Visibility and the Birch and Swinnerton-dyer conjecture for analytic rank one

Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e. such that the L-function LE (s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K vanishes to order one at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mordell–Weil rank is greater than one and whose associated newform is congruent to the newform associated to E modulo an integer r. The theory of visibility then shows that under certain additional hypotheses, r divides the product of the order of the Shafarevich–Tate group of E over K and the orders of the arithmetic component groups of E. We extract an explicit integer factor from the Birch and Swinnerton–Dyer conjectural formula for the product mentioned above, and under some hypotheses similar to the ones made in the situation above, we show that r divides this integer factor. This provides theoretical evidence for the second part of the Birch and Swinnerton–Dyer conjecture in the analytic rank one case.