A complex lattice is determined by its fourth and sixth Eisenstein sums ($G_4(\Lambda )=G_4({\Lambda }^{\prime }),G_6(\Lambda )=G_6({\Lambda }^{\prime })\Rightarrow \Lambda ={\Lambda }^{\prime }$)

Equal sums give equal Klein j-invariants, hence the complex lattices are homothetic. If both sums are nonzero, the scaling factor obeys $a^4=a^6=1$ and is $\pm 1$. If only $G_6$ vanishes, the lattice is homothetic to the square lattice and the possible fourth roots of unity already preserve it. If only $G_4$ vanishes, it is homothetic to the hexagonal lattice and the possible sixth roots of unity preserve it. The two sums cannot vanish together because the elliptic discriminant is nonzero.