Mordell Curves with Ordinates in Arithmetic Progression

We show that Mordell curves with arithmetic progressions in the $y$-coordinate of length $7$ have not been ruled out by previous work, and we give non-isomorphic families of Mordell curves with $y$-arithmetic progressions of length $6$. We also construct parametric families of elliptic curves of moderate rank, with subfamilies possessing rational points in arithmetic progression.