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Let $E$ be a smooth complex elliptic curve, and $\sigma$ translation of $E$ given by a point $p$ on $E$ of finite order, with respect to some fixed origin.

What are the $j$-invariants related with $E$ and $E/\langle \sigma\rangle$?. Or to be more precise of my question, if one knows the Weierstrass equation or Hesse equation for $E$ and the point $p$, can one write down that for $E/\langle \sigma\rangle$?

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The j-invariants satisfy a relation $F_n(j(E),j(E/\sigma))=0$ where $n$ is the order of $p$ and $F_n$ is the modular polynomial (see e.g. Silverman's "Advanced Topics ..." book pg 146). To get the Weierstrass equation of $E/\sigma$ there are the Vélu formulas (thanks Noam).

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    $\begingroup$ Right, though I believe it's Vélu (French, not Spanish). $\endgroup$ – Noam D. Elkies May 19 '16 at 3:10

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