  When studying families in the moduli space of dynamical systems, choosing an
appropriate representative function for a conjugacy class can be a delicate
task. The most delicate questions surround rationality of the conjugacy class
compared to rationality of the defining polynomials of the representation. We
give a normal form for degree three polynomials which has the property that the
set of fixed points is equal to the set of fixed point multipliers. This normal
form is given in terms of moduli space invariants and, hence, has nice
rationality properties. We further classify all degree three rational maps
which can be conjugated to have a similar relationship between the fixed points
and the fixed point multipliers.
