One-parameter families of elliptic curves over Q with maximal Galois representations

Let E be an elliptic curve over Q and let Q(E[n]) be its nth division field. In 1972, Serre showed that if E is without complex multiplication, then the Galois group of Q(E[n])/Q is as large as possible, that is, GL(2)(Z/nZ), for all integers n coprime to a constant integer m(E, Q) depending ( at most) on E/Q. Serre also showed that the best one can hope for is to have | GL(2)(Z/nZ) : Gal(Q(E[n])/Q)| <= 2 for all positive integers n. We study the frequency of this optimal situation in a one-parameter family of elliptic curves over Q, and show that in essence, for almost all one-parameter families, almost all elliptic curves have this optimal behavior.