On the number of rational points on certain elliptic curves

Let $E$ be an elliptic curve defined over the rationals, with rational 2-torsion. We prove a uniform bound for the number of rational points on $E$ of height $\leqslant B$ of the form $\#\{P\in E({\mathbb Q})\colon H(P)\leqslant B\}\leqslant c(\varepsilon)(\max(H(E),B))^\varepsilon$, valid for every fixed $\varepsilon>0$ and a suitable positive computable constant $c(\varepsilon)$. We give an application of this result to the counting of quadruples $(p_1,p_2,p_3,p_4)$ of distinct primes that do not exceed $X$ and satisfy $p_i^2\Delta_{jk}-p_j^2\Delta_{ik}+p_k^2\Delta_{ij}=0$ for all $1\leqslant i<j<k\leqslant 4$, where $\Delta_{ij}$ are given integers. This is applied by Konyagin (in the paper [3], which is published simultaneously with the present one) to a problem on the large sieve by squares.