# finding rational points on an elliptic curve Consider the following elliptic curve

$y^2=(x+1540)(x-508)(x-65024)$.

It is trivial that the points $P_1(-1540,0)$, $P_2(508,0)$ and $P_3(65024,0)$ lie on this curve. It is also quite easy to find four other integer points $P_4(-508, 262128)$, $P_5(-508, -262128)$, $P_6(130556, 33552384)$ and $P_7(130556, -33552384)$.

I want to find some other rational points on this curve. If one uses the usual group law we obtain that for every $1\le i \le j \le 7$ we have $P_i+P_j=P_k$ for some $1\le k\le 7$. Thus we fail to obtain any new points.

Any suggestion would be appreciated.

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sage: x,y = polygens(QQ,'x,y')
sage: E = EllipticCurve(-y**2+(x+1540)*(x-508)*(x-65024))
sage: E.integral_points()
[(-1540 : 0 : 1),
(-508 : 262128 : 1),
(508 : 0 : 1),
(65024 : 0 : 1),
(130556 : 33552384 : 1)]
sage: P = E.gens(); P
(100132172429824908929/1508738252550400 : 143687064081412107244001809983/58603135399923860992000 : 1)
sage: P*2
(23926617756252939195845889933899553715162280807826884830539689900801/28781599829126697172901832423192736087492173270011136379110400 : -112435953838617002324224529091205045118375173739751380123760462569961416430316858965047000875038947199/154408923747010694464616627108267136335941190834695015240452304070119936139205075365384192000 : 1)

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