= Vélu's formulas
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= Vélu formulas
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For an <elliptic curve> $E$ and a finite <subgroup> $G$ of its geometric <points>, <Jacques Vélu>'s formulas construct the <separable isogeny> with <kernel> $G$. On a <Short Weierstrass form> away from <characteristics> two and three, its normalized coordinates are
$$
x'(P)=x(P)+\sum_{Q\in G\setminus\{O\}}(x(P+Q)-x(Q)),\qquad y'(P)=y(P)+\sum_{Q\in G\setminus\{O\}}(y(P+Q)-y(Q)).
$$
These <rational functions> extend to a <morphism> of the smooth projective <curves>, sending $G$ to the new identity. Its <degree> is $\#G$. If $G$ is stable under the <absolute Galois group>, the summed expressions and quotient <curve> are defined over the base <field>, even when the individual <points> of $G$ are not rational.<Isogeny kernels and division polynomials by Andrew Sutherland>[Pages 5–7, finite-subgroup quotient and Vélu coordinate sums, Theorems 5.11, 5.13 and 5.15]{ref}
For $E:y^2=x^3-x$ and $G=\{O,(0,0)\}$, they give $x'=x-1/x$ and $y'=y(1+1/x^2)$, with quotient $y'^2=x'^3+4x'$. The apparent pole at $(0,0)$ represents its image at infinity, not a missing value of the projective <isogeny>.
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