The congruences between elliptic curves have been studied for a long time. A notable conjecture of Frey and Mazur regarding those congruences predicts that for two elliptic curves E,E' over Q, there is a constant C>=19 such that if E[l] and E'[l] are isomorphic Galois modules for a prime l>= C, then $E$ and $E'$ are isogeneous. In other words, by the much appreciated modularity theorem, one does not expect to observe congruences between the corresponding newforms f_E and f_{E'} modulo l for a sufficiently large prime l. In a more formal way, if we attach a mod l Galois Representation to f,
the Frey-Mazur Conjecture predicts the set of weight 2 forms g with isomorphic mod l representation is finite.

One can prove that congruence primes are in a close relationship with obstructed deformation problems, those deformation problems where the adjoint cohomology of the residual representation is not trivial. In this talk, we are going to observe possible congruences between elliptic curves and compare their deformation rings.