Rational points on certain del Pezzo surfaces of degree one

Let f(z) = z(5) + az(3) + bz(2) + cz + d is an element of Z[z] and let us consider a del Pezzo surface of degree one given by the equation epsilon(f) : x(2) - y(3) - f(z) = 0. In this paper we prove that if the set of rational points on the curve E(a,b) : Y(2) = X(3) + 135(2a - 15)X - 1350(5a + 2b - 26) is infinite then the set of rational points on the surface epsilon(f) is dense in the Zariski topology.