Torsion group Z/9Z, rank = 4


Fisher (2009)

y2 + xy = x3 - 210121912561076041141788727170x + 37072777483029442821111037453964917510678212

	Torsion points:

O, [277805513426364, -373688671662118340682], [264708943136484, -1413461833874258322], 
[497030711756964, 7444592848125263795118], [277805513426364, 373688393856604914318], 
[497030711756964, -7444593345155975552082], [264708943136484, 1413197124931121838], 
[265488709619364, 23578368188322342318], [265488709619364, -23578633677031961682]

	Independent points of infinite order:

P1 = [264624185910084, 161163173132978478] 
P2 = [255996480229028, 242551810327737066926] 
P3 = [278107166057028, 382329690016779874446] 
P4 = [256715819708964, 222495334012116159918] 

van Beek (2015)

y2 + xy = x3 + 2721419042645095050024690463410x + 70269004493489754502184256716133269311066066692

	Torsion points:

O, [19471437840246444, -2739634905721415222436942], [-3315317249972676, 157502052841253182426338], 
[-3315317249972676, -157502049525935932453662], [19471437840246444, 2739634886249977382190498], 
[4354838741124924, 405842589804263655207138], [4354838741124924, -405842594159102396332062], 
[8781565515324, 265128088832506426330338], [8781565515324, -265128088841287991845662]

	Independent points of infinite order:

P1 = [-1755829888879476, -245107199875348858468062]
P2 = [-2099693655522576, -235155076594372628504562]
P3 = [3412344195582694979150364/648669961, 7925940328633136022396959079148665642/16520975236709]
P4 = [1602723222576422901739032156/5377728889, 64164530908803512802156196914932573033466/394364992617037]

van Beek (2015)

y2 + xy = x3 - 178713244550246332063776938470605x + 28147864949458728271564240536379650884906835995025

	Torsion points:

O, [133912708523091330, -49047127766138759038618305], [-27152868437091390, 3602946370788681562165695], 
[-27152868437091390, -3602946343635813125074305], [133912708523091330, 49047127632226050515526975], 
[30955139510942010, -7230326827650168773227305], [30955139510942010, 7230326796695029262285295], 
[94583894516610, 5303862969614130650917695], [94583894516610, -5303862969708714545434305]

	Independent points of infinite order:

P1 = [101629293809588610, 32552494216369407363685695]
P2 = [-15202948761734070, -5229816529060178755649625]
P3 = [17334183174096114397199346/271227961, 74445992775872903454362900009494808403/4466853289709]
P4 = [9862972980678310875764382210/47782025281, 979029617841809233055721209528661420333345/10444720688199071]

Some curves with torsion group Z/9Z and rank = 3
High rank curves with prescribed torsion Andrej Dujella home page