We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much… CONTINUE READING