  For a hyperbolic link complement with a triangulation, there are
hyperbolicity equations of the triangulation, which guarantee the hyperbolic
structure of the link complement. In this paper, we explain that the number of
the essential solutions of the equations is equal to or bigger than the
extension degree of the invariant trace field of the link.
  On the other hand, Yokota suggested a potential function of a hyperbolic
knot, which gives the hyperbolicity equations and the complex volume of the
knot. Applying the fact above to his theory, we explain that the potential
function also gives all the values of the Borel regulator map and the complex
volumes of the parabolic representations. Furthermore, we explain the maximum
value of the imaginary parts of the complex volumes is the volume of the
complete hyperbolic structure of the knot complement.
  Especially, if the number of the essential solutions of the hyperbolicity
equations and the extension degree of the invariant trace field are the same,
then the evaluation of all essential complex solutions of the hyperbolicity
equations to the imaginary part of the potential function is the same with the
Borel regulator map. We show these actually happens in the case of the twist
knots.
