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.