An Application of the Arithmetic of Elliptic Curves to the Class Number Problem for Quadratic Fields

Let l be the prime 3, 5 or 7, and let m(1), m(2), n(1) and n(2) be non-zero rational numbers. We construct an infinite family of pairs of distinct quadratic fields Q(root m(1)D + n(1)) and Q(root m(2)D + n(2)) with D is an element of Q such that both class numbers are divisible by l, using rational points on an elliptic curve with positive Mordell-Weil rank to parametrize such quadratic fields.