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.