Torsion group Z/5Z, rank = 7

Eroshkin (2009)


y2 + xy = x3 - 2197114487075051237175634730x 
         + 118333571404896316649459410056137259532452

	Torsion points: 

O, [59463233460844, -444905264294837261522], [59463233460844, 444905204831603800678], 
[-29585624156876, -396787090739098935482], [-29585624156876, 396787120324723092358]

	Independent points of infinite order: 

P1 = [-378868717196, -345204192961172336762]
P2 = [-18123611174456, -390128486461247649722]
P3 = [25352870402644, 280938463349715427078]
P4 = [-6136039401356, -362745258200756284922]
P5 = [5106719426164, 327485417244493724038]
P6 = [17027927411344, 293016233815608523978]
P7 = [33939272137984, 287852161454273240218]

Eroshkin (2009)


y2 + xy = x3 - 217355654325805035467922425515x 
         + 42013652306493555363669753849175483887204017

	Torsion points: 

O, [-103188168425626, -7958860255592416399087], [342056119662974, -2772580877384580891487], 
[342056119662974, 2772580535328461228513], [-103188168425626, 7958860358780584824713]

	Independent points of infinite order: 

P1 = [188844878934974, 2775213317289131180513]
P2 = [86424485294864, 4886135730598232401433]
P3 = [105071256648074, 4509523645187814424313]
P4 = [-59941952884006, -7404525680681083827367]
P5 = [233638545003734, 1996149592252359197873]
P6 = [102255533637974, 4566950528612304203513]
P7 = [177878745378974, 2996487115973348984513]

Eroshkin (2009)


y2 + xy = x3 - 25985723533607031680666573307358665x 
         + 1777284485143248504295316827635107337913198661235225

	Torsion points: 

O, [-40882024231352070, 52643192418015210755896035], [-40882024231352070, -52643192377133186524543965], 
[121439602850462130, 20310796338887521474159035], [121439602850462130, -20310796460327124324621165]

	Independent points of infinite order: 

P1 = [4298590594051680, 40812521694358653560877285]
P2 = [17699871658437930, 36371494681679962171246035]
P3 = [78787005095985420, 14798952334535535860795985]
P4 = [8802508579777146, 39360220850496902632460067]
P5 = [-19386810987848070, -47684151372081292027807965]
P6 = [87283873682182950, 13195448026304205597402735]
P7 = [-68935033021558470, -56930038126087142850281565]

Dujella - Lecacheux (2009)


y2 + y = x3 + x2 + 21684687816792871940x 
         + 536323633330486820975424434082364

	Torsion points: 

O, [-44542393429, -21142008695255333], [-44542393429, 21142008695255332], 
[109043977586, 42840230433141487], [109043977586, -42840230433141488]

	Independent points of infinite order: 

P1 = [14602668701, 23232609206364442]
P2 = [-26663475064, 22733000003801587]
P3 = [12940574036, 23211446619966412]
P4 = [-67481833414, 15085147779935887]
P5 = [25151681021, 23511277519972357]
P6 = [43444943771, 24885064864904107]
P7 = [1428145783223, 1706872570530131119]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 279021136261247540446385495x 
         + 1793387038596711777513523387558820786025

	Torsion points: 

O, [9961734198790, -1554603022023059195], [9961734198790, 1554593060288860405], 
[16200652404310, 39052575274144704085], [16200652404310, -39052591474797108395]

	Independent points of infinite order: 

P1 = [-5948999789450, -56945102666568630155]
P2 = [4319183174680, 25861542387494115115]
P3 = [-7446369281978, -58806391157676338363]
P4 = [9786751791190, 237729005307986005]
P5 = [-251236942310, -43167946288761192995]
P6 = [9892302676630, 1123837020359445205]
P7 = [5263206579862, 21694214378887910101]

Eroshkin (2009)


y2 + xy = x3 - 4401046837403916631215x 
         + 113247068123398333719870687876921

	Torsion points: 

O, [42403442826, -1694460286433787], [1071123798, 10417975804721613], 
[1071123798, -10417976875845411], [42403442826, 1694417882990961]

	Independent points of infinite order: 

P1 = [19668144020, 5856182535325679]
P2 = [27196268844, 3697334491928205]
P3 = [23690800878, 4720083672205029]
P4 = [-71804280804, -7684274698415187]
P5 = [-5714779326, -11756333405379435]
P6 = [16469438730, 6725429087398149]
P7 = [31099970646, 2540611635246813]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 150896123934128959243712285x 
         + 844403124386456716996509171987377289009

	Torsion points: 

O, [9870859113370, -17795640513230511713], [9870859113370, 17795630642371398343], 
[-3956691250334, 37141742528679301039], [-3956691250334, -37141738571988050705]

	Independent points of infinite order: 

P1 = [6185569421986, 12152873534380947055]
P2 = [7399018466710, 11531808471208398823]
P3 = [6877371129730, 11485726171880266063]
P4 = [5996702988790, 12456642032009666983]
P5 = [3438036957610, 19137777873360242623]
P6 = [-14425657901400, -4382373755072136297]
P7 = [-13501730569340, -20504642078577659777]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 597872211893293896943612690x 
         + 6922653506619993467506840401756825604100

	Torsion points: 

O, [20228774992580, -55732525780462251490], [-8539067008780, 106795568943311265710], 
[-8539067008780, -106795560404244256930], [20228774992580, 55732505551687258910]

	Independent points of infinite order: 

P1 = [136948832900, 82708996863243207710]
P2 = [12658290255380, 37187096661453156110]
P3 = [-8085873615820, -105963703424465145490]
P4 = [1502180413580, 77639752721509657910]
P5 = [12034546795460, 38347087967452878110]
P6 = [4298443999850, 66574422404286487460]
P7 = [13688354493380, 36104952169567028510]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 235156296239005741872850045635x 
         + 43891763533599932513239182886961588521260225

	Torsion points: 

O, [280013549576070, -1154724224541667935], [280013549576070, 1154444210992091865], 
[268503905229870, 330141353210013565065], [268503905229870, -330141621713918794935]

	Independent points of infinite order: 

P1 = [234336284957370, 1286209419221406542565]
P2 = [200466598749870, 2192465003743342405065]
P3 = [-2834127485880, -6675193156218681124185]
P4 = [279920814527370, 1546651453491280065]
P5 = [248556346294800, 893335622087535108015]
P6 = [42471250346120, 5829321684481105923815]
P7 = [279929510903066, 1295779862979508037]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 3315884079090771665081738064855x 
         + 2323793946140310258573734535551993806310705769

	Torsion points: 

O, [1074668151366810, 1210654284981453877257], [1074668151366810, -1210655359649605244067], 
[1629880818005286, 35342540073250423143669], [1629880818005286, -35342541703131241148955]

	Independent points of infinite order: 

P1 = [895567291160850, 8513406983981821272813]
P2 = [489239829401070, 28611775007953371909477]
P3 = [1029838889959110, 1087143528194427522357]
P4 = [1041205220283690, 239510725686093270717]
P5 = [1068653764541286, 828759087493249419045]
P6 = [771016537342590, 15017855263894303405893]
P7 = [1318981694121630, 15647771154938585882373]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 1017843352326919892221795x 
         + 583997132716712282566571150568433025

	Torsion points: 

O, [-442309545850, 973481872856250305], [933608052590, 668943151205532905], 
[933608052590, -668944084813585495], [-442309545850, -973481430546704455]

	Independent points of infinite order: 

P1 = [60009577390, 723279229166022505]
P2 = [509759245670, 444527923405600505]
P3 = [-121890379090, -840387639572936215]
P4 = [207429320080, 617892786310079905]
P5 = [320127896840, 539410987983979505]
P6 = [-384838786210, -958492251841957495]
P7 = [704156584190, 465212037243711305]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 2073251774144136812897272941735x 
         + 1432876183537108072707643893400054387212141897

	Torsion points: 

O, [-515002840869066, 48621123256021381616133], [-515002840869066, -48621122741018540747067], 
[1202612554944174, 26055102491519409375573], [1202612554944174, -26055103694131964319747]

	Independent points of infinite order: 

P1 = [118329949700094, 34484855437424298847413]
P2 = [867058115530134, 16943784073694556490533]
P3 = [909384906347634, 17307115669822286446833]
P4 = [-1407270578953866, -39541483865560829324667]
P5 = [706412551709238, 17911397578027180382661]
P6 = [-355318770906726, -46094272277206204884327]
P7 = [379801868296224, 26462000875001288849373]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 4174902332253438226592458632430x 
         + 13497276399741906319596161056780175385144581252

	Torsion points: 

O, [-1438431165027556, 128554855957315582361378], [-1438431165027556, -128554854518884417333822], 
[2939345086547324, 163159439248843133534018], [2939345086547324, -163159442188188220081342]

	Independent points of infinite order: 

P1 = [436263387243644, 108438699147934027373378]
P2 = [-1308685355618956, -129304215583417012171222]
P3 = [-94773236670196, -117864728164768401462142]
P4 = [129855565474364, 113830276144389006060098]
P5 = [-1144767405136516, -129523586127315792602302]
P6 = [985939082533244, 101683255849646905939778]
P7 = [621525711651068, 105558320254423194976706]

Dujella - Lecacheux (2009)


y2 + xy = x3 - 104247069206207225692790x 
         + 13077998295796069821444447267212100

	Torsion points: 

O, [-884672030, -114761587511772080], [-884672030, 114761588396444110], 
[208048362280, 19868382873528310], [208048362280, -19868590921890590]

	Independent points of infinite order: 

P1 = [116272397230, 50287710041280910]
P2 = [135871495882, 37711009485178696]
P3 = [-209059328720, -160420326788933090]
P4 = [111490681480, 53303507561825110]
P5 = [179713181770, 12148632954661870]
P6 = [174119066110, 14333157496354510]
P7 = [130979046880, 40875709416792910]

Dujella - Lecacheux (2009)


y2 + xy = x3 + 50627101565668207228579525x 
         + 31560876466833383058857093536380500625

	Torsion points: 

O, [425783366050, -7293438707812816475], [425783366050, 7293438282029450425], 
[11454599000110, 45982677929730807265], [11454599000110, -45982689384329807375]

	Independent points of infinite order: 

P1 = [-605169073250, -837429992429581775]
P2 = [-364576031330, -3613168901523318095]
P3 = [-235888035350, -4427801072954045975]
P4 = [1546697018650, 10656724003864614025]
P5 = [-574263218390, -1515986080954691735]
P6 = [3969736276270, 17178339680519480305]
P7 = [-367692673010, -3591093088057271375]

Torsion group Z/5Z, rank = 6

Dujella - Lecacheux (2001)


y2 + y = x3 + x2 - 1712371016075117860x + 885787957535691389512940164

	Torsion points: 

O, [888689186, 8116714362487], [-139719349, -33500922231893],
[-139719349, 33500922231892], [888689186, -8116714362488]

	Independent points of infinite order: 

P1 = [624069446, 7758948474007]
P2 = [763273511, 4842863582287]
P3 = [680848091, 5960986525147]
P4 = [294497588, 20175238652299]
P5 = [-206499124, 35079702960532]
P6 = [676477901, 6080971505482]

Eroshkin (2009)


y2 + xy = x3 - 18179020279645248226680989180x 
         + 943130631992337822189098312617324575056400

	Torsion points: 

O, [80426476207360, -35891019822063144380], [80426476207360, 35890939395586937020], 
[130967719720120, 899274821884379441860], [130967719720120, -899274952852099161980]

	Independent points of infinite order: 

P1 = [-18761661361160, -1130307537589895283260]
P2 = [78963564397000, 2742997696968750580]
P3 = [79138318576600, 10334820762960314980]
P4 = [25180979020480, 708047939044305400900]
P5 = [45575351353360, 457471871820804920620]
P6 = [81784220164072, 58312634986292135236]

Eroshkin (2009)


y2 + xy = x3 - 86360896643095981339889711579335x 
         + 309665302194639946752926689756764603565256969225

	Torsion points: 

O, [5731791556264070, -54513140432723098112635], [1056726531333770, -468599474785909158178135], 
[1056726531333770, 468599473729182626844365], [5731791556264070, 54513134700931541848565]

	Independent points of infinite order: 

P1 = [4158217830147062, 149855571799355801199557]
P2 = [4549821734431270, 104517562985058018809365]
P3 = [4980535896409370, 55566390628513872120365]
P4 = [4383994008062966, 123763475455989744380357]
P5 = [7574149731348770, 300111348963862047534365]
P6 = [4196269588368770, 145473225310741094934365]

Eroshkin (2009)


y2 + xy = x3 - 203194603752496447018924708520235x 
         + 1114929529839196014201757121107587089072126910897

	Torsion points: 

O, [4718983640383614, 511022162467774249866273], [4718983640383614, -511022167186757890249887], 
[8369986802710134, 23808762469535901991833], [8369986802710134, -23808770839522704701967]

	Independent points of infinite order: 

P1 = [8027600100620988, 32894236019256470217201]
P2 = [-3829999133799666, -1355353397506211679356367]
P3 = [8061961722017574, 27774856907561242483473]
P4 = [7946968104936462, 45096768529877802703665]
P5 = [8034202165161774, 31905090505386929804673]
P6 = [-3374408499549486, -1327466750275500216108027]

Eroshkin (2009)


y2 + xy = x3 - 351977692642507927577441218602598795x 
         + 80382547686623410309989730337857042541667137846263025

	Torsion points: 

O, [349028910355532990, -7160374460319340991434495], [188740966643636990, -143782773666365128409818495], 
[188740966643636990, 143782773477624161766181505], [349028910355532990, 7160374111290430635901505]

	Independent points of infinite order: 

P1 = [326200176684289490, 16649302728555119433106505]
P2 = [-27389228425951810, -300003997523526814668053695]
P3 = [338181175716791990, 5188233499156614642031505]
P4 = [308881601145739490, 33657716433371790114294005]
P5 = [329547875183614670, 13361888436752341653463505]
P6 = [463278590311144190, 129425409263297735417125505]
High rank curves with prescribed torsion Andrej Dujella home page