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

Let E/Q be an elliptic curve of rank 1. We describe an algorithm which uses the value of L'(E, 1) and the theory of canonical heghts to efficiently search for points in E(Q) and E(Z S ). For rank 1 elliptic curves E/Q of moderately large conductor (say on the order of 10 7 to 10 10 ) and with a generator having moderately large canonical height (say between 13 and 50), our algorithm is the first practical general purpose method for determining if the set E(Z S ) contains non-torsion points.

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