A bstract We present generic expressions for the integrands of canonical bases under maximal cut in elliptic Feynman integral families with multiple kinematic scales. Such integrals frequently arise in phenomenologically relevant scattering processes. The derivation of our results starts from the Legendre normal form of elliptic curves, where the geometric properties of the curves are simple and explicit, and further kinematic singularities are presented as marked points. The simplicity of the normal form allows a straightforward construction of canonical bases with an arbitrary number of marked points. They can then be mapped into any univariate elliptic integral families via an appropriate Möbius transformation, leading to universal expressions for the integrands. As a demonstration, we discuss the application of our method to several concrete examples, including two new integral families whose canonical bases were not available in the literature. In several examples, we derive canonical bases for the full integral families without any cuts, demonstrating the simplicity of the sub-sector dependence of our canonical bases.
Chen et al. (2026) studied this question.