  If $M$ is a finite volume complete hyperbolic 3-manifold with one cusp and no
2-torsion, the geometric component $X_M$ of its $\SL(2,\BC)$-character variety
is an affine complex curve, which is smooth at the discrete faithful
representation $\rho_0$. Porti defined a non-abelian Reidemeister torsion in a
neighborhood of $\rho_0$ in $X_M$ and observed that it is an analytic map,
which is the germ of a unique rational function on $X_M$. In the present paper
we prove that (a) the torsion of a representation lies in at most quadratic
extension of the invariant trace field of the representation, and (b) the
existence of a polynomial relation of the torsion of a representation and the
trace of the meridian or the longitude. We postulate that the coefficients of
the $1/N^k$-asymptotics of the Parametrized Volume Conjecture for $M$ are
elements of the field of rational functions on $X_M$.
