  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 describe 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.
