  We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve
$X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it.
When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational
torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much
smaller than the classical Jacobian.
