In this paper we study the asymptotic behaviour of the number of square-free integers \(q \in \mathbb{Z}\) for which the curve \(qy^2 = (x^2-x-3)(x^2+2x-12) \) is everywhere locally solvable, i.e. for which it has a solution in \(\mathbb{Q}p\) for every prime number \(p\). The process for obtaining this asymptotic follows the methods used in the classical problem of determining the asymptotic of numbers that can be written as the sum of two squares. The solution for obtaining the wanted asymptotic is divided in several steps. The first step consists of finding the necessary and sufficient conditions on \(q\) for which the given curve is everywhere locally solvable. In the second step we define a certain sequence of numbers by which we then define the counting function for our \(q\)’s and a Dirichlet’s series that belongs to that sequence of numbers. After that we study the properties of this Dirichlet’s series : absolute and uniform convergence in region \(\operatorname{Re}(s)\geq c>1\), analytical continuation to \(\operatorname{Re}(s)>\frac{1}{2}\), residue in \(s=1\) and upper bound on \(\operatorname{Re}(s)=\frac{1}{2}+\delta\). In the last step we use Perron’s formula and introduce the key-hole contour which helps us to assess the integral obtained from Perron’s formula. From there we finally get the wanted asymptotic behaviour.