Let $N$ be a positive integer and $X_0(N)_{\mathbb {Q}}$ the modular curve over $\mathbb {Q}$ parametrizing cyclic isogenies of degree $N$ between elliptic curves. Let $J_0(N)_{\mathbb {Q}}$ be the Jacobian variety of $X_0(N)_{\mathbb {Q}}$. The Hecke algebra acts on $J_0(N)_{\mathbb {Q}}$ hence also on its Néron model $J_0(N)$ over $\mathbb {Z}$. Let $p$ be a prime number and $\Phi _{N,p}$ the group of connected components of the geometric fiber $J_0(N)_p$ of $J_0(N)$ in characteristic $p$. We prove that for $p>3$ the action of the Hecke algebra on $\Phi _{N,p}$ is “Eisenstein”.