Ramification in division fields and sporadic points on modular curves

Consider an elliptic curve E over a number field K. Suppose that has supersingular reduction at some prime $$\mathfrak {p}$$ of K lying above the rational p. We completely classify valuations $$p^n$$ -torsion points by valuation coefficient $$p{\text {th}}$$ division polynomial. This classification corrects error in earlier work Lozano-Robledo. As application, we find minimum necessary ramification order for to have point exact . Using this bound show sporadic on modular $$X_1(p^n)$$ cannot correspond curves without canonical subgroup. generalize our methods $$X_1(N)$$ with N composite.