ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 03 Dec 2019 22:34:27 +0100Mistake in SageMathCell code, finding integral points on elliptic curveshttps://ask.sagemath.org/question/48933/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?Tue, 03 Dec 2019 17:13:08 +0100https://ask.sagemath.org/question/48933/mistake-in-sagemathcell-code-finding-integral-points-on-elliptic-curves/Comment by John Palmieri for <p>I've the following number:</p>
<p>$$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$</p>
<p>Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has a given value) besides that $x\in\mathbb{N}^+$ and $x\ge2$.</p>
<p>I want to check if the number is a perfect square, so I can rewrite $(1)$ as follows:</p>
<p>$$y^2=12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag2$$</p>
<p>Where $y\in\mathbb{Z}$.</p>
<p>In this problem I've: $n=71$, the number is equal to;</p>
<p>$$y^2=57132x^3+2484x^2-54648x+40401\tag3$$</p>
<p>So, I used SageMathCell to look for the integral points on the elliptic curve and the code that was used is the following:</p>
<pre><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)
</code></pre>
<p>I found the coeficients I need to use using equation $(2)$ and $(3)$ (but I do not know if they are corect):</p>
<ul>
<li>$$\alpha=12(71-2)^2=57132\tag4$$</li>
<li>$$\beta=36(71-2)=2484\tag5$$</li>
<li>$$\gamma=-144(71-5)(71-2)^3=-3122149536\tag6$$</li>
<li>$$\delta=1296(71-4)^2(71-2)^4=131871507195024\tag7$$</li>
</ul>
<p>So the final code looks like:</p>
<pre><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)
</code></pre>
<p>But I found no solutions and it should give at least one solution at $x=1585$.</p>
<blockquote>
<p>What mistake have I made?</p>
</blockquote>
https://ask.sagemath.org/question/48933/mistake-in-sagemathcell-code-finding-integral-points-on-elliptic-curves/?comment=48934#post-id-48934Asked and answered at https://mathoverflow.net/questions/347543/mistake-in-sagemathcell-code-finding-integral-points-on-elliptic-curves.Tue, 03 Dec 2019 22:34:27 +0100https://ask.sagemath.org/question/48933/mistake-in-sagemathcell-code-finding-integral-points-on-elliptic-curves/?comment=48934#post-id-48934