Elliptic Curves and Their Isomorphisms

Abstract

Using the results of the previous chapter, we complete the unfinished business concerning elliptic curves as cubic curves, namely the associative law, the transformation into normal form, and the discriminant criterion for non-singularity or smoothness. At the same time admissible changes of variables are introduced; these are equivalent to isomorphisms from an elliptic curve defined by one cubic onto another given by a change of variable in the first equation.