An octic diophantine equation and related families of elliptic curves

Abstract

We obtain two parametric solutions of the diophantine equation $\phi(x_1, x_2, x_3)=\phi(y_1, y_2, y_3)$ where $\phi(x_1, x_2, x_3)$ is the octic form defined by $\phi(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank~$5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.