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