Multiplicative group law on the folium of Descartes

The folium of Descartes is still studied and understood today. Not only did it provide for the proof of some properties connected to Fermat's Last Theorem, or as Hessian curve associated to an elliptic curve, but it also has a very interesting property over it: a multiplicative group law. While for the elliptic curves, the addition operation is compatible with their geometry (additive group), for the folium of Descartes, the multiplication operation is compatible with its geometry (multiplicative group). The original results in this paper include: points at infinity described by the Legendre symbol, a group law on folium of Descartes, the fundamental isomorphism and adequate algorithms, algorithms for algebraic computation.