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

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Given a non-CM elliptic curve E over Q, define the ``torsion conductor'' mₑ to be the smallest positive integer so that the Galois representation on the torsion of E has image Pi⁻¹(Gal(Q(E[mₑ])/Q), where Pi denotes the natural projection GL₂(Z) onto GL₂(Z/mₑ Z). We show that, uniformly for semi-stable non-CM elliptic curves E over Q, mₑ is less than a constant times the 5th power of the conductor of E.