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Let $E/\mathbb{Q}$ be an elliptic curve with a 3-torsion point $T$. One can write a Weierstrass equation for $E$. If I define $C := E/\langle T \rangle$, what is the Weierstrass equation for $C$? Is it just the equation given at the top of page 4 in this paper here?

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Yes, the curve $\hat{E}$ in that paper is the correct curve. It is an elliptic curve $\hat{E}$ with an isogeny $E \to \hat{E}$ whose kernel is $\langle T \rangle$, so it's the right curve for $E/\langle T \rangle$.

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