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Given a non-CM elliptic curve E over Q, define the ``torsion conductor'' m_Eto be the smallest positive integer so that the Galois representation on thetorsion of E has image Pi^{-1}(Gal(Q(E[m_E])/Q), where Pi denotes the naturalprojection GL_2(\hat{Z}) onto GL_2(Z/m_E Z). We show that, uniformly forsemi-stable non-CM elliptic curves E over Q, m_E is less than a constant timesthe 5th power of the conductor of E.

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