Let E be an elliptic curve
having Complex Multiplication by the ring 𝒪K of integers of K =Q(), let
H = K(j(E)) be the Hilbert class field of K. Then the Mordell–Weil group E(H) is
an 𝒪K-module. Its Steinitz class St(E) is studied here. In particular, when D is a
prime number, St(E) is determined: If D ≡ 3 (mod4) then St(E) = 1;
if D ≡ 1 (mod4) then St(E) = [𝒫]t, where 𝒫 is any prime-ideal factor
of 2 in K, [𝒫] the ideal class of K represented by 𝒫, t is a fixed integer.
In addition, general structure for modules over Dedekind domain is also
discussed. These results develop the results by D. Dummit and W. Miller for
D = 10 and specific elliptic curves to more general D and general elliptic
curves.
