# 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)

edit retag close merge delete

Sort by ยป oldest newest most voted
more

Actually no -- as just explained by me also on Math.Stackexchange!

more