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Supersingular points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ and their Frobenius action

I am trying to understand certain points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ (where $p$ is a rational prime) via their moduli description. A point $P \in X_0(p)$ can be viewed as $P:=(E, ...
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