  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).
