Mistake in SageMathCell code, finding integral points on elliptic curves

asked 5 years ago

Jan123 gravatar image

I've the following number:

12(n2)2x3+36(n2)x212(n5)(n2)x+9(n4)2

Now I know that nN+ and n3 (and n has a given value) besides that xN+ and x2.

I want to check if the number is a perfect square, so I can rewrite (1) as follows:

y2=12(n2)2x3+36(n2)x212(n5)(n2)x+9(n4)2

Where yZ.

In this problem I've: n=71, the number is equal to;

y2=57132x3+2484x254648x+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(712)2=57132
  • β=36(712)=2484
  • γ=144(715)(712)3=3122149536
  • δ=1296(714)2(712)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?

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Comments

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John Palmieri gravatar imageJohn Palmieri ( 5 years ago )