We consolidate a collection of recent developments in the study of scattering amplitudes for supersymmetric field theories. We attempt to answer the questions of systematically and algorithmically setting up bases of loop integrands that amplitudes at higher orders in perturbation theory can be decomposed into by preparing a complete set of cuts or contours. Such problems get increasingly more complicated by the appearance of non-algebraic elliptic integrands. We extend the notion of a leading singularity to include any full dimensional compact contour integral of a scattering amplitude. By focusing on the massive double box integrand we find that the maximal co-dimension residue results in the inverse of the square root of a quartic, which defines an elliptic curve. Choosing to integrate the elliptic curve, (over one of its cycles) yields what we call an elliptic leading singularity. A concept that only refers to the geometry of the elliptic curve. We also expand upon the method of prescriptive unitarity to include such leading singularities. This redefinition hints on unitarity based on diagonalization with respect to the differential forms that appear. We emphasize the benefits of choosing a good basis of integrands by making manifest the local finiteness of the ratio function of N = 4 sYM at 2 loops. By chosing a spanning set of cuts and writing numerators dual to these cuts we prescriptively propose a bases of integrands at one loop for non-maximal supersymmetric theories. Finally, we show that all tree-level amplitudes in pure (super-)gravity can be expressed as term-wise, double-copies of those of pure (super-)Yang-Mills obtained via BCFW recursion.