Torsion group Z/5Z, rank = 8


Dujella - Lecacheux (2009)


y2 + xy = x3 - 1346404541224901580948752030x 
         + 19318553476047119421649468184366353981476

	Torsion points: 

O, [-1939073747114, -148060933257135625244], [24206518675528, -30178326846762062486], 
[24206518675528, 30178302640243386958], [-1939073747114, 148060935196209372358]

	Independent points of infinite order: 

P1 = [12942920076352, 63720747352258155334] 
P2 = [15622376802430, 45796178770313537884]
P3 = [17888256277234, 30948043911702748390]
P4 = [20046753870640, 19589500696221641206]
P5 = [25985838078466, 43340480124212287576]
P6 = [-3156496010696, -153417808323866764034]
P7 = [2984082857728, 123803646589650757114]
P8 = [3752540626408, 119661844030558924654]

Eroshkin (2009)


y2 + xy = x3 - 244225635073705856645045305358516300x 
         + 47380375332133580095127079115821390516194400266490000

	Torsion points: 

O, [329811726249331000, -52032904039793805545264300], [-37196964010191080, 237514970589466021499754100], 
[-37196964010191080, -237514970552269057489563020], [329811726249331000, 52032903709982079295933300]

	Independent points of infinite order: 

P1 = [215794269656731132, 68751803745826886566570852]
P2 = [171914770006422040, 102348760844784979939372660]
P3 = [168441145048264840, 104984820306486247545709300]
P4 = [221867993820059800, 64156572929980734493827700]
P5 = [179306589562588750, 96715921232165023539184450]
P6 = [234348514776775150, 54924490017815272873228450]
P7 = [-287753405371733000, -306317759077334372898344300]
P8 = [63621460365170200, 179164459984599002204298100]

Dujella - Lecacheux (2009)


y2 + xy + y = x3 + x2 - 257204508165423954816402995x 
             + 1748673531007907498834205901854033996657

	Torsion points: 

O, [12071024330787, -20070285668205695634], [12071024330787, 20070273597181364846], 
[-4050510939053, 52192223042379056926], [-4050510939053, -52192218991868117874]

	Independent points of infinite order: 

P1 = [-2689602117573, -49203604700818714234]
P2 = [1495205841187, 36978947192609990766]
P3 = [3864476855661, 28503076152165380054]
P4 = [6710474255907, 18024565289380447086]
P5 = [5067406549627, 23988279478024579006]
P6 = [4009191443107, 27963074034705378286]
P7 = [-895848773853, -44478881229791043474]
P8 = [11572113738915, 17942670234858378606]

Dujella - Lecacheux (2009)


y2 + y = x3 + x2 - 1805632198953220354072743054330x 
             + 937164323059943920996847199260009653476285734

	Torsion points: 

O, [114135566422431, 27065925164925932985472], [114135566422431, -27065925164925932985473], 
[836722030653426, 3484870128565476333637], [836722030653426, -3484870128565476333638]

	Independent points of infinite order: 

P1 = [494276905555896, 12862295658942711715597]
P2 = [737119316569371, 2589973408237602569227]
P3 = [826315889039376, 3057538411111718341762]
P4 = [-233877846248124, 36696986856509114936362]
P5 = [508285609696236, 12276205155103716481882]
P6 = [-67908599195559, 32549487101362118890957]
P7 = [575624023395156, 9408934695207694614922]
P8 = [-348343747461429, 39036851906891076445252]

Some curves with torsion group Z/5Z and rank = 6 or 7
High rank curves with prescribed torsion Andrej Dujella home page