We study elliptic curves of the form x^3+y^3=2^p and x^3+y^3=2p^2 where p is any odd prime satisfying p ≡ 2 mod 9 or p ≡ 5 mod 9. We first show that the 3-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their 2-Selmer group to the 2-rank of the ideal class group of Q(\sqrt[3]{p}) to obtain some examples of elliptic curves with rank one and non-trivial 2-part of the Tate-Shafarevich group.