Introduction

Let $E$ be an elliptic curve over ${\bf {Q}}$ , and and let $N$ be the conductor of $E$ . By [BCDT01], we may view $E$ as a quotient of the modular Jacobian $J_0(N)$ . After possibly replacing $E$ by an isogenous curve, we may assume that the kernel of the map $J_0(N)\to E$ is connected, i.e., that $E$ is an optimal quotient of $J_0(N)$ .

Let $\omega$ be the unique (up to sign) rational $1$ -form on a minimal Weierstrass model of $E$ over ${\bf {Z}}$ that restricts to a nowhere-vanishing $1$ -form on the smooth locus. The pullback of $\omega$ is a rational multiple of the differential associated to the normalized new cuspidal eigenform $f_E\in S_2(\Gamma_0(N))$ associated to $E$ . The Manin constant $c_E$ of is $E$ is the absolute value of this rational multiple. The Manin constant plays a role in the conjecture of Birch and Swinnerton-Dyer (see, e.g., [GZ86, p. 310]) and in work on modular parametrizations (see [Ste89,SW04,Vat05]). It is known that the Manin constant is an integer; this fact is important to Cremona's computations of elliptic curves (see [Cre97, pg. 45]), and algorithms for computing special values of elliptic curve $L$ -functions. Manin conjectured that $c_E=1$ . In Section 2, we summarize known results about $c_E$ , and give the new result that $2\nmid c_E$ if if $2$ is not a congruence prime and $4 \nmid N$ .

I made some modifications in the paragraph below. -AmodIn Section 3, we generalize the definition of the Manin constant and many of the results mentioned so far to optimal quotients of $J_0(N)$ and $J_1(N)$ of any dimension associated to ideals of the Hecke algebra. The generalized Manin constant comes up naturally in studying the conjecture of Birch and Swinnerton-Dyer for such quotients (see [AS05, §4]), which is our motivation for studying the generalization. We state what we can prove about the generalized Manin constant, and make a conjecture that the constant is also $1$ for quotients associated to newforms. The proofs of the theorems stated in Section 3 are in Section 4. Section 5 is an appendix written by J. Cremona about computational verification that the Manin constant is $1$ for many elliptic curves.

Acknowledgments. The authors are grateful to A. Abbes, K. Buzzard, R. Coleman, B. Conrad, B. Edixhoven, A. Joyce, L. Merel, and R. Taylor for discussions and advice regarding this paper. The authors wish to thank the referee for helpful comments and suggestions.