complete characterization of the mordell-weil group of some families of elliptic curves

the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$ the rank of elliptic curve $y^2=x^3-3px$ is at most two‎. ‎in this‎ ‎paper we go further‎, ‎and using height function‎, ‎we will determine the mordell-weil group of a‎ ‎family of elliptic curves of the form $y^2=x^3-3nx$‎, ‎and give‎ ‎a set of its generators under certain conditions‎. ‎we will‎ ‎introduce an infinite family of elliptic curves with rank at least‎ ‎two‎. ‎the full mordell-weil group and the generators of a‎ ‎family (which is expected to be infinite under the assumption of a standard conjecture) of elliptic curves with exact rank two will be described‎.