question on heightmatrix for algebraic points

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

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)

I did not divide the heights by the degree of the number field which is 
2 for both cases. Should I have done this?

We will use the addition law of points on an elliptic curve (2) to add 
these two algebraic points on E.

The two points are [x1,y1] and [x2,y2] and we're finding the third point 
[x3,y3].

(2)  x3 = m^2 - x1 - x2, y3 = m*(x1-x3) - y1   [Addition Law]

where m is the slope of the line through [x1,y1] and [x2,y2],i.e m = 
(y2-y1)/(x2-x1)

Using p,q to set the values of x1,y1 and x2,y2 respectively, we find:

  x3 = polroots(x^2 - 612*x + 9604)[1] = 
16.1172650881049797735668656860516831247...
  y3 = polroots(x^4 - 211649184*x^2 + 1227203115264)[2] = 
-76.1475394339292918879099932432709243972125533391...
  r  = [x3, y3]

Checking these 3 points using ellisoncurve(e,pt) shows that they are 
indeed on the curve.

Does GP-Pari have an easy way to add algebraic points on an elliptic 
curve?  (2) is simple enough, but we get decimals as output, and have to 
use algdep(results,degree) to recover the fields.

And after the 3rd point is found, can we find its height? x3 and y3 are 
each in different fields.

I am curious as if we can define an ellheightmatrix() function for two 
or more algebraic points on an elliptic curve to experiment with the 
height-(partial)regulator matrix determinant in hopes of pinning down 
the true regulator size (without the SHA component) for all curves which 
are in the isogenous group of the given curve.

This is to avoid two things, non-trivial SHA giving an overly high value 
regulator size versus the true size for the elliptic curve, and most 
importantly, avoiding have to compute the L-series for the curve, when 
the conductor is very large. This L-series computation for me right now 
is the bottle neck.