  We study the rigidity of the local conditions in two well-known local-global
principles for elliptic curves over number fields. In particular, we consider a
local-global principle for torsion due to Serre and Katz, and one for isogenies
due to Sutherland. For each of these local-global principles, we prove that if
an elliptic curve $E$ over a number field $K$ is such that it fails to satisfy
the local condition for at least one prime ideal of $K$ of good reduction, then
$E$ can satisfy the local condition at no more than 75% of prime ideals. We
also give for (conjecturally) all elliptic curves over the rationals without
complex multiplication, the densities of primes that satisfy the local
conditions mentioned above.
