Visibility for analytic rank one

Let E be an optimal elliptic curve of conductor N , 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. The Gross-Zagier theorem gives a formula that expresses the Birch and Swinnerton-Dyer conjectural order the Shafarevich-Tate group of E over K as a rational number. We extract an integer factor from this formula and relate it to certain congruences of the newform associated to E with eigenforms of odd analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime q divides this factor, then q divides the actual order of the Shafarevich-Tate group. This provides theoretical evidence for the Birch and Swinnerton-Dyer conjecture in the analytic rank one case.