Two-isogeny formula
= Two-isogeny formula
{title2=$\phi:E\longrightarrow E^{\prime},\quad\deg\phi=2$}
For $E:y^2=x(x^2+ax+b)$, set $b^{\prime}=a^2-4b$ and $E^{\prime}:Y^2=X(X^2-2aX+b^{\prime})$. The <isogeny of elliptic curves> with kernel $\{O,(0,0)\}$ is $\phi(x,y)=(x+a+b/x,\ y(1-b/x^2))$. Its dual is $\widehat\phi(X,Y)=((X-2a+b^{\prime}/X)/4,\ Y(1-b^{\prime}/X^2)/8)$. The maps extend over their exceptional affine points and satisfy $\widehat\phi\phi=[2]$.