Let E/k be an elliptic curve over a number field. We obtain some quantitative refinements of results of Hindry-Silverman, giving an upper bound for the number of k-rational torsion points, and a lower bound for the canonical height of nontorsion k-rational points, in terms of expressions depending explicitly on the degree d=[k:Q] of k and the Szpiro ratio σ of E/k. The bounds exhibit only polynomial dependence on both d and σ.