  This paper analyzes U(1) F-theory models admitting matter with charges $q=3$
and $4$. First, we systematically derive a $q=3$ construction that generalizes
the previous $q=3$ examples. We argue that U(1) symmetries can be tuned through
a procedure reminiscent of the SU(N) and Sp(N) tuning process. For models with
$q=3$ matter, the components of the generating section vanish to orders higher
than 1 at the charge-3 matter loci. As a result, the Weierstrass models can
contain non-UFD structure and thereby deviate from the standard Morrison-Park
form. Techniques used to tune SU(N) models on singular divisors allow us to
determine the non-UFD structures and derive the $q=3$ tuning from scratch. We
also obtain a class of a $q=4$ models by deforming a prior U(1)$\times$U(1)
construction. To the author's knowledge, this is the first published F-theory
example with charge-4 matter. Finally, we discuss some conjectures regarding
models with charges larger than 4.
