Summary: For each of $n =1,2,3$ we find the minimal height ${\widehat{h}}(P)$ of a nontorsion point $P$ of an elliptic curve $E$ over ${\bold C} (T)$ of discriminant degree $d =12n$ (equivalently, of arithmetic genus $n$), and exhibit all $(E, P)$ attaining this minimum. The minimal ${\widehat{h}}(P)$ was known to equal 1/30 for $n =1$ (Oguiso-Shioda) and $11/420$ for $n =2$ (Nishiyama), but the formulas for the general $(E, P)$ were not known, nor was the fact that these are also the minima for an elliptic curve of discriminant degree $12n$ over a function field of any genus. For $n =3$ both the minimal height (23/840) and the explicit curves are new. These $(E, P)$ also have the property that that $mP$ is an integral point (a point of naïve height zero) for each $m =1,2,\ldots, M$, where $M =6,8,9$ for $n =1,2,3$; this, too, is maximal in each of the three cases.