A bound for the torsion conductor of a non-CM elliptic curve

Given a non-CM elliptic curve $E$ over $\mathbb{Q}$ of discriminant $\Delta_E$, define the ``torsion conductor'' $m_E$ to be the smallest positive integer so that the Galois representation on the torsion of $E$ has image $\pi^{-1}(\operatorname{Gal}(\mathbb{Q}(E[m_E])/\mathbb{Q}))$, where $\pi$ denotes the natural projection $GL_2(\hat{\mathbb{Z}}) \rightarrow GL_2(\mathbb{Z}/m_E\mathbb{Z})$. We show that, uniformly for semi-stable non-CM elliptic curves $E$ over $\mathbb{Q}$, one has ${m_E \ll \left( \prod_{p \mid\Delta_E} p\right)^5}$.