I am working with a quartic
(1) $ \quad y^2 = -x^4 + 1504271405904482x^2 - 16053296232241^2 $
A known solution is $ [x,y] = [1853^2, 131674446875520] .$
The ellfromeqn() function in GP-Pari takes the affine definition (very close to 1) and finds the elliptic curve from the Jacobian connected to the quartic:
(2) $ \quad ellfromeqn((1)) = [0,1504271405904482,0,-32106592464482^2,-1504271405904482\times32106592464482^2] $
Using mwrank for this elliptic curve, we find that it is rank 2 with one known point:
(3) $ \quad [X,Y] = [3707691537441922/9,485429905397399652368720/27] $
However in doing the reverse transform to take this point to the quartic, an imaginary point pair results.
Let $ A = -1 ; B = 0 ; C = 1504271405904482 ; D = 0 ; q^2 = -16053296232241^2$
for the general quartic equation $ y^2 = Ax^4 + Bx^3 + Cx^2 + Dx + q^2 $
then $ q = 16053296232241\sqrt{-1} $ and $ Q = 2q = 32106592464482\sqrt{-1} $
The reverse transforms from Ian Connell or Dale Husemoller books on elliptic curves are
(4a) $ x = (Q*(X+C)-D^2/Q)*(1/Y) $
(4b) $ y = -q + x*(x*X-D)/Q $
and using the values in (3) above, we obtain
(5) $ x = \frac{6844011\sqrt{-1}}2 $ and $ y = \frac{536805907785085\sqrt{-1}}4 $
which does satisfy (1), so we know that mwrank got the point okay.
Questions:
How do we avoid complex points on the quartic coming from real points on the elliptic curve?
Are the transforms in (4a) and (4b) at fault here?
Thanks for explaining how to avoid complex numbers when we go from (1) to (2) or vice versa. This happens when we have a negative integer for the last coefficient in (1).