The equation E: x^3+y^3=N defines a classical family of elliptic curves as N varies over cube-free positive integers. They admit complex multiplication, which allows us to tackle the conjecture of Birch and Swinnerton-Dyer for E effectively. Indeed, using Iwasawa theory, Rubin was able to show the p-part of the conjecture for E for all primes p, except for the primes 2 and 3. The theory becomes much more complex at these small primes, but at the same time we can observe some interesting phenomena. I will explain a method to study the p-adic valuation of the algebraic part of the central L-value of E, and I will establish the 3-part of the conjecture for E in special cases. I will then explain a relation between the 2-part of a certain ideal class group and the Tate-Shafarevich group of E. Part of this talk is based on joint work with Yongxiong Li.