  We present an algorithm for the construction of lower dimensional elliptic
tori in parametric Hamiltonian systems by means of the parametrization method
with the tangent and normal frequencies being prescribed. This requires that
the Hamiltonian system has as many parameters as the dimension of the normal
dynamics, and the algorithm must adjust these parameters. We illustrate the
methodology with an implementation of the algorithm computing $2$--dimensional
elliptic tori in a system of $4$ coupled pendula (4 degrees of freedom).
