The tropical $j$-invariant
  If (Q,A) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support A defines an elliptic curve C on the toric surface X_A. If K has a non-archimedean valuation into the real numbers we can tropicalize C to get a tropical curve Trop(C). If the Newton subdivision induced by f is a triangulation, then Trop(C) will be a graph of genus one and we show that the lattice length of the cycle of that graph is the negative of the valuation of the j-invariant of C. 
Eric Katz, Hannah Markwig, Thomas Markwig