| 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
? 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?
Randall