Is there a nice description of all real algebraic curves of $\mathbb R^2$ which have the property that every such curve contains all zeroes (after the obvious identification of $\mathbb C$ with $\mathbb R^2$) of infinitely many monic rational polynomials of $\mathbb Q[z]$ having only simple roots.

Examples are the real line, the unit circle and there are many hyperelliptic curves (obtained by taking preimages of such curves with respect to suitable rational functions). (It is of course easy to construct also non-simple examples of such curves, eg. by considering all real lines determined by sets of zeroes of cyclotomic polynomials.)

I guess that there exists a curve $\mathcal C\subset \mathbb R^2\sim\mathbb C$ defined by a polynomial in $\mathbb Z[x,y]$ which contains the rootsets of only finitely many rational monic polynomials without multiple roots. Can someone exhibit such a curve?