In this post, we considered the elliptic curve,
$$x\big(x+(m-1)^2\big) \big(x+(m+1)^2\big) = y^2\tag1$$
Online Magma was able to solve the case $m=2^7$ but timed-out on $m=5^7$,
$$x\big(x+78124^2\big) \big(x+78124^2\big) = y^2$$
However, Allan MacLeod in this answer was able to find a rational point on it.
This paper "Finding rational points on elliptic curves using $6$-descent and $12$-descent" (2007) by Fisher includes the elliptic curve,
$$x\big(x+(n-2)^2\big) \big(x+(n+2)^2\big) = y^2\tag2$$
This has at least five torsion points,
$$u = 0,\quad -(n-2)^2,\quad -(n+2)^2,\quad \pm(n-2)(n+2)$$
I tried the case $n=809$ of $(2)$,
$$x\big(x+807^2\big) \big(x+811^2\big) = y^2$$
on online Magma using the commands,
Q<x> := PolynomialRing(Rationals()); n:=809; E00 := EllipticCurve(x *(x + (n + 2)^2)*(x + (n - 2)^2)); Q00 := Generators(E00); Q00;
and just got [ (-651249 : 0 : 1), (654477 : 1058943786 : 1) ] which are two of the torsion points.
However, according to the paper, $n=809$ does have a non-torsion rational point of huge height.
Q: What are the assumptions Magma uses to find the non-torsion point of $m=2^7=128$, but not $n=809\,$? Why didn't it just time-out like the case $m=5^7=79125$, or is it a bug?