  Motivated by the theory of Diophantine $m$-tuples, we study rational points
on quadratic twists $H^d:d y^2=(x^2+6x-18)(-x^2+2x+2)$, where $|d|$ is a prime.
If we denote by $S(X)=\{ d \in \mathbb{Z}: H^d(\mathbb{Q})\ne \emptyset, |d|
\textrm{ is a prime}\textrm{ and } |d| < X\},$ then, by assuming some standard
conjectures about the ranks of elliptic curves in the family of quadratic
twists, we prove that as $X \rightarrow \infty$ $$\frac{43}{256}+o(1)\le
\frac{\#S(X)}{2\pi(X)}\le \frac{46}{256}+o(1).$$
