It is well known that isomorphism class of elliptic curve and lattice up to homeothetic corresponds bijectively.
But I don't have concrete examples. Can we figure out lattice from given weierstrass equation?
For example,
What is a lattice of elliptic curve $y^2＝x^3-x$ ?
The curve you asked about ($y^2 = x^3 - x$) is LMFDB 32.a3. Here the Weierstrass invariants are $\,g_2=-1/4,\, g_3=0\,$ and this is known as the pseudo-lemniscate case. See DLMF 23.5.iv for some details. The lattice for this curve is a square lattice with side length $$\Gamma(1/4)^2/\sqrt{2\pi}=5.244115108584239620\dots$$ and the period parallelogram is a square one of whose diagonals is along the real axis.