  Tropical geometry is sensitive to embeddings of algebraic varieties inside
toric varieties. The purpose of this paper is to advertise tropical
modifications as a tool to locally repair bad embeddings of plane curves,
allowing the re-embedded tropical curve to better reflect the geometry of the
input curve. Our approach is based on the close connection between analytic
curves (in the sense of Berkovich) and tropical curves. We investigate the
effect of these tropical modifications on the tropicalization map defined on
the analytification of the given curve. Our study is motivated by the case of
plane elliptic cubics, where good embeddings are characterized in terms of the
j-invariant. Given a plane elliptic cubic whose tropicalization contains a
cycle, we present an effective algorithm, based on non-Archimedean methods, to
linearly re-embed the curve in dimension 4 so that its tropicalization reflects
the j-invariant. We give an alternative elementary proof of this result by
interpreting the initial terms of the A-discriminant of the defining equation
as a local discriminant in the Newton subdivision.
