  We show that a two dimensional $\ell $-adic representation of the absolute
Galois group of a number field which is locally potentially equivalent to a
$GL(2)$-$\ell$-adic representation $\rho$ at a set of places of $K$ of positive
upper density is potentially equivalent to $\rho$.
  For an elliptic curver \( E \) defined over a number field \( K \) and a
finite place \( v \) of \( K \) of good reduction for \( E \), let \( F(E,v) \)
denote the Frobenius field of \( E \) at \( v \), given by the splitting field
of the characteristic polynomial of the Frobenius automorphism at \( v \)
acting on the Tate module of \( E \).
  As an application, suppose \( E_1 \) and \( E_2 \) defined over a number
field \( K \), with at least one of them without complex multiplication. We
prove that the set of places \( v \) of \( K \) of good reduction such that the
corresponding Frobenius fields are equal has positive upper density if and only
if \( E_1 \) and \( E_2 \) are isogenous over some extension of \( K \).
  We show that for an elliptic curve \( E \) defined over a number field \( K
\), the set of finite places of \( K \) such that the Frobenius field \( F(E,
v) \) at $v$ equals a fixed imaginary quadratic field \( F \) has positive
upper density if and only if \( E \) has complex multiplication by \( F \).
