Let K=ℚ(√−3). Assume E is an elliptic curve defined over K, with complex multiplication by OK, the ring of integers of K. Ω∈[Copf ]X is a minimal period of E. For each prime v of K, let cv denote the Tamagawa number [E(Kv)[ratio ] E0(Kv)]. Let L(E/K, s) denote the Hasse–Weil L-function of E/K. [lfloor ][mid ][rfloor ](E/K)= ker{H1(K, E)→ [oplus ]vH1(Kv, E)} denotes the Tate-Shafarevic group of E/K.