Finding integral-points on elliptic curves

Question

Well, I am trying to find the perfect squares in the following number:

$$57132x^3+2484x^2-54648x+40401\tag1$$

Now, when $(1)$ has to be a perfect square we can write:

$$\text{n}^2=57132x^3+2484x^2-54648x+40401\tag2$$

Where $\text{n}\in\mathbb{Z}$.

I would like to use SageMathCell to find the integral points to equation $(2)$ for some values of $\text{n}$ and $x$.

So, I searched online and I found that I can use the following code:

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)

Where $\alpha$, $\beta$, $\gamma$ and $\delta$ are constants that can be found using the information given by SageMathCell using this documentation.

I was going trough the information and I used the following code:

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)

And I should find that $x=1585$ is a solution. But I found no solution.

I did something wrong but I can not find where I get wrong?!

Answer 1

$y^2=57132x^3+2484x^2-54648x+40401 \implies \\(2116 y)^2 = (6348 x)^3 + 276 (6348 x)^2 - 38545056 (6348 x) + 180893699856$

Magma code:

S:= IntegralPoints(EllipticCurve([0,276,0,-38545056,180893699856]));
for s in S do
q:= s[1]/6348;
if q eq Floor(q) then
print "(",q,", ",s[2]/2116,")";
end if;
end for;