Families of elliptic curves with rational 3-torsion

In this paper we look at three families of elliptic curves with rational 3-torsion over a finite field. These families include Hessian curves, twisted Hessian curves, and a new family we call generalized DIK curves. We find the number of -isogeny classes of each family, as well as the number of -isomorphism classes of the generalized DIK curves. We also include some formulas for efficient computation on these curves, improving upon known results. In particular, we find better formulas for doubling and addition on the original tripling-oriented DIK curves and also for addition and tripling on elliptic curves with -invariant 0.