interesting discovery about elliptic curve [0,0,0, 393129,0]

[American Citizen <website.reader3@gmail.com>]

To all:

While working with Heron triangles with 2 square sides (the 3rd is not 
necessarily a square) I came across an interesting elliptic curve

E = [0, 0, 0, 393129, 0]

I checked the first 618 rational points on this curve (sorting by 
height) and every x-coordinate is a square.

Can anyone explain why this rank 3 curve has these first 618 
x-coordinates as rational (or integer) squares? I would assume that all 
x-coordinates are squares for this particular elliptic curve, but that 
would have to be an inductive proof.

Randall

P.S. this also provides a list of 318 rational squares from 627^2 + 
x[i]^2 which I also find interesting

P.P.S: 627 = 3 * 11 * 19 and 627^2 = 393129