  Let E be an elliptic curve of conductor N and rank one over Q. So there is a
non-constant morphism X+0(N) --> E defined over Q, where X+0(N) = X0(N)/wN and
wN is the Fricke involution of the modular curve X+0(N). Under this morphism
the traces of the Heegner points of X+0(N) map to rational points on E. In this
paper we study the index I of the subgroup generated by all these traces on
E(Q). We propose and also discuss a conjecture that says that if N is prime and
I > 1, then either the number of connected components of the real locus
X+0(N)(R) is greater than 1 or (less likely) the order S of the Tate-Safarevich
group is non-trivial. This conjecture is backed by computations performed on
each E that satisfies the above hypothesis in the range N < 129999. This paper
was prepared for the proceedings of the Conference on Algorithmic Number
Theory, Turku, May 8-11, 2007. We tried to make the paper as self contained as
possible.
