  In this paper we introduce a new problem called the Isogenous Embedding
Problem (IEP). The existence of solutions to this problem is related to the
primes of bad reduction of CM curves of genus $3$ and we can detect potentially
good reduction in absence of solutions. We propose an algorithm for computing
the solutions to the IEP and run the algorithm through different families of
curves. We were able to prove the reduction type of some particular curves at
certain primes that were open cases in [LLLR21].
