Abstract: We study the rational points on the elliptic surface given by the equation: y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3, where 4A^3-27B^2 is nonzero and Q(u,v) is a positive-definite quadratic form. We prove asymptotics for a special subset of the rational points of increasing height, specifically those that are integral with respect to the singularity. This method utilizes Mordell's parameterization of integral points on quadratic twists on elliptic curves, which is based on a syzygy for invariants of binary quartic forms. We reduce the point-counting problem to the question of determining an asymptotic formula for the correlation sums of representation numbers of binary quadratic and binary quartic forms, where the quartic forms have certain invariants. These sums are then treated using a connection to modular forms.