On elliptic curves y2= x3− n2x with rank zero

Abstract

In this paper we determine all elliptic curves E n : y 2= x 3− n 2 x with the smallest 2-Selmer groups S n = Sel 2( E n ( Q))={1} and S n ′= Sel 2( E n ′( Q))={±1,± n}( E n ′: y 2= x 3+4 n 2 x) based on the 2-descent method. The values of n for such curves E n are described in terms of graph-theory language. It is well known that the rank of the group E n ( Q) for such curves E n is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers.