Summary: 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 \(c\geq 1\) and a set \(\Lambda\) of prime numbers such that \([K(E_l,E_l'): K(E_l)\cap K(E_l')]\leq c\) for all \(l\in\Lambda\). We prove that \(E'\) and \(E\) are isogenous over the algebraic closure of \(K\) in each of the following cases: (a) \(\Lambda\) is infinite and \(E\) has no complex multiplication. (b) \(\Lambda\) is infinite, \(E\) has complex multiplication, and \(\text{char} (K)=0\). (c) \(\Lambda\) has Dirichlet density \(>3/4\), \(E\) has complex multiplication, and \(\text{char}(K)>0\).