Elliptic curves coming from Heron triangles

Abstract

Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quartic v^2=u(u-a)(u-b)(u-c). The Heron formula states that Q=sqrt{; ; ; ; P(P-a)(P-b)(P-c)}; ; ; ; where P is the semi-perimeter of the triangle, so the point (u, v)=(P, Q) is a rational point on the quartic. Also the point of infinity is on the quartic. By a standard construction it can be proved that the quartic is equivalent to the elliptic curve y^2=(x+ a b)(x+ b c)(x+c a). The point (P, Q) on the quartic transforms to (x, y)= ((-2 a b c)/(a + b +c), (4 Q a b c)/(a+ b+ c)^2) on the cubic, and the point of infinity goes to (0, a b c). Both points are independent, so the family of curves induced by Heron triangles has rank >= 2. In this note we construct subfamilies of rank at least 3, 4 and 5. For the subfamily with rank >= 5, we show that its generic rank is exactly equal to 5 and we find free generators of the corresponding group. By specialization, we obtain examples of elliptic curves over Q with rank equal to 9 and 10. This is an improvement of results by F. Izadi et al., who found a subfamily with rank >= 3, and several examples of curves of rank 7 over Q.