ELLIPTIC CURVES COMING FROM HERON TRIANGLES

Triangles having rational sides a, be 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 = root 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 + ab)(x + bc)(x + ca). The point (P, Q) on the quartic transforms to (x, y) = (-2abc/a + b + c, 4Qabc/(a + B + c)(2)) on the cubic, and the point of infinity goes to (0, abc). 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 Izadi et al., who found a subfamily with rank >= 3 and several examples of curves of rank 7 over Q.