Trisections and Cubes in Elliptic Curves

We show how to compute the pre-images of multiplication by 3 in the group of points of an elliptic curve E over a field k of characteristic different from 2 and 3 such that E[3](k)=E[3]. We show Q=(xQ,yQ)∈[3]E(k) if and only if yQ−d−m(xQ−c) is a cube in k for every 3-torsion point (c,d)∈E[3], where m is the slope of the tangent to E at (c,d). We reduce our problem to the computation of at most 2 independent cubic roots plus a few polynomial operations. © 2022 The Author(s)