Constructing elliptic curves over finite fields with prescribed torsion

Abstract

We present a method for constructing optimized equations for the modular curve X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F_q to efficiently generate elliptic curves with nontrivial N-torsion by searching for affine points on X_1(N)(F_q), and we give a fast method for generating curves with (or without) a point of order 4N using X_1(2N).

Corrected typos in the statement of Lemma 2, 17 pages

Cited by in corpus (7)

• Sporadic Cubic Torsion
• Gonality of the modular curve X1(N)
• Torsion points on elliptic curves over number fields of small degree
• Computation on Elliptic Curves with Complex Multiplication
• Sato-Tate distributions of twists of the Fermat and the Klein quartics
• Classification of torsion of elliptic curves over quartic fields
• Torsion of rational elliptic curves over different types of cubic fields