The Classification of Supersingular Elliptic Curves in Characteristic 3

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.