From: "Gregg Kelly" <greggk@ozemail.com.au>
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