Motivated by the theory of Diophantine m-tuples, we study rational points on quadratic twists \(H^d:d y^2=(x^2-x-3)(x^2+2x-12)\) , where |d| is a prime. If we denote by \(S(X)=\{ d \in \mathbb {Z}: H^d(\mathbb {Q})\ne \emptyset , |d| \text { is a prime}\text { and } |d| < X\},\) then, by assuming some standard conjectures about the ranks of elliptic curves in the family of quadratic twists, we prove that as \(X \rightarrow \infty \) \(\frac{43}{256}+o(1)\le \frac{\#S(X)}{2\pi (X)}\le \frac{46}{256}+o(1),\) where \(\pi (X)\) is the prime-counting function. A novel aspect of our work is the computation of the Cassels-Tate pairing by studying the splitting behavior of primes in the governing fields defined by Smith [21], which plays a key role in analyzing the existence of rational points on the curve \(H^d\) .
