Torsion group Z/2Z, rank = 17

Elkies (2005)

y2 + xy + y = x3 -  957089489055751752507625259831765957846101x 
       + 351598252970651757672333752869879740192822872602430248013582348

	Torsion points: 

O, [2541668599439235342183/4, -2541668599439235342187/8] 

	Independent points of infinite order:

P1 = [1037048102780198794447, 21779881979625846052436063081576]
P2 = [640151922319155456727, 1116379497785052017163204160436]
P3 = [8215515531545032362283/4, 671712937153262899818205610666563/8]
P4 = [-252494436924143857397/4, 162335292364600094400633862613143/8]
P5 = [3033356276097950346763/4, 62943339550041852920918618404183/8]
P6 = [-697039650804322492943, 26077977524885678965262644558346]
P7 = [11414518769372982291382, 1215170223607965741833261815562846]
P8 = [891674633090402503696, 14392420123651702644251835214211]
P9 = [94919125996596744048847, 29242036615099659730101568678875176]
P10 = [636565095929048595313, 542487929999796513742831489802]
P11 = [4202001410067480893647, 265563310660870839720550898955176]
P12 = [438987621400369262071, 4005643151170292277973367502092]
P13 = [-1055345243070301537073, 13647840402839172835783080260936]
P14 = [-1120995419674665243803, 3976668559063776088092945926576]
P15 = [476001270991734096802, 1968115763191782636541113312386]
P16 = [639949947842762264623, 1091505880400660592954070253744]
P17 = [45374293429909818890233/64, 2776529523851458968917039502401287/512]

Dujella (2009)

y2 + xy = x3 - 1243378215801409573671681394724148505836465x 
       + 533568551747456170418786926292796873632980274337049541031491817 

	Torsion points: 

O, [2549822399914115274671/4, -2549822399914115274671/8]

	Independent points of infinite order:

P1 = [295909366620588641274, 13840224417503771788066682779563]
P2 = [635444448565646044074, 238026802184285253010435263963]
P3 = [406841929406614922574, 9749398968236802857137233294963]
P4 = [-498536763048246361848, -32086329206297967537307675521387]
P5 = [365329877709573516324, 11317439122188943552599543969963]
P6 = [653658047088929911074, 334802329486502314680260310963]
P7 = [390116221121923660074, 10386461778331482325086438819963]
P8 = [4686357172719668406771/4, 209378289880549899790913148373779/8]
P9 = [620651162007351845724, 971661597627620215237774115163]
P10 = [-174087993049324255086, -27290103511304743453323169025397]
P11 = [601743991009112691324, 1806148426566753856090607319963]
P12 = [-69410472226008425466, -24890512532673385235973153079977]
P13 = [-334709979328169335446, -30203341402018178778521393923557]
P14 = [628923493523439835074, 588364888705804863742467594963]
P15 = [594390299829369312882, 2124756989643788803088946333003]
P16 = [659600354729670357324, 640233767104009049626213294713]
P17 = [137924634886574871255456/169, 17355263577043625203177043015047311/2197]

Torsion group Z/2Z, rank = 15

Dujella (2002)

y2 + xy + y = x3 + 34318214642441646362435632562579908747x 
       + 3184376895814127197244886284686214848599453811643486936756

	Torsion points: 

O, [-55741267008740887705/4, 55741267008740887701/8] 

	Independent points of infinite order:

P1 = [-5955399047526089895, -52619192486073556789679851928]
P2 = [-10883488374931039920, -39009259582579792447480800428]
P3 = [25617164053798897605, -144480091250102706688189014428]
P4 = [-10819998365744200320, -39323416370158251425370822053]
P5 = [104461756233244297605, 1070832856583711138856374116822]
P6 = [-13178335910050336770, -21058433294831012498764155053]
P7 = [-8109021285488360520, -48712125024735484500144150053]
P8 = [-13641790592099985510, 13322871549061184750149766947]
P9 = [17051811768980000730, 93421745224403634416501304322]
P10 = [51413108001886672605, 375299141884454862856563616822]
P11 = [-9527137759746081954, -44639433008625651474894783026]
P12 = [-674553190484964645, -56222064156796770815845330928]
P13 = [60487341240781250730, 475989858966422000049387054322]
P14 = [44737088527034836586505/1681, 
      10440384606973435833942470793114062/68921]
P15 = [3423934952363100338459817/36481, 
      6360102793003685525437050804199384514/6967871]

Dujella (2009)

y2 + xy = x3 - 44983303148372813062091588735006222790x 
       + 113817052678905764015135842208733506900505149372185755492 

	Torsion points: 

O, [13670258741780008607/4, -13670258741780008607/8]

	Independent points of infinite order:

P1 = [4310210790883112544, 65365365737680503958746198]
P2 = [3256139879023839684, 1366812206363889258442661898]
P3 = [5606214825113296074, 6150799258812661160218467438]
P4 = [24740932855465046124, 118933399713162424765457508138]
P5 = [115043084731737275724, 1231878170189110085685888799338]
P6 = [6084305921271097704, 8084457212260904108353542318]
P7 = [814735121550003833076/169, 6591145014838387448887374359946/2197]
P8 = [4325066798743811724, 408458629406023762754275338]
P9 = [-6887211948173828136, -9845848358426608735316715222]
P10 = [1565261985677118444, 6873235096567408584547967658]
P11 = [616492241812365224364/121, 5469682223035425660234155027838/1331]
P12 = [2143543503151209224, 5219434221398996545403197838]
P13 = [-4942390918231726236, -14676961940075347315520232942]
P14 = [4823556538171617775944/841, 162676940434775957768151444511122/24389]
P15 = [-3224165652616605156, -15011152001960539717353346902]

Dujella (2009)

y2 + xy + y = x3 - 9787472643107005523075719618421934671816478x 
       + 1666408105881456642404124109941261415712089105209504669167254756 

	Torsion points: 

O, [-12841521034486825188905/4, 12841521034486825188901/8]

	Independent points of infinite order:

P1 = [-6176007349090217626405/4, -915553551467723487022948702104849/8]
P2 = [-3129727543028407345720, -40524570541991864371797835504653]
P3 = [3461438776471679393555, 96234617652133597957415588750097]
P4 = [-304569520786114509444805/4489, -14518337791411701360968690696166211989/300763]
P5 = [46749918438362505836941130/3721, 309717635823882664652843137775449108032/226981]
P6 = [-121760299909016401445, -53444630640731736532288009219903]
P7 = [-3878023321719040442905/4, -809706755031980940882066233773599/8]
P8 = [-650904952869654088945, -88098518147939984998018731159903]
P9 = [3211833539877933133055, 57996099141689105147032333358847]
P10 = [-279627254130111308410, -66192052084066431198211238763528]
P11 = [4778427566087150721680, 252992655387344035936908771921972]
P12 = [-22549528799987020100995/16, -7201356571428059433520262702341917/64]
P13 = [-41484235392297509524075/16, -6275050846011773962948320830988897/64]
P14 = [11404299380469664557060755/3364, 16838047702372204271654455398073697139/195112]
P15 = [2032370808462899194583150/529, 1753546464147333979091664135941424159/12167]

Dujella (2009)

y2 + xy = x3 - 38064598122520711827608887857833864769365x 
       + 2856369890490724644127902394897782785656508815105789929216225 

	Torsion points: 

O, [440616430188382357359/4, -440616430188382357359/8]

	Independent points of infinite order:

P1 = [154978462370139206530, 824317835312651521651728277735]
P2 = [101211988731799110310, 201440640099151015743920851435]
P3 = [-138339729063805884254, -2339804876291888838641359535273]
P4 = [107195862460181729490, 88233073199137259807678945175]
P5 = [118199632472186128210, 92377192999174562695465598935]
P6 = [109228167835768103350, 42687234750831717124370829115]
P7 = [1606735493653863109090/9, 35725359320226464578835504931245/27]
P8 = [319346546500341238540, 4823716478183218456447348148125]
P9 = [1632745931431399908160/9, 37427654249574472129716075752725/27]
P10 = [2968406979076876830646, 161387128120275934793381147733307]
P11 = [50435845020951603864809650/458329, 4266112332467988872880575035369208795/310288733]
P12 = [205650416099455169679121330/1874161, 71005882980840361899501165201016633615/2565726409]
P13 = [1020136311332667375161410/9409, 56764307939951866079331723615072775/912673]
P14 = [112794531630621535030942506847930/358368652321, 1007521941982262938831142487558547877562152870065/214533451656791119]
P15 = [166817185151770979365206811379605/1169624946064, 725173124355794556504624330436224982365418744135/1264940022168647488]

Elkies (2009)

y2 + xy = x3 - x2 - 47377997631934616131259094x 
       + 122064326885475393485584567015835076600 

	Torsion points: 

O, [13690851677475/4, -13690851677475/8]

	Independent points of infinite order:

P1 = [4500895301116, 17895577089353792]
P2 = [-7879047179700, -2496274582710240120]
P3 = [-7791526340259, -4266631333507969158]
P4 = [-7816770890871, -3845470103309695578]
P5 = [-7919982209319, -712375905043009743]
P6 = [-7861912714884, -2933213810436686208]
P7 = [-7921327767924, -563899328680245528]
P8 = [-7679614164996, -5743906466119247676]
P9 = [-7898698253999, -1869092036434260498]
P10 = [-7858400307321, -3014687899238193876]
P11 = [-7918000894850, -886766356327884100]
P12 = [-7916835300134, -974847693981397408]
P13 = [-7902140608214, -1735516496710803088]
P14 = [-7787026675214, -4337047328172214698]
P15 = [-7830443686839, -3595099957696460463]

Torsion group Z/2Z, rank = 14

Fermigier (1996)

y2 = x3 + x2 - 1692310759026568999140789578145x
      + 839379398840982294584587970773038145228669599

	Torsion points: 

O, [809823326908353, 0]

	Independent points of infinite order:

P1 = [866442291552879, 4852554862534554468168]
P2 = [643619051198181, 4097833397957401859316]
P3 = [310789314615903, 18532307930418477983880]
P4 = [935470575993669, 8654920131154858324092]
P5 = [988927329611391, 11530632350958995292984]
P6 = [832699306754277, 2752675553616194984436]
P7 = [2252513103656559, 91957939555201321662648]
P8 = [690291245594778, 342430653544628328855]
P9 = [2191409680250085/4, 70052106447307921537881/8]
P10 = [681925764652050, 1568465432141985518043]
P11 = [244337019116010, 20987431811752700667807]
P12 = [982479697930607, 11183605599461162465976]
P13 = [75293021661996670215/22201, 610918230634190643034903681368/3307949]
P14 = [1177083061299989778618/1261129, 
      12096101176020871187811506551077/1416247867]

Kulesz - Stahlke (2001)

y2 = x3 - 12980282175768890811438819732x 
      + 116802985717915987899448766677826872704944 

	Torsion points:

O, [-118188654139412, 0]

	Independent points of infinite order:

P1 = [6607378832738, 176990975394487960300]
P2 = [-103557136527662, 591987135009508933500]
P3 = [623445113629378, 15308407213006350954420]
P4 = [-655755381332, 353998013583629010960]
P5 = [-54341822899562, 813450641210644821300]
P6 = [-101353861608362, 625489006156183529100]
P7 = [-71527319807162, 824198245522092094500]
P8 = [-52956155250062, 809741315448326067300]
P9 = [-18026547302282, 587311513851766998060]
P10 = [-69415283384042, 826653910581533974860]
P11 = [-7975654090532, 468851667964437988440]
P12 = [373258347587308, 6875669199226331967840]
P13 = [148248807617938, 1204433799812463413700]
P14 = [579564746423857/64, 139391042715474957225/512]

Dujella (2001)

y2 + xy = x3 - 7372911492530406268416156245x 
       + 243594391906613268628507257344604677608161 

	Torsion points: 

O, [195391032383343/4, -195391032383343/8]

	Independent points of infinite order:

P1 = [-72376481830226, -630940588803637101569]
P2 = [56312693537242, 83547350623177255879]
P3 = [50545746944914, 7938107321308587511]
P4 = [-24664074799982, -640653456150957147401]
P5 = [48768170604178, 4243475038421673079]
P6 = [44275623883186, 62841111181852200343]
P7 = [44033644751050, 65709829875594008719]
P8 = [48781448921950, 3856666655607214531]
P9 = [48831932859328, 1852759677330985393]
P10 = [226433646572535934/361, 106787838011896189595371765/6859]
P11 = [153942078863442, 1660342552186148579319]
P12 = [2829817753498450/9, 145424908163363854584413/27]
P13 = [6692769349018554/289, 1434645063716592596334423/4913]
P14 = [270782451612808972/1849, 120702043132790721191878453/79507]

Dujella (2002)

y2 + xy = x3 - 9468594009199910821775026235x 
       + 391555244342706113038825183927385153742225 

	Torsion points: 

O, [-454561868024561/4, 454561868024561/8]

	Independent points of infinite order:

P1 = [47837672254570, 219256739278969879465]
P2 = [159012385189570, 1704860597983625569465]
P3 = [113203019208730, 877707099875585260105]
P4 = [-379785557030, -628610550798257040035]
P5 = [150694553832070, 1544923707786326053465]
P6 = [4423977031795/4, 4938562614739635671195/8]
P7 = [-105239107838570, -471667325486574870335]
P8 = [-97378850096330, -624649787995923803135]
P9 = [2781894749213470, 146638885030002257302465]
P10 = [5956132655951830/169, 700376526525629066927605/2197]
P11 = [5727420806916370, 433387520147639397982165]
P12 = [2510868216033430/49, 69378911457566195617495/343]
P13 = [29411611466070, 372170566570883535465]
P14 = [189299897497282, 2320045014085200589957]

Dujella (2002)

y2 + xy = x3 + 3316490747528662771547963488767495x 
       + 73682323027772693525742362632573935085905483165177 

	Torsion points: 

O, [-79425459402809553/4, 79425459402809553/8]

	Independent points of infinite order:

P1 = [42094543239920394, 16966966004633078487992403]
P2 = [27540919289973114, 13634932178069643502918323]
P3 = [44598680591423634, 17615393735713734372514443]
P4 = [-1136456995004838, -8361328012699544626237965]
P5 = [-15098621421260856, -4490643229989030974701347]
P6 = [26579506093367994, 13439145664988645954960403]
P7 = [-1917696386868006, -8204587024706744054523597]
P8 = [419077623404760282, 273979011551621447610165555]
P9 = [-2044470545809506, -8178833031796563463824597]
P10 = [359645054924887740306/169, 6822943308980403701468300739351/2197]
P11 = [207192901970717874, 98261888491687585278481443]
P12 = [196062307481841535386/49, 2745594468228521388878051373909/343]
P13 = [282299098535186380866/289, 4751544447032516633024737563339/4913]
P14 = [65052368021618394, 23763779112715350509257203]

Dujella (2002)

y2 + xy + y = x3 - 12949314925486190388475166345996148x 
       + 567175248136512872178139054853895538293674461867078 

	Torsion points: 

O, [263129338838302711/4, -263129338838302715/8]

	Independent points of infinite order:

P1 = [65601277223770979, 23563368510169401694305]
P2 = [66125889214142084, 185860513983389743561770]
P3 = [-69485744108634566, -33637363010212406099385505]
P4 = [-32331867075002041, -30855350764246689879332355]
P5 = [65800723636725209, 25822487830178626473645]
P6 = [65788305691643459, 14208407210362871438145]
P7 = [63871447749001739, 807017461103066030534475]
P8 = [66025536242433209, 140081934594192315807645]
P9 = [64736942716038959, 424751832934349364408645]
P10 = [-14167608711707416, -27345791928090461105989230]
P11 = [2020573482310072484, 2867722481686210471818712170]
P12 = [-43599334996233586, -32386396874479931764271565]
P13 = [23541087505805298449/361, 1465209842401319333313932055/6859]
P14 = [39933036650693051/4, 167598351724390091167288575/8]

Dujella (2002)

y2 + xy = x3  - 199429807965769527099340083105123887569092655x 
       + 946696238553698249107052023781560988843839015226125020059567966777 

	Torsion points: 

O, [-64290689047140226735153/4, 64290689047140226735153/8]

	Independent points of infinite order:

P1 = [5639318228208776052212, 37271684216749719028429712708303]
P2 = [15704314119697504774094, 1299181253016152729254527327109253]
P3 = [5646086610056830399244, 26189293677432863920539989800103]
P4 = [8483687132452927799891/4, 5841982001348824946858354165397079/8]
P5 = [5291742343771787729174, 198865004663386643794432850951933]
P6 = [563809874246061125624, 913474120253834186871652935806243]
P7 = [986165512420371070374946/9, 971508041876787699448435768073304131/27]
P8 = [2129108697489953593600229/16, 3089765187069780494937336263780777717/64]
P9 = [55517454259089667567694, 12688186520170214767232133089825453]
P10 = [12733217275021471264502, 686885056742081034745356367894925]
P11 = [87980925084778734722696966/6889, 
      397287764237169029978161387104794075311/571787]
P12 = [820513627981586695423544, 743129902037346668451025626696015203]
P13 = [5258429421462479540535254, 12058200726631987221580325174355239213]
P14 = [2872756878358176252959674442/508369, 
      4912881888067244704270816163064058926397/362467097]

Dujella (2002)

y2 + xy + y = x3  - 1773871677446826525005545991723918613373997023x 
       + 28749498729292324150415916003880104290244501502154076094544626253078 

	Torsion points: 

O, [96051482732372697840711/4, -96051482732372697840715/8]

	Independent points of infinite order:

P1 = [-47044310997606458664291, -2843086997644656346080364015494980]
P2 = [23745203486730985504984, 130093056208578535670863801614170]
P3 = [205573821123733735733881/9, 10414838935169185953308935885141790/27]
P4 = [15696202411860401264209, 2184842169148806945167348397048020]
P5 = [-17291536024932555789791, -7365619021860070584720462749450230]
P6 = [39131903091467280157879, 4388329507640944379855417135264990]
P7 = [20079751914859584086119, 1107555842610401088236193721385750]
P8 = [24010761905126646071959, 9651119797335609647441543876270]
P9 = [31504114780415394538839079/1156, 
     31628049196167576342016172027834709605/39304]
P10 = [26921537270076604490689, 711369417084865847526978127961300]
P11 = [95284429954576796577751/4, 842207664122997953807518180209025/8]
P12 = [23195498903484427434279, 289070198352919962487571429038390]
P13 = [116445120400296998088803386/5329, 
      252612413972703463025231084884895504840/389017]
P14 = [8327200301669504405477307229/281961, 
      218023730294084529820717977462112475769460/149721291]

Dujella (2002)

y2 + xy + y = x3  - 26363867662083075045300295608x 
       + 1621855500711191032644827748500946408874618 

	Torsion points: 

O, [335995856219831/4, -335995856219835/8]

	Independent points of infinite order:

P1 = [-13523214151261, -1405669456443029816745]
P2 = [108064574225729, -186623470209939903855]
P3 = [-9283400340886, 1365943554420666161130]
P4 = [45091038864014, 724403525277098828655]
P5 = [-39381624677386, -1612150715652115017870]
P6 = [103202418965714, 14921148085177587930]
P7 = [229522556724614, 2768056987261476921255]
P8 = [196487519758739, -2006874802546559401995]
P9 = [248201700083864, -3220021411044645365370]
P10 = [121414876687139, 459059783092694170455]
P11 = [105113501512604, 109727698460088223125]
P12 = [-88622695886761, -1806170337626894748495]
P13 = [-152427507622741, 1448762069584790066925]
P14 = [82158932261969, 102021930719234171685]

Watkins (2002)

y2 = x3 + 402599774387690701016910427272483x 

	Torsion points:

O, [0, 0]

	Independent points of infinite order:

P1 = [17715373576525779, 3562569314711466369088086]
P2 = [2626434695669379, 1037072601415883504491614]
P3 = [2569230493256067, 1025344316293086716196318]
P4 = [235538747268099, 307962520197557881526046]
P5 = [72777729441003372, 20366017444893924849237282]
P6 = [208383733118864688, 95565185470960061947766676]
P7 = [36178079522739, 120686925577870348570566]
P8 = [103189419061250643, 33768487838255557704513174]
P9 = [1751414347117072176, 2317991574180462284959749972]
P10 = [306104494367228425/4, 175082211930567255911081155/8]
P11 = [54693351931994304/25, 118007688830447299097189592/125]
P12 = [2696555916804876, 1051304226981395145047478]
P13 = [2842774711299072, 1080497092155012281695968]
P14 = [46439279877409015377/1681, 391130341466321391183789029622/68921]
High rank curves with prescribed torsion Andrej Dujella home page