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When studying families in the moduli space of dynamical systems, choosing anappropriate representative function for a conjugacy class can be a delicatetask. The most delicate questions surround rationality of the conjugacy classcompared to rationality of the defining polynomials of the representation. Wegive a normal form for degree three polynomials which has the property that theset of fixed points is equal to the set of fixed point multipliers. This normalform is given in terms of moduli space invariants and, hence, has nicerationality properties. We further classify all degree three rational mapswhich can be conjugated to have a similar relationship between the fixed pointsand the fixed point multipliers.

New Normal Forms For Degree Three Polynomials and Rational Functions