Modular Degrees of Elliptic Curves and Some Quotient of $L$-values

By the modular degree we mean the degree of a modular parametrization of an elliptic curve, namely the mapping degree of the surjection from a modular curve to an elliptic curve. Its arithmetic significance is discussed by Zagier and Agashe-Ribet-Stein et al. in terms of the congruence of modular forms. Given an elliptic curve $E_f$ attached to a rational newform $f$, we explicitly relate its modular degree to a quotient of special values of some two $L$-functions attached to $f$. We also provide several numerical examples of the formula.