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

Silverman, Joseph

doi:10.1090/S0025-5718-99-01068-6

Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights
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Math. Comp. 68 (1999), 835-858 Request permission

Abstract:

Let $E/\mathbb {Q}$ 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 $E(\mathbb {Q})$ and $E(\mathbb {Z}_{S})$. For rank 1 elliptic curves $E/\mathbb {Q}$ 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 $E(\mathbb {Z}_{S})$ contains non-torsion points.

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