Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

Abstract In the present paper, we generalize celebrated classical lemma of Birch and Heegner on quadratic twists elliptic curves over ℚ {{\mathbb{Q}}} . We prove existence explicit infinite families with analytic ranks 0 1 for a large class curves, use points to explicitly construct rational order rank 1. addition, show that these satisfy 2-part Swinnerton-Dyer conjecture when original curve does. also new result in direction Goldfeld conjecture.