  In this paper, we investigate two-loop non-planar triangle Feynman integrals
involving elliptic curves. In contrast to the Sunrise and Banana integral
families, the triangle families involve non-trivial sub-sectors. We show that
the methodology developed in the context of Banana integrals can also be
extended to these cases and obtain $\varepsilon$-factorized differential
equations for all sectors. The letters are combinations of modular forms on the
corresponding elliptic curves and algebraic functions arising from the
sub-sectors. With uniform transcendental boundary conditions, we express our
results in terms of iterated integrals order-by-order in the dimensional
regulator, which can be evaluated efficiently. Our method can be
straightforwardly generalized to other elliptic integral families and have
important applications to precision physics at current and future high-energy
colliders.
