Smallest representatives of $\mathrm{SL}(2,\mathbb{Z})$-orbits of binary forms and endomorphisms of ${\mathbb{P}}^1$

We develop an algorithm that determines, for a given squarefree binary form $F$ with real coefficients, a smallest representative of its orbit under $\mathrm{SL}(2,\mathbb{Z})$, either with respect to the Euclidean norm or with respect to the maximum norm of the coefficient vector. This is based on earlier work of Cremona and Stoll. We then generalize our approach so that it also applies to the problem of finding an integral representative of smallest height in the $\mathrm{PGL}(2,\mathbb{Q})$ conjugacy class of an endomorphism of the projective line. Having a small model of such an endomorphism is useful for various computations.