2
$\begingroup$

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!

$\endgroup$
1
  • 3
    $\begingroup$ There's an algorithm implemented in the maple-package algcurves. $\endgroup$ Jun 24, 2015 at 13:59

1 Answer 1

2
$\begingroup$

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).

$\endgroup$
1
  • $\begingroup$ There might be a typo here: should the Fermat cubic be defined by $x^3 + y^3 = 1$? $\endgroup$
    – user321945
    Oct 3, 2022 at 2:36

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .