We describe the geometry of the leading singularity locus of the traintrack integral family directly in momentum twistor space. For the two-loop case, known as the elliptic double box, the leading singularity locus is a genus one curve, which we obtain as an intersection of two quadrics in P-3. At three loops, we obtain a K3 surface which arises as a branched surface over two genus-one curves in P(1)x P-1. We present an analysis of its properties. We also discuss the geometry at higher loops and the supersymmetrization of the construction.