On the Manin constants of modular elliptic curves

For M a positive integer, let X 0 (M) Q be the modular curve over Q classifying elliptic curves with a given cyclic subgroup of order M and let J 0(M)Q be the jacobian of X 0(M)Q. An elliptic curve E over Q is said to be modular if it is an isogeny factor (isogenics over Q) of some J 0(M)Q; the smallest M for which this happens is then called the level of E. The Shimura-Taniyama conjecture states that every elliptic curve E over Q is modular, and that the level of E equals the conductor of E. A modular elliptic curve of level M is called strong if there exists a closed immersion E ↪ J 0(M)Q. It follows from the multiplicity one principle for modular forms that such an immersion is unique up to sign.