The Future

Based on computations, Amod and I conjectured and Ribet proved the following theorem.

Theorem 5.1 (Ribet, 2001)   Let $E$ be an elliptic curve of conductor $N$. If $p^2\nmid N$ then $\ord_p(m_E) = \ord_p(c_E)$.

New Version of ``Question 4.4. For all $N_E\leq 539$, we have

$$2\cdot \ord_p(c_E/m_E) \leq \ord_p(N_E).$$

In particular, for $p\geq 5$, do we have

$$\ord_p(c_E/m_E) \leq 1?$$

Is this true in general?