Conductor of elliptic curves with complex multiplication and elliptic curves of prime conductor

On the conjecture of Birch and Swinnerton-Dyer

“…The family of curves E d is exactly the class of elliptic curves over Q with complex multiplication by the ring of integers O of K = Q( √ −7) (see [15]). Suppose now that L(E…”

Group order formulas for reductions of CM elliptic curves

“…In addition to d = 1, 2, 3 given above, the remaining d's for which the ring of integers O K of K = Q( √ −d) has class number one are d ∈ {7, 11, 19, 43, 67, 163}. The elliptic curves with CM by these O K are the curves A(d) (in the notation of Dick Gross's thesis [9]) in Table 1 below and their quadratic twists (see [11] or §24 of [9]). For these d, if p is a prime = d, and…”

Solutions of Specific Diophantine Equations and their Relationship to Complex Multiplication

“…THEOREM 2. In the process of describing any given rational solution (x(0, y(f)) of diophantine equation 1, i.e., p(\u + p) / x p f (Xu + n) x(t) = ---,;y(0 = -rr\->' = P("X M = O,UJ U UJ 2 ,LU 3 p(u) p'(u) we find that the unique complex (or real) multiplier X, associated with the given (x(t),y(t)) satisfies (5) Xy(t) = tx\t)+x(t) = d(tx(t))/dt PROOF. Differentiate p(u)X(t) = p(Xu + /x) with respect to u.…”