The Definitions

Let $E/\mathbf{Q}$ be an elliptic curve that is an optimal quotient of $J_0(N_E)$, where $N=N_E$ is the conductor of $E$. Here $J_0(N)$ is the Jacobian of the algebraic curve $X_0(N)$ and a deep theorem implies that there is a surjective morphism $\pi: X_0(N)\rightarrow E$. The condition that $E$ is optimal means that the induced map $\pi_* : J_0(N) \rightarrow E$ has (geometrically) connected kernel.

Definition 1.1   The modular degree of $E$ is

$$m_E = \deg(\pi).$$

One reason that the modular degree is well worth thinking about is that an assertion about how $m_E$ grows relative to $N_E$ is equivalent to the ABC Conjecture.

Let $f=f_E=\sum a_n q^n \in S_2(\Gamma_0(N))$ be the newform attached to $E$.

Definition 1.2   The congruence modulus of $E$ is

$$c_E = \char93 \left(\frac{S_2(\Gamma_0(N),\mathbf{Z})}{\mathbf{Z}{}f + (\mathbf{Z}{}f)^{\perp}}\right),$$

where $(\mathbf{Z}{}f)^{\perp}$ is the unique $\mathbf{T}=\mathbf{Z}[\ldots T_n \ldots]$-module complement of $\mathbf{Z}{}f$ in $S_2(\Gamma_0(N),\mathbf{Z})$. Equivalently,

$$c_E = \max\{ c : f\equiv g \pmod{c}$$    for some $g&isin#in;(Zf)^&perp#perp;$ $$\}.$$