Mistake in SageMathCell code, finding integral points on elliptic curves
I've the following number:
12(n−2)2x3+36(n−2)x2−12(n−5)(n−2)x+9(n−4)2
Now I know that n∈N+ and n≥3 (and n has a given value) besides that x∈N+ and x≥2.
I want to check if the number is a perfect square, so I can rewrite (1) as follows:
y2=12(n−2)2x3+36(n−2)x2−12(n−5)(n−2)x+9(n−4)2
Where y∈Z.
In this problem I've: n=71, the number is equal to;
y2=57132x3+2484x2−54648x+40401
So, I used SageMathCell to look for the integral points on the elliptic curve and the code that was used is the following:
E = EllipticCurve([0, β, 0, γ, δ])
P = E.integral_points()
for p in P:
if p[0] % α == 0:
print(p[0] // α, p[1] // α)
I found the coeficients I need to use using equation (2) and (3) (but I do not know if they are corect):
- α=12(71−2)2=57132
- β=36(71−2)=2484
- γ=−144(71−5)(71−2)3=−3122149536
- δ=1296(71−4)2(71−2)4=131871507195024
So the final code looks like:
E = EllipticCurve([0, 2484, 0, -3122149536, 131871507195024])
P = E.integral_points()
for p in P:
if p[0] % 57132 == 0:
print(p[0] // 57132, p[1] // 57132)
But I found no solutions and it should give at least one solution at x=1585.
What mistake have I made?
Asked and answered at https://mathoverflow.net/questions/34....