  The existence problem for maximal sets of equiangular lines (or SICs) in
complex Hilbert space of dimension $d$ remains largely open. In a previous
publication (arXiv:2112.05552) we gave a conjectural algorithm for how to
construct a SIC if $d = n^2+3 = p$, a prime number. Perhaps the most surprising
number-theoretical aspect of that algorithm is the appearance of Stark units in
a key role: a single Stark unit from a ray class field extension of a real
quadratic field serves as a seed from which the SIC is constructed. The
algorithm can be modified to apply to all dimensions $d = n^2+3$. Here we focus
on the case when $d= n^2+3 = 4p$, $p$ prime, for two reasons. First, special
measures have to be taken on the Hilbert space side of the problem when the
dimension is even. Second, the degrees of the relevant ray class fields are
`smooth' in a sense that facilitates exact calculations. As a result the
algorithm becomes easier to explain. We give solutions for seventeen different
dimensions of this form, reaching $d = 39604$. Several improvements relative to
our previous publication are reported, but we cannot offer a proof that the
algorithm works for any dimensions where it has not been tested.
