On the squareness of the discriminant of elliptic curves with an isogeny

We establish a classification of the values of $N$ for which an elliptic curve defined over $\mathbb{Q}$ with square discriminant admits an $N$-isogeny. Furthermore, we determine the values of $N$ for which two elliptic curves defined over $\mathbb{Q}$, both possessing square discriminants, are $N$-isogenous. In both cases, we explicitly parametrize the corresponding $j$-invariants of the elliptic curves associated with these problems.