A sextic diophantine chain and a related Mordell curve

In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain $\phi(x_1,\,y_1,\,z_1)=\phi(x_2,\,y_2,\,z_2)=\phi(x_3,\,y_3,\,z_3)=k$ where $\phi(x,\,y,\,z)$ is a sextic form defined by $\phi(x,\,y,\,z)$ $=x^6+y^6+z^6-2x^3y^3-2x^3z^3-2y^3z^3$ and $k$ is an integer. Each numerical solution of such a sextic chain yields, in general, nine rational points on the Mordell curve $y^2=x^3+k/4$. While all of these nine points are not independent in the group of rational points of the Mordell curve, we have constructed a parameterized family of Mordell curves of generic rank $\geq 6$ using the aforementioned parametric solution of the sextic diophantine chain. Similarly, the numerical solutions of the sextic chain yield additional examples of Mordell curves whose rank is $\geq 6$.