Computations on an equation of the Birch and Swinnerton-Dyer type

Let us assume that E/Q is an elliptic curve of level N and rank equal to 1. Let q be a prime that does not divide the conductor. We study conjecture 4 of B. Mazur and J. Tate in [MT87]. This conjecture relates to the Birch and Swinnerton-Dyer problem in the q-adic case. We produce a lot of numerical evidence towards the conjecture. We also propose a refinement of the conjecture in the rank 1 case in section 2.3.