Integral points on a cubic curve in 2 variables which is not in Weierstrass form
How to find all integral points on a cubic projective curve in 2 variables -5 + x + x^2 + x^3 + y + 2*y^2 - y^3=0
I converted the equation eq to its Weierstrass from el and since it is not integral I had to convert it further to its integral form eli.
Then I found integral points on eli.
But how I find integral points on the original eq? Is it possible to deduce the points form integral pints of eli?
R.<x,y,z> = QQ[]
eq=-5 + x + x^2 + x^3 + y + 2*y^2 - y^3
el=EllipticCurve_from_cubic(eq.homogenize(z), [1,2,1],morphism=False)
print(el)
print(el.is_integral())
eli=el.integral_model()
print(eli)
eli.integral_points()
Elliptic Curve defined by y^2 + 2052480/29309*x*y + 6846114120597504/25176943350629*y = x^3 - 17235873792/16207877*x^2 over Rational Field
False
Elliptic Curve defined by y^2 + 2052480*x*y + 6846114120597504*y = x^3 - 913501310976*x^2 over Rational Field
[(0 : 0 : 1),
(9928949760 : -3803184916070400 : 1),
(105246867456 : -52575457042759680 : 1),
(264110063616 : -101164718767472640 : 1),
(913501310976 : 0 : 1)]
If you do not say "morphism=False", then you will get the isomorphism and can ask for its inverse map.
@FrédéricC This would give me map between
elandeqbut how do I get map betweeneliandeqoreliandel?@FrédéricC OK, I can use
el.isomorphism_to(eli). But the maps does not preserve rationality of points so how can I be sure I found all points on the originaleq?In your case, the elliptic curve
elhas rank one, so you can definea, = el.gens()and look at the inverse images of the multiples ofa.@FrédéricC It should be a method that would work for any cubic curve not just some specific family of curves.