The Examples

These examples were computed by myself and Amod Agashe.

• 54B: Let $E$ be the elliptic curve $y^2 + xy + y = x^3 - x^2 + x -1$. Then $m_E = 2$ and $c_E = 6$. In fact, it's easy to see that $3\mid c_E$ ``by hand'' by writing down the form $f$ corresponding to 54B and the form $g$ corresponding to $X_0(27)$ and noting that $f(q) \equiv g(q) + g(q^2)\pmod{3}$. (Because of the ``Sturm Bound'', it suffices to check this up to $O(q^{19})$.)

  Hey $c_E \neq m_E$!! In fact, $c_E \not\vert \,\,m_E$!! When we first did this computation, Ribet had already mentioned to us that he had really proved that $m_E\mid c_E$, not vice-versa. We were, however, extremely surprised to find so quickly an example in which $c_E \neq m_E$.

• T-shirt: My t-shirt has 243A and 243B on it. For 243A, we have $m_E = 9$ and $c_E=27$. For 243B, we have $m_E = 6$ and $c_E=54$. I designed the t-shirt many months before I knew that question 4.4 had a negative answer.

• 242B: $N=2\cdot 11^2$.
  $$m_E = 2^4 \neq c_E = 2^4\cdot 11$$
  The failure is probably not just a ``small primes'' phenomenon.

Moral: A little computation sometimes greatly cleans the air.