If we take our elliptic curve over $K$ to be the zero set of $ F(X_1, X_2, X_3) = X_2^2 X_3 - (X_1^3 + AX_1X_3^2 + BX_3^3), $ which is in projective form with $X = X_1, Y = X_2, Z=X_3$, then I have been able to show that for any point $P$ on the curve, if $3P = \mathbf{o}$ then the Hessian matrix $ \bigg(\frac{\partial F}{\partial X_i \partial X_j}\bigg) $ has determinant $0$ at $P$.
I am then asked on this exercise to show that there are at most nine 3-torsion points over $K$. Is this an obvious deduction? I am afraid I cannot see how to do it.