  In this article, we study Lehmer-type bounds for the N\'eron-Tate height of
$\bar{K}$-points on abelian varieties $A$ over number fields $K$. Then, we
estimate the number of $K$-rational points on $A$ with N\'eron-Tate height
$\leq \log B$ for $B\gg 0$. This estimate involves a constant $C$, which is not
explicit. However, for elliptic curves and the product of elliptic curves over
$K$, we make the constant explicitly computable.
