  We give an elementary proof of the fact that any elliptic curve $E$ over an
algebraically closed non-archimedean field $K$ with residue characteristic
$\neq{2,3}$ and with $v(j(E))<0$ admits a tropicalization that contains a cycle
of length $-v(j(E))$. We first define an adapted form of minimal models over
non-discrete valuation rings and we recover several well-known theorems from
the discrete case. Using these, we create an explicit family of marked elliptic
curves $(E,P)$, where $E$ has multiplicative reduction and $P$ is an inflection
point that reduces to the singular point on the reduction of $E$. We then
follow the strategy as in \cite[Theorem 6.2]{BPR11} and construct an embedding
such that its tropicalization contains a cycle of length $-v(j(E))$. We call
this a numerically faithful tropicalization. A key difference between this
approach and the approach in \cite{BPR11} is that we do not require any of the
analytic theory on Berkovich spaces such as the {\it{Poincar\'{e}-Lelong
formula}} or \cite[Theorem 5.25]{BPR11} to establish the numerical faithfulness
of this tropicalization.
