Torus and Elliptic curves

If $\Lambda = \mathbb{Z}+\tau \mathbb{Z}, \tau \not\in \mathbb{R}$ is a lattice then $\mathbb{C}/\Lambda$ is a complex torus. We have the Weierstrass function of the lattice $$\wp(z) = \frac{1}{z^2}+\sum_{\lambda \in \Lambda^*} \frac{1}{(z-\lambda)^2} -\frac{1}{\lambda^2}$$ which is meromorphic and $\Lambda$ periodic, with a double pole on $\mathbb{C}/\Lambda$. From the Laurent expansion $\wp(z) = z^{-2}+0z^0+g_2 z^2+ g_3 z^4+\mathcal{O}(z^6)$ and because an entire doubly periodic function is constant, we find a non-linear relation : $$\wp'(z)^2 = 4 \wp(z)^3- g_2 \wp(z)-g_3$$ In other words, we have found an isomorphism (of Riemann surface and group) $$\varphi : \mathbb{C}/\Lambda \to E, \qquad \varphi(z) = (\wp(z),\wp'(z))$$ where $E$ is the complex elliptic curve $$E = \{ (x,y) \in \mathbb{C}^2, y^2 = 4x^3-g_2 x-g_3\}$$ Finally, applying change of variables to $x,y $ we find any complex elliptic curve is isomorphic to such a complex torus by the mean of its Weierstrass function.