Let \(E/\mathbb {Q}\) be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the \(\ell \) -adic Galois representation \(\rho _{E,\ell ^\infty }\) is surjective for all but finitely many prime numbers \(\ell \) . Considerable work has gone into bounding the largest possible nonsurjective prime; a uniform bound of 37 has been proposed but is yet unproven. We consider an opposing direction, proving that the smallest prime \(\ell \) such that \(\rho _{E,\ell ^\infty }\) is surjective is at most 7. Moreover, we completely classify all elliptic curves \(E/\mathbb {Q}\) for which the smallest surjective prime is exactly 7.
