Let $p \geq 5$ be a prime, and let $(a,b,c) \in \mathbb{Z}^3$ be a primitive Fermat solution, meaning that $a,b,c$ are nonzero, pairwise coprime integers satisfying
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\begin{align*}
a^p + b^p = c^p.
\end{align*}
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Let $E_{a,b,p}$ be the Frey curve over $\mathbb{Q}$ given by