  R. Coleman and K. McMurdy compute the stable reduction of $X_0(p^3).$ On the
basis of their ideas, we compute the stable reduction of $X_0(p^4).$ As a
result, in the stable reduction of $X_0(p^4)$, we find irreducible components,
defined by $a^p-a=t^{p+1}$. These components are called Deligne-Lusztig curve
for ${\rm SL}_2(\mathbb{F}_p).$ We also compute the intersection multiplicity
datum in the stable reduction of $X_0(p^4)$.
