Geometry of the del Pezzo surface y^2=x^3+Am^6+Bn^6

In this paper, we give an effective and efficient algorithm which on input takes non-zero integers $A$ and $B$ and on output produces the generators of the Mordell-Weil group of the elliptic curve over $\mathbb{Q}(t)$ given by an equation of the form $y^2=x^3+At^6+B$. Our method uses the correspondence between the 240 lines of a del Pezzo surface of degree 1 and the sections of minimal Shioda height on the corresponding elliptic surface over $\overline{\mathbb{Q}}$. For most rational elliptic surfaces, the density of the rational points is proven by various authors, but the results are partial in case when the surface has a minimal model that is a del Pezzo surface of degree 1. In particular, the ones given by the Weierstrass equation $y^2=x^3+At^6+B$, are among the few for which the question is unsolved, because the root number of the fibres can be constant. Our result proves the density of the rational points in many of these cases where it was previously unknown.