Different forms for the short Weierstrass Equation

Question

I am reading about elliptic curves from Silverman's AEC. Initially we see that for a field $K$ whose characteristic is not 2 or 3, any elliptic curve is given by a (short) Weierstrass equation $y^2 = x^3 +Ax+B $. Also the allowed change of coordinates that preserve this form are $(x,y)\rightarrow (u^2x,u^3y)$ for $u \in K, u \neq 0.$ However later while looking at Elliptic curves over $\mathbb{C}$, we look at curves with equation $y^2 =4x^3 - g_2x - g_3$, which are also claimed to be elliptic. What I do not understand is - firstly how is $y^2 =4x^3 - g_2x - g_3$ elliptic and secondly can any elliptic curve in $K$ be reduced to such a form (as in can we go from $y^2 = x^3 +Ax+B $ to $y^2 =4x^3 - g_2x - g_3$). I do know that for any $A,B$ such that $A^3-27B^2\neq 0$, we have a unique lattice $\Lambda$ such that $g_2(\Lambda)=A$ and $g_3(\Lambda)=B$

Answer 1

$$y^2=4x^3-g_2x-g_3$$ is isomorphic to $$Y^2=X^3+AX+B$$ under the substitutions $x=X$, $y=2Y$, $A=-g_2/4$ and $B=-g_3/4$.