  Let $k$ denote an algebraically closed field. We revisit a construction of
the author of families of elliptic curves over the rational function field
$k(t)$. Combining a combinatorial analysis with a rank formula of Ulmer we
prove that, for all but finitely many families of these curves, the
Mordell-Weil groups over $k(t^{1/d})$ have rank zero, as $d$ ranges over
positive integers prime to the characteristic of $k$.
