  We consider the problem of counting and classifying symmetric informationally
complete positive operator-valued measures (SICs or SIC-POVMs), that is, sets
of $d^2$ equiangular lines in $\mathbb{C}^d$. For $4 \leq d \leq 90$, we show
the number of known equivalence classes of Weyl--Heisenberg covariant SICs in
dimension $d$ equals the cardinality of the ideal class monoid of (not
necessarily invertible) ideal classes in the real quadratic order of
discriminant $(d+1)(d-3)$. Equivalently, this is the number of
$\mathbf{GL}_2(\mathbb{Z})$ conjugacy classes in $\mathbf{SL}_2(\mathbb{Z})$ of
trace $d-1$. We conjecture the equality extends to all $d \geq 4$. We prove
that this conjecture implies more that one equivalence class of
Weyl--Heisenberg covariant SICs for every $d > 22$.
  Additionally, we refine the "class field hypothesis" of Appleby, Flammia,
McConnell, and Yard (arXiv:1604.06098) to predict the exact class field
generated by the ratios of vector entries for the equiangular lines defining a
Weyl--Heisenberg covariant SIC. The class fields conjecturally associated to
SICs in dimension $d$ come with a natural partial order under inclusion; we
show that the natural inclusions of these fields are strict, except in a small
family of cases.
