Let K be a field which is finitely generated over its prime field. Consider elliptic curves E and E′ defined over K. Suppose there exists c ≥: 1 and a set A of prime numbers such that [K(El, El′]: K(El′) ∩ K (El′) ≤, c for all l ∈ Λ. We prove that E′ and E are isogeneous over the algebraic closure of K in each of the following cases: (a) A is infinite and E has no complex multiplication. (b) A is infinite, E has complex multiplication, and char(K) = 0. (c) A has Dirichlet density >3/4, E has complex multiplication, and char(K) > 0.