The Classification of Supersingular Elliptic Curves in Characteristic 3

TL;DR

This paper provides an explicit, implementation-friendly classification of supersingular elliptic curves in characteristic 3, including formulas for point counts, supporting computational tools like Hecke.jl.

Contribution

It offers a self-contained, elementary classification and point count formulas for supersingular elliptic curves in characteristic 3, complementing prior work by Morain.

Findings

Explicit classification of supersingular elliptic curves in characteristic 3

Formulas for counting points on these curves

Implementation-oriented approach for computational applications

Abstract

We provide an explicit classification of supersingular elliptic curves in characteristic~3 into isomorphism classes, and give explicit formulae for their point counts. This report was written specifically to support implementation of point counting in Hecke.jl; accordingly, the reasoning is concrete and elementary. After the first version appeared, we learned that the explicit classification was already obtained by Morain. We believe the present note remains useful in offering a more self-contained and implementation-oriented treatment. The explicit connection to previous Morain's classificaion is given in the appendix.