without knowing any deeper theory, I am required to find the Weierstrass normal form of an elliptic curve, i.e. a representation of type $y^2z-x^3-axz-bz^3$ where $x,y,z $ are variables and $a,b$ are coefficients that have to be determined. The given curve is the vanishing set of $f=x^3+y^3+z^3$. My question: Is there a systematic way (beyond "guessing" coordinate transformations) to proceed in order to achieve this representation?
Thank you very much in advance!
There are indeed algorithms to put an elliptic curve into Weierstrass normal form by prescribing a particular change of coordinates, e.g. Nagell's algorithm. (As Jan-Magnus Okland has pointed in the comments, many of these algorithms are implemented in Maple and other softwares.)
In the special case of the Fermat cubic $x^3 + y^3 + 1$, Nagell's algorithm tells us that we should make the substitution $x = \frac{36-v}{6u}$ and $y=\frac{36+v}{6u}$ to get the Weierstrass form $$ v^2 = u^3-432. $$
For many families of elliptic curves (e.g. the twisted Fermat cubics, Selmer curves, and Desboves curves, just to name a few), Nagell's algorithm and others have been worked out in detail to provide us with the general change of coordinates to put a member of this family into Weierstrass form (this is what was used above for the standard Fermat cubic).
maple-packagealgcurves. – Jan-Magnus Økland Jun 24 '15 at 13:59