A way to compute $m_E$

Use the (not-exact!) sequence:

$$H_1(E,\mathbf{Z}) \rightarrow H_1(X_0(N),\mathbf{Z}) \rightarrow H_1(E,\mathbf{Z}).$$

The composition map from $H_1(E,\mathbf{Z})\rightarrow H_1(E,\mathbf{Z})$ is multiplication by $m_E$, and $H_1(E,\mathbf{Z})$ can be computed because its image in $H_1(X_0(N),\mathbf{Z})$ is saturated, as $E$ is optimal. This algorithm is described in detail in [Kohel-Stein, ANTS IV], and amounts to finding ``left and right eigenvectors'' and taking their dot product.