In this paper we obtain the following results: Let X=lim〖DD(X〗←〖DD)〗{Xα,πβα,Λ} and |Λ|=k, suppose that each projection πα：X→XIn 2002, Feng gave some necessary and sufficient conditions on the vanishing of the Selmer groups of congruent elliptic curves. In virtue of these conditions, many old results on non congruent numbers can be easily proved and more new non congruent numbers can be found. In this paper, the author will use the similar methods to compute the weak Mordell Weil groups of those elliptic curves and get some necessary conditions on the vanishing of this group. Moreover, we can also use our method to prove some known results, and exemplify the efficiency of our conditions by some new non congruent numbers.
