Relation between Weierstrass $ \wp $ function and elliptic curve

Question

I want to see how the Weierstrass $ \wp $ function and its derivative can define the curve $$ y ^ 2 = 4x^3 - g_2 x - g_3 $$ I think I can simulate the $ \wp $ function and its derivative properly, but I don't understand how could they define the curve mentioned above. If I make $x:=\wp(z)$ and $y:=\wp'(z)$ what do I do when $\wp(z)$ is not real? Here is the link to my worksheet: https://www.geogebra.org/m/eydcwzzt As you can see I am not a master of this topic, but I am very interested and I want to understand these relations. Thank you for your help!