Two-isogeny formula ($\varphi :E⟶E^{\prime },\mathrm{deg}\varphi =2$)

Assume $\mathrm{char}K\ne 2$, $b\ne 0$ and $b^{\prime }=a^2-4b\ne 0$. Thenare elliptic curves, andis a degree-two separable isogeny with kernel $\{O,(0,0)\}$. Substitution verifies the target equation. The poles at $O$ and $(0,0)$ extend to the target point at infinity, giving a morphism of the full projective curves. This is a special case of Vélu formulas.

Applying the same construction to $E^{\prime }$ gives a curve with coefficients $4a,16b$. Rescaling that curve to $E$ supplies the dual isogenyThe factors four and eight are essential to this dual normalization. The characteristic restriction permits these divisions, while $b{b}^{\prime }\ne 0$ excludes singular cubic models.