Classical Iwasawa theory and infinite descent on a family of abelian varieties

For primes $$q \equiv 7 \ \mathrm {mod}\ 16$$ , the present manuscript shows that elementary methods enable one to prove surprisingly strong results about Iwasawa theory of Gross family elliptic curves with complex multiplication by ring integers field $$K = {\mathbb {Q}}(\sqrt{-q})$$ which are in perfect accord predictions conjecture Birch and Swinnerton-Dyer. We also some interesting phenomena related a classical Greenberg, give new proof an old theorem Hasse.