Computing projective equivalences of planar curves birationally equivalent to elliptic and hyperelliptic curves

Abstract

We present algorithms to find the projective equivalences between two planar curves, birationally equivalent to either elliptic or hyperelliptic curves. The main idea, in both cases, is to find first a corresponding birational mapping between the Weierstrass normal forms of the curves. The computation of this mapping benefits from the abundant theory on elliptic and hyperelliptic curves available in the literature. In particular, we discuss cubic curves in detail, and completely characterize the projective equivalences in terms of the isomorphisms between the Weierstrass forms leaving the infinity point invariant, and the inflection points of one of the curves.