Weierstrass to general form

Question

Can we go from short Weierstrass equation equation $y^2=x^3+Ax^2+Bx+C$ to general $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$?

Answer 1

What you call the "general equation" is the Weierstrass equation, and what you call the "Weierstrass equation" is the so-called "short" Weierstrass equation, which is just the general Weierstrass equation with some coefficients $0$, so your question doesn't really make sense.

But maybe you meant to ask it the other way around: how to get a short Weierstrass equation from a general one? We perform a change of variables which preserves the behavior of the curve, called an admissible change of variables. If we are over a field of characteristic not $2$ or $3$, it is always possible to get a short equation from a general one through an admissible change of variables. Otherwise, not always.