On a conjecture of Agashe

Let E/Q be an optimal elliptic curve, −D be a negative fundamental discriminant coprime to the conductor N of E/Q and let E−D /Q be the twist of E/Q by −D. A conjecture of Agashe predicts that if E−D /Q has analytic rank 0, then the square of the order of the torsion subgroup of E−D /Q divides the product of the order of the Shafarevich-Tate group of E−D /Q and the orders of the arithmetic component groups of E−D /Q, up to a power of 2. This conjecture can be viewed as evidence for the second part of the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero. We provide a proof of a slightly more general statement without using the optimality hypothesis.