Explicit Lower Bounds on the Modular Degree of an Elliptic Curve

We give explicit lower bounds on the modular degree of a rational elliptic curve. The technique is via a convolution-type formula involving the symmetric-square L-function, for which an analogue of “no Siegel zeros” is known due to a result of Goldfeld, Hoffstein, and Lieman; our main task is to determine an explicit constant for their bound. Combined with an easy bound on the Faltings height in terms of the discriminant, this gives an explicit lower bound on the modular degree that is N logN √ log logN where N is the conductor. This improves previous explicit bounds that were linear in N . In an appendix, we calculate the Euler factors and local conductors for symmetric power L-functions of an elliptic curve, a topic of independent interest.