On elliptic curves y2=x3−n2x with rank zero

AbstractIn this paper we determine all elliptic curves En:y2=x3−n2x with the smallest 2-Selmer groups Sn=Sel2(En(Q))={1} and Sn′=Sel2(En′(Q))={±1,±n}(En′:y2=x3+4n2x) based on the 2-descent method. The values of n for such curves En are described in terms of graph-theory language. It is well known that the rank of the group En(Q) for such curves En is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers