2019-12-05 12:04:34 +0200 received badge ● Student (source) 2019-12-03 18:07:18 +0200 asked a question Mistake in SageMathCell code, finding integral points on elliptic curves I've the following number: $$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$ Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has a given value) besides that $x\in\mathbb{N}^+$ and $x\ge2$. I want to check if the number is a perfect square, so I can rewrite $(1)$ as follows: $$y^2=12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag2$$ Where $y\in\mathbb{Z}$. In this problem I've: $n=71$, the number is equal to; $$y^2=57132x^3+2484x^2-54648x+40401\tag3$$ 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): $$\alpha=12(71-2)^2=57132\tag4$$ $$\beta=36(71-2)=2484\tag5$$ $$\gamma=-144(71-5)(71-2)^3=-3122149536\tag6$$ $$\delta=1296(71-4)^2(71-2)^4=131871507195024\tag7$$ 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?