  Let $E$ be an elliptic curve defined over a number field $K$. We say that a
prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined
over $K$. The so-called exceptional set of all such prime numbers is finite if
and only if $E$ does not have complex multiplication over K. In this paper, we
prove that the exceptional set is included in the set of prime divisors of an
explicit list of integers (depending on $E$ and $K$), whose infinitely many of
them are non-zero. It provides an efficient algorithm for computing it in the
finite case. Other less general but rather useful criteria are given, as well
as several numerical examples.
