Bill Allombert on Sat, 02 Dec 2023 12:15:32 +0100 |
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Re: tossing out an idea to use heights of algebraic points on elliptic curves to help sort isogenies for point searches |
On Fri, Dec 01, 2023 at 05:26:34PM -0800, American Citizen wrote: > Bill > > Can you please explain some more on how you came up with P1? You gave me P1 = [-2641, (2641+sqrt(373230717))/2] by definition, sqrt(373230717) is a root of x^2-373230717 (which is irreducible over Q). Consider the map Q[x] -> Q(sqrt(373230717)) P(x) -> P(sqrt(373230717)) (P is a polynomial in Q[x]) its kernel is the principal ideal generated by x^2-373230717 so this define an isomorphism from Q[x]/(x^2-373230717) to Q(sqrt(373230717)) which send x mod (x^2-373230717) to sqrt(373230717) Mod(x, x^2-373230717) is just the GP syntax for "x mod (x^2-373230717)" In gp I use the variable 'a because it is safer to keep x and y for true polynomials, due to variable priority. So P1 = [-2641, (2641+sqrt(373230717)/2] algebraic form is P1 = [-2641, (2641+Mod(x, x^2-373230717))/2] Of course this is an easy case. When there are several algebraic numbers, it is more involved. Note that, since (2641+sqrt(373230717))/2 minimal polynomial is x^2-2641*x-91563959, you could also write P1 = [-2641,Mod(b,b^2-2641*b-91563959)] However, again if you start from scratch, you will directly find the equation satified by y since your curve is given by y^2+x*y=x^3-114223080*x -283150929600 and x=-2641, you get y^2-2641*y=91563959 directly. Cheers, Bill