From: "Gregg Kelly" Subject: Answer to old question found in Dave Rusin's pages Date: Sat, 29 Jul 2000 17:43:35 +1000 Newsgroups: sci.math Summary: [missing] Here is an extract from a '95 newsgroup posting I found in the Mathematical Atlas maintained by Dave Rusin. Ohn Christian (chohn@vub.ac.be) wrote: : Let : a x^3 + b y^3 + c z^3 + d x^2 y + e x^2 z + f y^2 x + g y^2 z + h z^2 x : + i z^2 y + k x y z = 0 : be an cubic curve in P^2(C). : 1) Can one express a "discriminant" D explicitely in the ten : coefficients a,b,c,d,e,f,g,h,i,k such that the curve is elliptic : iff D is not zero? : 2) If so, can one express explicitely the j-invariant of that elliptic : curve in terms of a,b,c,d,e,f,g,h,i,k? : For both questions, I mean: *without* reducing the equation of the curve : first. Well the answer to both questions is "yes". I have posted a web page containing the answer at http://www.ozemail.com.au/~greggk/cubicinv/cubicinv.htm In case you're wondering why I don't just type in the formula requested here read the web page and you will understand why. I don't think the qualification about "without reducing the equation..." is meaningful because any method which gives you the j-invariant also (with a little extra calculation) gives you the birational reduction transformation and vice versa. Gregg Kelly