  Let $N>1$ be an integer coprime to $6$ such that $N\notin\{5,7,13\}$ and let
$g=g(N)$ be the genus of the modular curve $X_0(N)$. We compute the
intersection matrices relative to special fibres of the minimal regular model
of $X_0(N)$. Moreover we prove that the self-intersection of the Arakelov
canonical sheaf of $X_0(N)$ is asymptotic to $3g\log N$, for $N\to+\infty$.
