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Let E=Q be an elliptic curve of rank 1. We describe an algorithm which uses the value of L0(E;1) and the theory of canonical heghts to e- ciently search for points in E(Q )a nd E ( Z S). For rank 1 elliptic curves E=Q of moderately large conductor (say on the order of 107 to 1010 )a nd with a generator having moderately large canonical height (say between 13 and 50), our algorithm is the rst practical general purpose method for determining if the set E(ZS) contains non-torsion points.

Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights