The History

• $\mathbf{<}$1984: ??
• 1984: Don Zagier wrote the often-cited paper Modular parametrizations of elliptic curves (1985), in which he gave an algorithm to compute $m_E$ (sometimes?). The paper incluced
  • A result of Ribet:
    Theorem 2.1 (Ribet)   If $N_E$ is prime, then

    $$m_E = c_E.$$

  • It also said
    $$c_E \mid m_E.$$

• 1998: Frey and Müller published a wonderful survey: Arithmetic of modular curves and applications.
  • They ask: Question 4.4: Let $E$ be an optimal quotient of any conductor. Does $m_E = c_E$?
  • They remark that $c_E \mid m_E$ and give two references [Ribet 83, Inventiones] and [Zagier 1985].

• 1995: Cremona wrote a Math. Comp. paper, and computed $m_E$ for every curve of conductor $\leq N$, where $N$ is a few thousand.

• 2001: Mark Watkins computed $m_E$ for some curves with $N_E$ HUGE, using an algorithm he created from a formula of M. Flach.¹