Exercise 1.10 from Silverman "The Arithmetic of Elliptic Curves "

Question

I am having trouble with Silverman's exercise 1.10(b). The converse of (a) is easy because there is no integer solution to the equation when $p \equiv 3$ mod $4$. However, this method does not work for (b) because we always get empty set when intersecting whatever $V_p (p \equiv 3 $ mod $4)$ with $\mathbb{Q}$. I tried to intersect two varieties $V_p, V_q$ with something else, for example $\mathbb{Q}({\sqrt{p}})$ and try to tell the difference, but this is hard since it is very possible for $V_q$ to have nonempty intersection with $\mathbb{Q}({\sqrt{p}})$ and I think it is true that if so then $V_p, V_q$ are isomorphic over $\mathbb{Q}({\sqrt{p}})$. Are there some invariants I can use in this circumstance to show that two varieties are not isomorphic over $\mathbb{Q}$?