Distinguishing elliptic curves modulo $p$ and identifying images of product representations

Given two elliptic curves defined over $\mathbb{Q}$ and a rational prime $p$, we study the product of their residual Galois representations. Using Goursat’s lemma, we explicitly enumerate and completely characterize all possible images of such product representations. We also define associated invariants to these image groups, which we call $\mathit{\text{witness}}\mathit{\text{ratios}}$, and we explain their computational utility and their relationship to the well-known Sturm bound for testing congruences between modular forms.