 
Summary: The Manin Constant, Congruence Primes, and the
Modular Degree
Amod Agashe Kenneth Ribet William Stein
January 16, 2005
Abstract
We obtain relations between the modular degree and congruence
modulus of elliptic curves, and answer a question raised in a paper
of Frey and M¨uller about whether or not the congruence number and
modular degree of elliptic curves are always equal; they are not, but
we give a conjectural relation between them. We also prove results
and make conjectures about Manin constants of quotients of J1(N)
of arbitrary dimension. For optimal elliptic curve, we prove that if 2
exactly divides N and the congruence number of E is odd, then the
Manin constant of E is also odd.
1 Introduction
Let N be a positive integer and E be an optimal elliptic curve quotient
of J0(N), where optimal means that ker(J0(N) E) is connected. The
Manin constant of E is an invariant associated to E that plays a role in the
Birch and SwinnertonDyer conjecture (see Section 3.2.1). Manin conjec
tured that it is always equal to 1. In Section 2.1, we recall the definition
