The field is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X (0)(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X (0)(49) by the quadratic extension , where M is any square free element of O with M ae 1 mod 4 and (M,7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F (a) = K(E (pa)), where E (pa) denotes the group of p(a)-division points on E. Moreover, writing B for the twist of X (0)(49) by , our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z(2)-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.
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