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A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map $\\gal(\\bar{K}/K) \\to \\gl_2(\\ff_\\ell)$ is surjective for all but finitely many $\\ell$.\n  We say that a prime number $\\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. 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