Re: question on heightmatrix for algebraic points

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Tue, Jun 24, 2025 at 12:30:16PM -0700, American Citizen wrote:
> Hello, all of you, I appreciate your patience, especially Bill's.
> 
> I have some questions about heights of algebraic points on an elliptic curve
> and the height matrix associated with them.
> 
> Let an elliptic curve E be expressed in Weierstrass format.
> 
> (1)  E = [0, 0, 0, 100, 0]
> 
> One Mordell-Weil basis for E is the point [5,25].
> 
> We define 2 algebraic points on E
> 
>    p = [1, sqrt(101)] with height ~= 4.69969906449875... using
> K1=nfinit(x^2-101) and ellinit(e,K1)
>    q = [2, sqrt(208)] with height ~= 2.37364501798303... using
> K2=nfinit(x^2-208) and ellinit(e,K2)

You need to construct the compositum of your fields, so that you
have a common field for both.
 Do this:

 ? [P,a,b]=polcompositum(x^2-101,x^2-208,1)[1];
 ? a^2
 %5 = Mod(101,x^4-618*x^2+11449)
 ? b^2
 %6 = Mod(208,x^4-618*x^2+11449)
 ? K=nfinit(P);
 ? E=ellinit([0, 0, 0, 100, 0],K);
 ? p=[1,a]; q=[2,b];
 ? ellheightmatrix(E,[p,q])
 %7 = [9.3993981289975127877576183833519503910,4.701977403289150032E-38;
       4.701977403289150032E-38,4.7472900359660764922371320463865657704]
 ? elladd(E,p,q)
 %8 = [Mod(x^2 - 3, x^4 - 618*x^2 + 11449), Mod(-213/214*x^3 + 345/214*x, x^4 - 618*x^2 + 11449)]

Cheers,
Bill