American Citizen on Sun, 03 Dec 2023 23:59:58 +0100

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Re: correction to previous post

In regard to heights of points across isogenous curves

On 12/2/23 11:08, Bill Allombert wrote:
? ellisomat(E)[2][,1]
%5 = [1,2,4,4,2,2,4,4]

As stated this should be the same in Q as in a number field.

I think that one of my pari programs was checking for saturation by a prime and when it used 2, it was able to find a smaller 1/2 height point on some of the isogenous curves, hence my relative vector did not match, but was altered from the normal result in %5.

This begs the question, given all the curves in the ellisomat() command can we ferret out those who have a 2-degree function such that we can reliably locate where the situation occurs and the relative height vector is altered?

My hunch is yes.

I have a follow up question, suppose we select a point on an elliptic curve in a number field, and in my case I picked a rational x-coordinate, and deriving y as a quadratic surd,  which gave me the number field to use. Are we able to find another independent point in the same number field also on the curve and I am NOT talking about doubling the given point and adding or subtracting multiples to give ht, 4ht, 9ht, 16ht etc, but a truly independent point in the same NF. I tried a naive approach and quickly realized the chances of hitting this was incredibly low.

Or is this as hard as trying to find a rational point on a curve in Q?