Let $p \ge 5$ be a prime. Let $\mathbb{T}$ be the weight $2$ Hecke algebra of level $\Gamma_0(p)$ acting on the cuspidal quotient, generated over $\mathbb{Z}$ by the Hecke operators $T_q$ for primes $q \ne p$ and the level-$p$ operator $U_p$. Define the Eisenstein ideal
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\begin{align*}
I_E := \left(T_q - (1+q) \text{ for all primes } q \ne p,\; U_p - 1\right) \subseteq \mathbb{T}.
\end{align*}
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Then $\mathbb{T}/I_E$ is finite, and its order is the numerator of
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\begin{align*}
\frac{p-1}{12}
\end{align*}
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when written in lowest terms. Equivalently, for a prime $\ell$, there exists a nonzero mod-$\ell$ congruence between a weight $2$ cuspidal Hecke eigenform of level $\Gamma_0(p)$ and the Eisenstein system of eigenvalues