Complete characterization of the Mordell-Weil group of some families of elliptic curves

The Mordell-Weil theorem states that the group of rational pointsý ýon an elliptic curve over the rational numbers is a finitelyý ýgenerated abelian groupý. ýIn our previous paper, Hý. ýDaghighý, ýand Sý. ýDidariý, On the elliptic curves of the form y2=x3−3pxý, ýýBullý. ýIranian Mathý. ýSocý.ýý 40 (2014)ý, noý. ý5ý, ý1119--1133ý.ý, ýusing Selmer groupsý, ýwe have shown that for a prime p the rank of elliptic curve y2=x3−3px is at most twoý. ýIn thisý ýpaper we go furtherý, ýand using height functioný, ýwe will determine the Mordell-Weil group of aý ýfamily of elliptic curves of the form y2=x3−3nx
ý, ýand giveý ýa set of its generators under certain conditionsý. ýWe willý ýintroduce an infinite family of elliptic curves with rank at leastý ýtwoý. ýThe full Mordell-Weil group and the generators of aý ýfamily (which is expected to be infinite under the assumption of a standard conjecture) of elliptic curves with exact rank two will be describedý.