On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties

Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $\Gamma_0(N)$, and its rational Eisenstein primes of $\mathbb{T}(N)$ containing $\ell$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\ell, ~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}] \neq 0$, where $\mathcal{C}(N)$ is the rational cuspidal group of $J_0(N)$. To do this, we compute the precise order of the cuspidal divisor $\mathcal{C}^D_{M, N}$, defined in Section 4, and the index of $\mathcal{I}^D_{M, N}$ in $\mathbb{T}(N)\otimes \mathbb{Z}_\ell$.