Ogg's cuspidal torsion conjecture for the Jacobian of X_0(p)

Let $p$ be prime, let $J_0(p)$ be the Jacobian of $X_0(p)$, and let $\mathcal{C}(p)$ be its rational cuspidal divisor class group. Ogg's cuspidal torsion conjecture. The cyclic group $\mathcal{C}(p)$ is the full torsion subgroup of $J_0(p)(\mathbb{Q})$:

$$\mathcal{C}(p)=J_0(p)(\mathbb{Q})_\mathrm{tor}.$$

This conjecture identifies all rational torsion on $J_0(p)$ with the subgroup generated by rational cuspidal divisor classes. The source says that Mazur proved it using the Eisenstein ideal.