Geometry ag.algebraica – Atlas for the stack of pulleys of rank 1 with torsion

I would like to build an atlas for the stack of E-sheaves of rank 1 and grade b on an elliptic curve C such that E has a range of torsion of at most 1. Am I allowed to fix both the determinant L of sheaf E and the point x in C where E possibly has torsion? In that case, I would build an atlas like PHom (L (-2x), F) where F is the unique non-trivial extension of L (-x) by L (-x). Would this help me build an atlas for the original stack?