Re: question on heightmatrix for algebraic points

[American Citizen <website.reader3@gmail.com>]

Thanks Bill, this is exactly what I need.

On 6/24/25 13:06, Bill Allombert wrote:
> 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