Torsion Points and Matrices Defining Elliptic Curves

Let k be an algebraically closed field, char k ≠ 2, 3, and let X ⊂ P^2 be an elliptic curve with defining polynomial f. We show that any non-trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Φr of size 3r×3r with linear polynomial entries such that det Φr = f^r. We also show that the identity, thought of as the trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Ψr of size (3r − 2) × (3r − 2) with linear and quadratic polynomial entries such that det Ψr = f^r.