For an amorphous material, the symmetry is much lower than that for an ordered phase, or very likely there is no point symmetry, so that you will have to calculate all the elastic constants, assuming no symmetry for your amorphous structure.

If you think that the symmetry is approximately preserved (or you want to enforce it) for the amorphous structure, you may do averaging (in a post-processing manner) over the elastic constants that are supposed to be identical for the ordered material. In this case, you will get three effective elastic constants.