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| We obtain relations between the modular degree and congruence modulus of elliptic curves, and answer a question raised in a paper of Frey and Muller about whether or not the congruence number and modular degree of elliptic curves are equal; they are not, but we prove a theorem relating them and make a conjecture. We also prove results and make conjectures about Manin constants of quotients of J1(N) of arbitrary dimension. For optimal elliptic curves E, we give a new condition under which the Manin constant of E is odd. |