A central problem in quantum information theory is to understand the apparent existence of d^2 equiangular lines in any d-dimensional complex vector space.  Such configurations of lines determine highly-symmetric optimal quantum measurements known as Symmetric Informationally-Complete Positive Operator-Valued Measures (SIC-POVMs).  Much evidence indicates that SIC-POVMs can always be obtained as special orbits of finite Heisenberg groups.  Exact constructions of Heisenberg-covariant SIC-POVMs are currently known in 22 different dimensions, while numerical evidence indicates that they exist for every dimension up to 67.  The elements of Heisenberg-covariant SIC-POVMs correspond to quantum states that are maximally localized in discrete phase space and can be viewed as finite-dimensional analogs of coherent states.  They also contain rich algebraic structure, as they are defined over number fields with nonabelian Galois groups - namely, over abelian extensions of quadratic fields.  In this talk, I will show how the mathematics of class field theory can explain some of the group-theoretic structure possessed by these known examples while offering predictions for their structure in arbitrary dimensions.