The tropical variety associated to a plane curve over the field of Puiseux series is a graph in the plane whose edges have rational slopes. A tropical cubic may have a cycle. We show how the valuation of the j-invariant can be read off of the tropical variety: it is the generalized lattice length of the cycle. We will also explain how this result is related to stable reduction.
This is joint work with Hannah Markwig and Thomas Markwig.