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-3 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-3 rational maps which can be conjugated to
have a similar relationship between the fixed points and the fixed point
multipliers.
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