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

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- by Joseph H. Silverman PDF
- 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.## References

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## Additional Information

**Joseph H. Silverman**- Affiliation: Mathematics Department, Box 1917, Brown University, Providence, RI 02912 USA
- MR Author ID: 162205
- ORCID: 0000-0003-3887-3248
- Email: jhs@gauss.math.brown.edu
- Received by editor(s): May 8, 1996
- Received by editor(s) in revised form: March 3, 1997
- Additional Notes: Research partially supported by NSF DMS-9424642.
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp.
**68**(1999), 835-858 - MSC (1991): Primary 11G05, 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-99-01068-6
- MathSciNet review: 1627825