Canonical liftings of Edwards curves

Twisted Edwards curves are genus $1$ curves given by equations of the form $bx^2 + y^2 = 1 + ax^2y^2$. Due to their simplified formulas for point arithmetic, they most often offer better performance in concrete applications, such as cryptography. Here we study the canonical liftings of such curves and their associated elliptic Teichmüller lifts. The coordinate functions of the latter are proved to be polynomials, and their degrees and derivatives are given. Moreover, an algorithm is described for explicit computations, and some properties of the general formulas are given.