The spectrum and essential minimum of heights of algebraic points on the projective line given by x+y+z=0 in the projective plane P^2 will be discussed. These notions are motivated by work of Zhang and Zagier. In the first part we prove that the spectrum of the heights is asymptotically dense in R after a certain threshold C(0). In the second part of the work, we prove that the six special points where Zagier showed that the height achieves a known sharp bound can be normalized to a single point. This makes it plausible to find all values in the spectrum before the threshold C(0) also and to hence find the essential minimum. We provide a conjecture about the essential minimum and the distribution of the spectrum of heights before this minimum.

The main ingredients in our analysis are potential theory, particularly the estimation of the capacity of the level curve and the theorem of Fekete-Szego. Our results were greatly motivated by numerical experiments with the capacity.