  Recently, several intriguing conjectures have been proposed connecting
symmetric informationally complete quantum measurements (SIC POVMs, or SICs)
and algebraic number theory. These conjectures relate the SICs and their
minimal defining algebraic number field. Testing or sharpening these
conjectures requires that the SICs are expressed exactly, rather than as
numerical approximations. While many exact solutions of SICs have been
constructed previously using Gr\"obner bases, this method has probably been
taken as far as is possible with current computer technology (except in special
cases where there are additional symmetries). Here we describe a method for
converting high-precision numerical solutions into exact ones using an integer
relation algorithm in conjunction with the Galois symmetries of a SIC. Using
this method we have calculated 69 new exact solutions, including 9 new
dimensions where previously only numerical solutions were known, which more
than triples the number of known exact solutions. In some cases the solutions
require number fields with degrees as high as 12,288. We use these solutions to
confirm that they obey the number-theoretic conjectures and we address two
questions suggested by the previous work.
