + xy = x3 - 40181448765949668810483386349055533876x 
	    + 87685454975186288849638211899417160265576039917893134480

	Torsion points: 

O, [-7232264724957374232, 3616132362478687116], 
[6115606257695808744, 8407087653678098186834947980], 
[6115606257695808744, -8407087659793704444530756724], 
[1513362989106346248, 5508388483134534423945289836], 
[1513362989106346248, -5508388484647897413051636084]

	Independent points of infinite order:

P1 = [5767574207154876648, 6913307257513139305206397836]
P2 = [-4392687999111426144, -13395148314120232071888737196]
P3 = [-508299732196550424, -10391262960665328649059773556]
P4 = [-2013136420006718136, -12665603742771148123065580404]
P5 = [873116869278439656, 7298490260347533004478059404]
P6 = [1196221225405469826, 6428940824958120242378308230]
P7 = [-4478026713934625982, -13335007517439310419079211784]
P8 = [73818477197226261511336/729, 20017664665370240627142983822734628/19683]

Dujella - Peral - Tadic (2015)

y2 + xy = x3 - 13579205648744547106359624860424940534765x 
	    + 568609448618954655318256769174781341052119102179499627318225

	Torsion points: 

O, [-534216794604535705105/4, 534216794604535705105/8], 
[105426657885843307730, 555691739321518888942958589935], 
[105426657885843307730, -555691739426945546828801897665], 
[33534357570174157490, 388523715989670962577616455935], 
[33534357570174157490, -388523716023205320147790613425]

	Independent points of infinite order:

P1 = [-68125728946350153670, -1085137499657173204590021465265]
P2 = [-2411792587660417630, -775464794409275143548509448865]
P3 = [47257826968730065250, 180074171077330885332711899615]
P4 = [37629831017824874930, 333031879544959852526337677135]
P5 = [13592542643486836130, 621727301230890014364854211935]
P6 = [13921403378216971430, 618276550882608461881691870435]
P7 = [81597993707839884530, 62224846045143291519345883535]
P8 = [46130304047697398630, 200903157606131590903799414435]

Dujella - Peral - Tadic (2015)

y2 + xy = x3 - 1161478661360056498450485718381297335x 
	    + 795675079452688771133200765624035786829237375610829225

	Torsion points: 

O, [1445245886909047470, -1461434043893853937205491635], 
[-1327049612953350634, 663524806476675317], 
[159697872862442070, -783748842682768308063711435], 
[159697872862442070, 783748842523070435201269365], 
[1445245886909047470, 1461434042448608050296444165]

	Independent points of infinite order:

P1 = [-431902785304108410, -1103065635140539084577485035]
P2 = [2254549066884405462, 3104338604644785612099564981]
P3 = [-4090237474027380, -894665152202297571183112635]
P4 = [999870563737515990, 796215106755089875498414965]
P5 = [77735311620644178472470/841, 21671961090277366062462349437791385/24389]
P6 = [-928617289694051277086/729, -8982321052449310891079230302749/19683]
P7 = [382333220754431321212710/573049, 244526314435395210191808302614541745/433798093]
P8 = [16543676786475853768766520/1776889, 66870680148155482396445570969792440185/2368593037]

Dujella - Peral - Tadic (2015)

y2 + xy + y = x3 - x2 - 574566017499116627722160048766372482x 
	    + 183318463515904127413857838802960720074418403931454081

	Torsion points: 

O, [-884241588715000959, 442120794357500479], 
[227571040211066641, 253671949238545255146888079], 
[227571040211066641, -253671949466116295357954721], 
[708727854212906421, 363453025212131634641373859], 
[708727854212906421, -363453025920859488854280281]

	Independent points of infinite order:

P1 = [327016118214648825, 174347550963201094598668087]
P2 = [524152945235151489, 161746218428062387756839871]
P3 = [358148635689684019, 153226346116986737191013701]
P4 = [429153480712042791, 125617740407867851514090479]
P5 = [20357270873756856289, 91787340742738526836614344671]
P6 = [-17116878043610652891/49, -200413324409213842532654013503/343]
P7 = [171495550578045804822301/370881, 29018928146389652669842676566852151/225866529]
P8 = [35316680003361131871639/39601, 4859197606767536054005218514568761/7880599]

Some curves with torsion group Z/6Z and rank = 6 or 7
High rank curves with prescribed torsion Andrej Dujella home page