Rational torsion on optimal curves and rank-one quadratic twists

When an elliptic curve E^′/Q of square-free conductor N has a rational point of odd prime order l?N, Dummigan (2005) in [Du] explicitly constructed a rational point of order l on the optimal curve E, isogenous over Q to E^′, under some conditions. In this paper, we show that his construction also works unconditionally. And applying it to Heegner points of elliptic curves, we find a family of elliptic curves E^′/Q such that a positive proportion of quadratic twists of E^′ has (analytic) rank 1. This family includes the infinite family of elliptic curves of the same property in Byeon, Jeon, and Kim (2009) [B-J-K].