In this article, we construct a family of elliptic curves \(E^A: y^2 = (x + A)(x^2 + A^2)\) with one \(2\) -torsion point over \(\mathbb {Q}\) and prove that there exist infinitely many square-free integers \( d \) such that the rank of the quadratic twists of \( E^A \) by \( d \) is zero. This work is a generalization of the result of M. Xiong: [On positive proportion of rank-zero twists of elliptic curves over \({\mathbb {Q}}\) , J Aust Math Soc 98:281–288, (2015)].
