In this paper. we study some properties of parametrizations of elliptic cur
ves by Shimura curves. Fix a square-free positive integer N and an isogeny
class E of elliptic curves of conductor N defined over Q. Consider a pair (
D, M) such that N = DM and the number of prime factors of D is even. Let J
be the Jacobian of Shimura curve X (D)(0)(M) associated with an Eichler ord
er of level M in an indefinite quaternion albebra of discriminant D defined
over Q. There is a unique E in E and a homomorphism J --> E having the con
nected kernel. For a prime r \N, we study the map on groups of connected co
mponents of Neron fibers at r induced from J --> E. We show that if r divid
es D. then the map is surjective. Moreover, we study some relations among d
egrees of parametrizations X-0(D)(M) --> E when D and M vary. Also. we desc
ribe a method of computing the degree of X-0(D)(M) --> E when D > 1. (C) 20
01 Academic Press.