## Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves

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## Abstract:

Let $E$ be an optimal elliptic curve defined over $\mathbb {Q}$. The*critical subgroup*of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\mathbb {Q})$ generated by traces of branch points under a modular parametrization of $E$. We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of

*critical polynomials*attached to $E$ and develop two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical polynomials. Finally, a table of critical polynomials is obtained for all elliptic curves of rank two and conductor smaller than 1000, from which we deduce our result.

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## Additional Information

**Hao Chen**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98115
- Email: chenh123@uw.edu
- Received by editor(s): January 17, 2015
- Received by editor(s) in revised form: March 1, 2015, and March 18, 2015
- Published electronically: December 9, 2015
- © Copyright 2015 Hao Chen
- Journal: Math. Comp.
**85**(2016), 2499-2514 - MSC (2010): Primary 11G05
- DOI: https://doi.org/10.1090/mcom3057
- MathSciNet review: 3511290