Counting rational points on elliptic curves with a rational 2-torsion point

Let $E/\\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by work of Bombieri and Zannier, that if $E$ has full 2-torsion, number $N_E(B)$ points with Weil height bounded $B$ $\\operatorname{exp}\\big(O \\big(\\frac{\\operatorname{log}B}{\\sqrt{\\operatorname{log}\\operatorname{log} B}}\\big)\\big)$. In this paper we exploit method descent via 2-isogeny to extend result curves just one nontrivial 2-torsion point. Moreover, make use a Petsche derive stronger upper bound $N_E(B) = \\operatorname{exp}\\big(O B}}\\big)\\big)$ for these remove deep transcendence theory ingredient from proof.