Quadratic twists of genus one curves and Diophantine quintuples

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 $\mathrm{\mid }d\mathrm{\mid }$ is a prime. If we denote by $S(X)=\{d\in \mathbb{Z}:H^d(\mathbb{Q})\mathrm{\ne }\mathrm{\varnothing },\mathrm{\mid }d\mathrm{\mid }\text{ is a prime and }\mathrm{\mid }d\mathrm{\mid }<X\},$ then, by assuming some standard conjectures about the ranks of elliptic curves in the family of quadratic twists, we prove that as $X\to \mathrm{\infty }$ $$\frac{43}{256}+o(1)\le \frac{\mathrm{\#}S(X)}{2\pi (X)}\le \frac{46}{256}+o(1)\mathrm{.}$$