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

[John Cremona <john.cremona@gmail.com>]

This is not an explanation, but the condition that x is a square is equivalent to the point (x,y) being in the image of the 2-isogeny from [0,0,0,-4n^2,0].    Calling the curves E and E' and looking at how descent by 2-isogeny works (e.g. in my book, other books are available!), roughly speaking the rank of E comes partly  from E/phi'(E') and part from the image inder phi of E'/phi(E).  (Here phi:E --> E') and phi' is the dual.)   Saying that "all" the points have square x-coordinates is therefore saying that E/phi'(E') is trivial and all the points are coming from phi'(E').

But why that should be, apart from chance, I don't know.

John

On Thu, 24 Oct 2024 at 02:16, American Citizen <website.reader3@gmail.com> wrote:

This might not be so unusual after all. I set up other [0,0,0,n^2,0]
curves and found that most had the x-coordinates a rational square.

Curious as to why?

Randall

On 10/23/24 17:02, American Citizen wrote:
> 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
>
>
>