engleski

On the high rank π/3 and 2π/3 - congruent number elliptic curves

Consider the elliptic curves given by E_{; ; ; ; ; n, theta}; ; ; ; ;  : y^2 = x^3 + 2snx^2 - (r^2 - s^2)n^2x where 0 < theta < pi, cos(theta) = s/r is rational with 0 <= |s| < r and gcd(r, s) = 1. These elliptic curves are related to the theta-congruent number problem as a generalization of the congruent number problem. For fixed theta this family corresponds to the quadratic twist by n of the curve E_{; ; ; ; ; theta}; ; ; ; ;  : y^2 = x^3 +2sx^2 - (r^2 - s^2)x. We study two special cases theta = pi/3 and theta = 2pi/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 E_{; ; ; ; ; n, theta}; ; ; ; ;  has rank up to 7 over Q in both cases.

theta-congruent number; elliptic curve; Mordell-Weil rank

nije evidentirano