TITLE: Are results found of an Elliptic Curve by SageMathCell proven (does there exists no more solutions)?
QUESTION [7 upvotes]: 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)

REPLY [2 votes]: The answer given would be correct in an ideal world.  Unfortunately there are (1) bugs in the implementation, and also, sadly (2) bugs in the description of the algorithm in Henri COhen's book, because he takes some formulas from Smart's book, which Smart gets from the monograph of Sinou David, but he (Smart) does so incorrectly.  (It's a question of which order you put the generating periods: is is w1/w2 or w2/w1 which is in the fundamental region.  HC knows about this (I informed him) but I don't think that there is yet a correction on his web page (which has 20 pages of errata for the two volumes).
I am preparing a better implementation -- have been doing so for a few years now -- but currently you can not assume that the list of integral points output by Sage for elliptic curves over Q is complete, even if you have the full Mordell-Weil group.  When I compute integral pints for the curve in the LMFDB I do so using both Sage and Magma and take the union.  It is some time since I saw any integral points found by Sage not found by Magma, but in fairness I should say that while I was testing the first implementation of integral points in Sage (in 2008) that happened a lot, and as a result the Magma code was completely rewritten by Steve Donnelly, using several tricks he thought up but which have never been published.