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, 09 Feb 2021 12:30:03 +0100Are results found of an Elliptic Curve by SageMathCell proven?https://ask.sagemath.org/question/49605/are-results-found-of-an-elliptic-curve-by-sagemathcell-proven/ Well, I have for example the following SageMathCell-code:
sage: E = EllipticCurve(QQ, [0,63,-2205,-12348,0])
sage: E
sage: for P in E.integral_points():
....: Q = -P
....: print( "P = %8s and -P = %8s" % (P.xy(), Q.xy()) )
This code computes the integral points of the Elliptic Curve that is defined by:
$$[0,63,-2205,-12348,0]\space\space\space\to\space\space\space y^2 - 2205y = x^3 + 63x^2 - 12348x\tag1$$
>Are these results I get, proven to be the only ones out there? Or can there be more solutions that SageMathCell did not find?
___
Bytheway, the code gives the following output:
P = (-174, 1161) and -P = (-174, 1044)
P = (-147, 2205) and -P = (-147, 0)
P = (-98, 2548) and -P = (-98, -343)
P = (-68, 2528) and -P = (-68, -323)
P = (-54, 2484) and -P = (-54, -279)
P = (0, 2205) and -P = (0, 0)
P = (57, 2052) and -P = (57, 153)
P = (84, 2205) and -P = (84, 0)
P = (147, 3087) and -P = (147, -882)
P = (231, 4851) and -P = (231, -2646)
P = (309, 6840) and -P = (309, -4635)
P = (375, 8730) and -P = (375, -6525)
P = (378, 8820) and -P = (378, -6615)
P = (711, 20691) and -P = (711, -18486)
P = (1176, 42336) and -P = (1176, -40131)
P = (2107, 99127) and -P = (2107, -96922)
P = (2886, 157716) and -P = (2886, -155511)
P = (5412, 401472) and -P = (5412, -399267)
P = (5572, 419293) and -P = (5572, -417088)
P = (37275, 7203735) and -P = (37275, -7201530)
P = (26162409, 133818797385) and -P = (26162409, -133818795180)
Sun, 19 Jan 2020 15:33:57 +0100https://ask.sagemath.org/question/49605/are-results-found-of-an-elliptic-curve-by-sagemathcell-proven/Answer by rburing for <p>Well, I have for example the following SageMathCell-code:</p>
<pre><code>sage: E = EllipticCurve(QQ, [0,63,-2205,-12348,0])
sage: E
sage: for P in E.integral_points():
....: Q = -P
....: print( "P = %8s and -P = %8s" % (P.xy(), Q.xy()) )
</code></pre>
<p>This code computes the integral points of the Elliptic Curve that is defined by:</p>
<p>$$[0,63,-2205,-12348,0]\space\space\space\to\space\space\space y^2 - 2205y = x^3 + 63x^2 - 12348x\tag1$$</p>
<blockquote>
<p>Are these results I get, proven to be the only ones out there? Or can there be more solutions that SageMathCell did not find?</p>
</blockquote>
<hr>
<p>Bytheway, the code gives the following output:</p>
<pre><code>P = (-174, 1161) and -P = (-174, 1044)
P = (-147, 2205) and -P = (-147, 0)
P = (-98, 2548) and -P = (-98, -343)
P = (-68, 2528) and -P = (-68, -323)
P = (-54, 2484) and -P = (-54, -279)
P = (0, 2205) and -P = (0, 0)
P = (57, 2052) and -P = (57, 153)
P = (84, 2205) and -P = (84, 0)
P = (147, 3087) and -P = (147, -882)
P = (231, 4851) and -P = (231, -2646)
P = (309, 6840) and -P = (309, -4635)
P = (375, 8730) and -P = (375, -6525)
P = (378, 8820) and -P = (378, -6615)
P = (711, 20691) and -P = (711, -18486)
P = (1176, 42336) and -P = (1176, -40131)
P = (2107, 99127) and -P = (2107, -96922)
P = (2886, 157716) and -P = (2886, -155511)
P = (5412, 401472) and -P = (5412, -399267)
P = (5572, 419293) and -P = (5572, -417088)
P = (37275, 7203735) and -P = (37275, -7201530)
P = (26162409, 133818797385) and -P = (26162409, -133818795180)
</code></pre>
https://ask.sagemath.org/question/49605/are-results-found-of-an-elliptic-curve-by-sagemathcell-proven/?answer=49629#post-id-49629Yes, [as explained on Math.StackExchange](https://math.stackexchange.com/questions/3514632/are-results-found-of-an-elliptic-curve-by-sagemathcell-proven-does-there-exists/3514706#3514706).Tue, 21 Jan 2020 22:22:43 +0100https://ask.sagemath.org/question/49605/are-results-found-of-an-elliptic-curve-by-sagemathcell-proven/?answer=49629#post-id-49629Answer by John Cremona for <p>Well, I have for example the following SageMathCell-code:</p>
<pre><code>sage: E = EllipticCurve(QQ, [0,63,-2205,-12348,0])
sage: E
sage: for P in E.integral_points():
....: Q = -P
....: print( "P = %8s and -P = %8s" % (P.xy(), Q.xy()) )
</code></pre>
<p>This code computes the integral points of the Elliptic Curve that is defined by:</p>
<p>$$[0,63,-2205,-12348,0]\space\space\space\to\space\space\space y^2 - 2205y = x^3 + 63x^2 - 12348x\tag1$$</p>
<blockquote>
<p>Are these results I get, proven to be the only ones out there? Or can there be more solutions that SageMathCell did not find?</p>
</blockquote>
<hr>
<p>Bytheway, the code gives the following output:</p>
<pre><code>P = (-174, 1161) and -P = (-174, 1044)
P = (-147, 2205) and -P = (-147, 0)
P = (-98, 2548) and -P = (-98, -343)
P = (-68, 2528) and -P = (-68, -323)
P = (-54, 2484) and -P = (-54, -279)
P = (0, 2205) and -P = (0, 0)
P = (57, 2052) and -P = (57, 153)
P = (84, 2205) and -P = (84, 0)
P = (147, 3087) and -P = (147, -882)
P = (231, 4851) and -P = (231, -2646)
P = (309, 6840) and -P = (309, -4635)
P = (375, 8730) and -P = (375, -6525)
P = (378, 8820) and -P = (378, -6615)
P = (711, 20691) and -P = (711, -18486)
P = (1176, 42336) and -P = (1176, -40131)
P = (2107, 99127) and -P = (2107, -96922)
P = (2886, 157716) and -P = (2886, -155511)
P = (5412, 401472) and -P = (5412, -399267)
P = (5572, 419293) and -P = (5572, -417088)
P = (37275, 7203735) and -P = (37275, -7201530)
P = (26162409, 133818797385) and -P = (26162409, -133818795180)
</code></pre>
https://ask.sagemath.org/question/49605/are-results-found-of-an-elliptic-curve-by-sagemathcell-proven/?answer=55635#post-id-55635Actually no -- as just explained by me also on Math.Stackexchange!Tue, 09 Feb 2021 12:30:03 +0100https://ask.sagemath.org/question/49605/are-results-found-of-an-elliptic-curve-by-sagemathcell-proven/?answer=55635#post-id-55635