On elliptic curves assigned to a pair of right triangles with an equal hypotenuse

Abstract

Consider a pair of right triangles with an equal hypotenuse. This turns out to solve the diophantine system of equations a ² + b ² = c ² + d ² = e ² in integers. To this system we associate a family of elliptic curves with defining equation y ² = (x−a ²)(x−b ²)(x−c ²)+a ² b ² c ². We show that there exists a subfamily of rank ≥ 3 over ℚ(m, n, k, ℓ) and obtain a subfamily of rank (exactly) four over ℚ(k) and determine a set of its free generators. Besides, we show there exist infinitely many elliptic curves of rank ≥ 5 parameterized by a rank five quartic elliptic curve. We also find a few particular examples with higher ranks. The families we construct have ℤ/2ℤ torsion subgroups in general. The previous work [13] has obtained similar Pythagorean quadruplet elliptic curve families in two parameters with rank ≥ 3. (Recall that by a Pythagorean quadruplet (a, b, c, d), we mean an integer solution to the quadratic equation a ² + b ² = c ² + d ².)