  An algebraic function of the third order plays an important role in the
problem of asymptotics of Hermite-Pad\'e approximants for two analytic
functions with branch points. This algebraic function appears as the Cauchy
transform of the limiting measure of the asymptotic distribution of the poles
of the approximants. In many cases this algebraic function can be determined by
using the given position of the branch points of the functions which are
approximated and by the condition that its Abelian integral has purely
imaginary periods. In the present paper we obtain a hyperelliptic
uniformization of this algebraic function. In the case when each approximated
function has only two branch points, the genus of this function can be equal to
0, 1 (elliptic case) or 2 (ultra-elliptic case). We use this uniformization to
parametrize the elliptic case. This parametrization allows us to obtain a
numerical procedure for finding this elliptic curve and as a result we can
describe the limiting measure of the distribution of the poles of the
approximants.
