Dixon elliptic functions

Summary

The Dixon elliptic functions $\operatorname{sm}$ and $\operatorname{cm}$ are two elliptic functions that parametrise the Fermat cubic $x^3 + y^3 = 1$, as $\sin$ and $\cos$ parametrise the circle $x^2 + y^2 = 1$. They satisfy $\operatorname{sm}' = \operatorname{cm}^2$ and $\operatorname{cm}' = -\operatorname{sm}^2$. Their period lattice has a threefold symmetry: rotation by a primitive cube root of unity $\omega$ maps it to itself. They are named after Alfred Cardew Dixon.