Serre's Constant of Elliptic Curves Over the Rationals

Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre's constant associated to E, that gives necessary conditions to conclude that the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying then is non-surjective. Conditionally under Serre's Uniformity Conjecture, we determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constants.