Consider two (primitive) lattices $\pi_{i} \in \mathbb{C}$, such that $\pi_{1} = M \pi_{2}$ for $M \in \mathcal{S}_{m}$ with $$\mathcal{S}_{m}:=\Big\{\begin{pmatrix} A & B \\ 0 & D \end{pmatrix} \in \mathrm{GL}_{2}(\mathbb{Z}) : AD = m >0,\;D>0,\;0 \leq B \leq D\Big\}.$$ Does this imply that the elliptic curves that correspond to these lattices are isogenous and therefore one lattice corresponds to an order that it is an over-order (or the same order if the lattices are homothetic) to the order of the other lattice?