Teichmüller curves are algebraic curves in the moduli space of genus $g$ Riemann surfaces that are isometrically immersed for the Teichmüller metric. These curves arise from $\mathrm{SL}(2,\mathbb{R})$-orbits of highly symmetric translation surfaces, and the underlying surfaces have remarkable dynamical and algebro-geometric properties. A Teichmüller curve is algebraically primitive if the trace field of its affine symmetry group has degree $g$. In genus $2$, Calta and McMullen independently discovered an infinite family of algebraically primitive Teichmüller curves. However, in higher genus, such curves seem to be much rarer. We will discuss a result that shows that the regular $14$-gon yields the unique algebraically primitive Teichmüller curve in genus $3$ of a particular combinatorial type. All relevant notions will be explained during the talk.