Torsion group Z/7Z, rank = 5


Dujella - Kulesz (2001)

y2 + xy = x3 - 3871874529581532682991685x 
	   + 2932443695831973034287157248711276225

	Torsion points: 

O, [1171933025970, -66562907187908985], [1137976122990, 3199206958195395],
[1910775530370, 1584461941569375015], [1137976122990, -3200344934318385],
[1910775530370, -1584463852344905385], [1171933025970, 66561735254883015]

	Independent points of infinite order: 

P1 = [1139468193570, 6113353595217015]
P2 = [423997075362, 1169189107911863031]
P3 = [1157854869570, 40341192423477015]
P4 = [1137065493270, 1069297314954015]
P5 = [1137272072370, 1646036791995015]

Elkies (2006)

y2 + xy = x3 - 801634269837107139166772851280x 
	   + 315710247481803582042676623278248394848262400

	Torsion points: 

O, [1131186190931680, -29263601581724789037200], [430440545011360, -7099716846165341836880], 
[-561233063314040, -24265928141999279774480], [430440545011360, 7099716415724796825520], 
[-561233063314040, 24265928703232343088520], [1131186190931680, 29263600450538598105520]

	Independent points of infinite order: 

P1 = [85526630898592, 15740863331343335618992]
P2 = [290385834223569760/961, -299500046651904120799989680/29791]
P3 = [-178019446528040, -21278513390150019041480]
P4 = [-81372900992864, -19503915948192485780048]
P5 = [-437834763635540, -24140442806878317303980]

Elkies (2006)

y2 + xy = x3 + 17189192408099124368974280x 
	   + 14347563590500778387882617467320974400

	Torsion points: 

O, [13227279881840, -50556618967055108920], [2286184756880, -8099031927845206360], 
[-454714844560, -2537195715703556920], [2286184756880, 8099029641660449480], 
[-454714844560, 2537196170418401480], [13227279881840, 50556605739775227080]

	Independent points of infinite order: 

P1 = [-733027479160, -1163415328491323920]
P2 = [-683839158160, -1507690912971332920]
P3 = [1495149953554160/1849, 426501021351985991673560/79507]
P4 = [55827804602887280/2209, -13373046199076994518617160/103823]
P5 = [2543774259440, 8633260650255723080]

Elkies (2006)

y2 + xy = x3 - 1644173544974177009215399260x 
	   + 29144075820496678960101949181880262179600

	Torsion points: 

O, [50748118673640, -276406650897190527180], [19543376894520, -66901727956450562460], 
[-25215304199880, -233602705412996604060], [19543376894520, 66901708413073667940], 
[-25215304199880, 233602730628300803940], [50748118673640, 276406600149071853540]

	Independent points of infinite order: 

P1 = [18840522713400, 69675800769119471460]
P2 = [-5279882261448, -194108035447655037084]
P3 = [28556991474360, 74025000283910777700]
P4 = [14185096013112, 93142882933735615332]
P5 = [-16480379716140, -227518256916697261800]

Elkies (2006)

y2 + xy = x3 - 239247306678234007210828097520x 
	   + 45246626407933026379552521501618249969926400

	Torsion points: 

O, [-148947654166560, -8807808260645246196720], [393999910993440, -3485133109850043996720], 
[267568080668640, 622574473646697227280], [-148947654166560, 8807808409592900363280], 
[393999910993440, 3485132715850133003280], [267568080668640, -622574741214777895920]

	Independent points of infinite order: 

P1 = [46074969993440, 5858423537724469003280]
P2 = [275652423665370, 492718372260784145100]
P3 = [138768613071840, 3836513831575606436880]
P4 = [297508887793440, 633557750590112003280]
P5 = [169967679183690, 3080992491904517095530]

Eroshkin (2009)

y2 + xy = x3 - 11181845072095618101356644660230x 
	   + 14391914697117329050689039214352461501292556900

	Torsion points: 

O, [1986637187437500, -4284940246971133320870], [1928016592364460, -218287646293676691030], 
[897878946226860, -71244855727391880665430], [1986637187437500, 4284938260333945883370], 
[1928016592364460, 218285718277084326570], [897878946226860, 71244854829512934438570]

	Independent points of infinite order: 

P1 = [-50212338713940, -122283501274174214105430]
P2 = [1494991187264172, 31881939944292796642986]
P3 = [1933393961761860, 230632595187600474570]
P4 = [2937317706484836, 83005881436952963158602]
P5 = [469571691228966885/16, 319772096187718923698671605/64]

Dujella - Lecacheux (2009)

y2 + xy = x3 - 23029452969111267963793629026710x 
	   + 43247274604914893171276548913972152943570776100

	Torsion points: 

O, [-2026713362963980, -285651033709562007205510], [4404650476318220, 165121071122022034846490], 
[2534623395651020, -34051511381888809345510], [4404650476318220, -165121075526672511164710], 
[2534623395651020, 34051508847265413694490], [-2026713362963980, 285651035736275370169490]

	Independent points of infinite order: 

P1 = [-2346860782356820, -290462146379303405728870]
P2 = [2850183100882700, 27617093411134421129690]
P3 = [2758937068611020, 26660034889979335294490]
P4 = [-1201537965455980, -263027344254022693525510]
P5 = [-5139499726938280, -160779489389448463993510]

Eroshkin (2009)

y2 + xy = x3 - 19970723901492180991598876405614595610x 
	   + 1128216419636366078345077292747293694901309237422300864100

	Torsion points: 

O, [27699579731035398420, 147743024764975177294792323990], [27699579731035398420, -147743024792674757025827722410], 
[4234627545077060820, 33460177546079624864074621590], [-8213766415332397140, 27168026339294265704253226590], 
[-8213766415332397140, -27168026331080499288920829450], [4234627545077060820, -33460177550314252409151682410]

	Independent points of infinite order: 

P1 = [-2703095212592975148, -34094697795764678945874029418]
P2 = [44901815012942018820, 301266196829776702902849541590]
P3 = [44605145162873126520, 298302971098885510345090573590]
P4 = [15097279409901820820, 65328425588859962336358621590]
P5 = [7179393227630076720, 36808845915857031738953928090]

Dujella - Lecacheux (2009)

y2 + xy = x3 - 36180559909936665984616479270x 
	   + 2653605664924418914307133525288436083708900

	Torsion points: 

O, [105804676040940, -99886935778126468470], [-42659775320220, 2029636825097341138290], 
[142595695818540, 627604708716620443530], [-42659775320220, -2029636782437565818070], 
[142595695818540, -627604851312316262070], [105804676040940, 99886829973450427530]

	Independent points of infinite order: 

P1 = [47596634272620, 1019490087452772331530]
P2 = [65357521331220, 753734338701716054250]
P3 = [37071254323980, 1167601425603747441930]
P4 = [110821323250140, 71171122159540704330]
P5 = [20835879745863744/169, 568498652465192600946354/2197]

Dujella - Eroshkin (2009)

y2 + xy = x3 - 2989806201437063120537582346x 
	   + 62923591421774788114221127970660801865252

	Torsion points: 

O, [31499319107676, 761568248418148038], [11931636316404, 170143891921344990054], 
[32852997138612, 12584684220719275686], [32852997138612, -12584717073716414298], 
[31499319107676, -761599747737255714], [11931636316404, -170143903852981306458]

	Independent points of infinite order: 

P1 = [6853784163492, 206770538800519125222]
P2 = [30065337157044, 14520823416137528742]
P3 = [32052598075764, 4730940164143391142]
P4 = [30446686071732, 10862750159321762982]
P5 = [18923480796347904/11449, 294994843954559765609898354/1225043]

Eroshkin (2009)

y2 + xy + y = x3 - x2 - 81998483989093847324602383387x 
	     + 9039662557314583503984236650851472967245899

	Torsion points: 

O, [191882826583267, -608686499672848318814], [162720369372107, -73045620933554030854], 
[-17238856037493, -3232352015313135149254], [162720369372107, 73045458213184658746], 
[-17238856037493, 3232352032551991186746], [191882826583267, 608686307790021735546]

	Independent points of infinite order: 

P1 = [120758100624827, 947965475556972223546]
P2 = [-117778126528243, -4130799992267648976004]
P3 = [166003611484257, 47038619859647754246]
P4 = [292645587321907, 3178955262932679668346]
P5 = [-95474599855473557/289, -1331648350736416603324902/4913]

Eroshkin (2009)

y2 + xy = x3 - 12151449801116771364158721464095x 
	     + 16303853171210911871908176340222708651618600025

	Torsion points: 

O, [2013059167335440, -35199376003741729045], [2696250702242090, 56050108202283692852705], 
[2029966459194590, -1352774570748645034795], [2696250702242090, -56050110898534395094795], 
[2029966459194590, 1352772540782185840205], [2013059167335440, 35197362944574393605]

	Independent points of infinite order: 

P1 = [2012780061897290, 9918904791182773505]
P2 = [295193067596090, 112882913713614812821205]
P3 = [2797902037885640, 64868942660403853981205]
P4 = [2005584289115840, 543238680070107868955]
P5 = [521254825746693017/256, 7516252265355007774821851/4096]

Eroshkin (2011)

y2 + xy + y = x3 - x2 - 273298230287404986977345675193402x 
	     + 1763992752893124742187852765930832732626192628729

	Torsion points: 

O, [-6753079378194263, -1817038635834474322965369], [14916706572653387, -1003180850734141911526369], 
[8774156170053237, 203757145601492595687131], [8774156170053237, -203757154375648765740369], 
[-6753079378194263, 1817038642587553701159631], [14916706572653387, 1003180835817435338872981]

	Independent points of infinite order: 

P1 = [2917519700212787, 995727611790608931699581]
P2 = [9749082105993237, 161813140897747451659631]
P3 = [16971589375305737, 1419188683076734957659631]
P4 = [-29969163375951083/16, 96411464282442301352179509/64] 
P5 = [105832944850010347003/1089, 1073950829336930074855988831717/35937]


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