: Galois representations with non surjective traces.

Let be an elliptic curve over , and let be an integer. According to the Lang-Trotter conjecture, the number of primes such that is either finite, or is asymptotic to where is a non-zero constant. A typical example of the former is the case of rational -torsion, where is impossible if . We prove in this paper that, when has a rational -isogeny and , the number of primes such that is finite (for some modulo ) if and only if has rational -torsion over the cyclotomic field . The case is special, and is also treated in the paper. We also classify all those occurences.