Does r_E = deg(phi_E)?

William A. Stein

(This is joint with Amod Agashe and Ken Ribet.)

Question 4.4 of Frey and Muller, Arithmetic of Modular Curves and Applications, 1998, asks whether or not the equality "r_E = deg(phi_E)" holds. The answer is no; the first counterexample occurs for an elliptic curve of conductor 54. Here is some additional information related to their question:

Table of r and deg(phi) for all elliptic curves of conductor < 540.
In July 2001, I gave a talk at Oberwolfach about the congruence modulus and modular degree.
Ken Ribet recently proved that if p exactly divides the conductor of E, then ord_p(r) = ord_p(N).
Additional analysis of the above table.

The Modular Forms Database

Does rE = deg(phiE)?

William A. Stein

Does r_E = deg(phi_E)?