Let $E/K$ an elliptic curve defined by $f(x,y)=0$. I have read that there always exists a birational transformation $(X,Y)=(X(x,y),Y(x,y))$ such that it can be written in Weierstrass form (I'm mainly interested in $K=\mathbb{R},\mathbb{C}$, so let us assume $\text{char}(K)\neq2,3$)

$$Y^2=X^3+f X+g\,.$$ I am wondering if there is a standard way to find said birational transformation, as well as $f$ and $g$ in terms of the original coefficients.

For cubics, I know that Nagell's algorithm allows one to find the form of the transformation for the cubic (see e.g. this question), and similarly for polynomials quartic in $x$, but what about higher degree polynomials in $x$ (provided they are elliptic curves), or more exotic examples, such as Watt's curve? I would like to use this to see what are the conditions for a curve of the form $y^2=h(x)$, with $h(x)$ a degree six polynomial to be elliptic. For instance, in the case where $h(x)$ is degree four, one can deduce that the constant term must be a perfect square.

I am also not quite sure about the nomenclature: When one says any elliptic curve is birationally equivalent to a Weierstrass form, does one mean that it is the same curve, i.e. simply a coordinate redefinition, or does it change something?



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