On the High Rank Π/3 and 2π/3-congruent Number Elliptic Curves

Consider the elliptic curves given by En,θ : y 2 = x + 2snx − (r − s)nx where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For xed θ this family corresponds to the quadratic twist by n of the curve Eθ : y 2 = x +2sx − (r − s)x. We study two special cases θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that En,θ has rank up to 7 over Q in both cases.