  Given a non-CM elliptic curve E over Q, 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}(Gal(Q(E[m_E])/Q), where Pi denotes the natural
projection GL_2(\hat{Z}) onto GL_2(Z/m_E Z). We show that, uniformly for
semi-stable non-CM elliptic curves E over Q, m_E is less than a constant times
the 5th power of the conductor of E.
