I'm studying the proof of Mazur's theorem on the torsion of elliptic curves. He first show that, for a prime $N > 7$ and $N\neq 13$, there's no elliptic curve over $\mathbb{Q}$ with a rational point of order $N$. The proof uses Jacobian $J_{0}(N)$ of modular curve $X_{0}(N)$, and when $N = 13$, $X_{0}(13) \simeq \mathbb{P}^{1}$ so that $J_{0}(13)$ is trivial. This gives a reason to treat $N=13$ case separately.
The proof for $N = 13$ can be found in Mazur and Tate's paper Points of order 13 on elliptic curves, and you can also find it in Snowden's lecture note. To show that there's no non-cuspidal points on $X_{1}(13)$ which are rational over $\mathbb{Q}$, they consider an action of so-called twisted dihedral group on $X_{1}(13)$. They show that, under the embedding $X_{1}(13)(\mathbb{C}) \hookrightarrow J_{1}(13)(\mathbb{C})$, the image of modular curve intersects with $J_{1}(13)(\mathbb{Q})_{\mathrm{tors}}$ only at the cuspidal points, so that it is enough to show that $J_{1}(13)(\mathbb{Q})$ has rank 0.
I think I don't get why the number $N=13$ is special for the proof, and I wonder if the method of the proof can be also applied to other $N$'s. The Mazur and Tate's paper is mentioning about forthcoming work of D. Kubert that seems to study $X_{1}(N)$ for other $N$'s, but I can't find the paper they mentioned.