2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76000Let 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 non-torsion k-rational points, in terms of expressions depending explicitly on the degree of k and the Szpiro ratio of E/k. The bounds exhibit only polynomial dependence on both quantities.11 pages. In this revision, an irritating LaTeX error has been correctedNumber Theory14H52; 11G50; 14G05text