An octic diophantine equation and related families of elliptic curves

Abstract

We obtain two parametric solutions of the diophantine equation $\varphi (x_1,x_2,x_3)=\varphi (y_1,y_2,y_3)$ where $\varphi (x_1,x_2,x_3)$ is the octic form defined by $\varphi (x_1,x_2,x_3)=x_1^8+x_2^8+x_3^8-2x_1^4{x}_2^4-2x_1^4{x}_3^4-2x_2^4{x}_3^4$. 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.

8 pages