Elliptic curves coming from Heron triangles

Abstract

Abstract. Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quarticThe Heron formula states that Q = √ 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 innity is on the quartic. By a standard construction it can be proved that the quartic is equivalent to the elliptic curveThe point (P, Q) on the quartic transforms toon the cubic, and the point of innity 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 nd 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. The Heron formula states that the area Q of a triangle with sides {a, b, c} is equal toSo, a triangle with sides a, b, c is a Heron triangle when

High-rank elliptic curves with torsion induced by Diophantine triples

“…Using this algorithm we can show that rank(E(Q(t))) = 4 and that the four points P 1 , P 2 , P 3 , P 4 are free generators of E(Q(t)). We will sketch the application of this algorithm (for a detailed example of such application see, for example, [9]). To apply the algorithm, we write E in the form y 2 = (x − e 1 )(x − e 2 )(x − e 3 ), with e 1 , e 2 , e 3 ∈ Z[t], and consider the factorization…”

On rational triangles via algebraic curves

“…A Heron triangle is a rational triangle whose area is rational. In [1,2,3,6,9], Rational and Heron triangles with certain properties are investigated via studying algebraic curves and surfaces. Moreover, rational triangles are used to explore the size of the sets of rational points of some algebraic curves.…”

Integral isosceles triangle–parallelogram and Heron triangle–rhombus pairs with a common area and common perimeter

“…As a second example, several researchers have related various types of triangles and quadrilaterals to the theory of elliptic curves. Both Goins and Maddox [5] and Dujella and Peral [4] constructed elliptic curves over ℚ coming from Heron triangles. Izadi, Khoshnam, and Moody later generalized their notions to Heron quadrilaterals [7].…”