ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 19 Jan 2020 08:33:57 -0600Are results found of an Elliptic Curve by SageMathCell proven?http://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)
Jan1997Sun, 19 Jan 2020 08:33:57 -0600http://ask.sagemath.org/question/49605/Mistake in SageMathCell code, finding integral points on elliptic curveshttp://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?Jan123Tue, 03 Dec 2019 10:13:08 -0600http://ask.sagemath.org/question/48933/