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

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