general question relating to elliptic curves and their rational points and creating a pool of rational points of count N

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

Hello:

I am currently working with rational Diophantine sextuples, example: 
[5/4, 5/36, 32/9, 189/4, 665/1521, 3213/676] where the product of any 
pair + 1 is a rational square.

If we select 3 of the ratios, making a triple, say [5/4, 5/36, 32/9], we 
can create an elliptic curve associated with the triple [a,b,c]

E:  [0,(a*b+a*c+b*c),0,(a*b*c)*(a+b+c),(a*b*c)^2]

I have been working with 758 sextuples, and found elliptic curves of 
rank =1 through rank = 10 for Diophantine triples derived from the 
sextuples. (20 triples per sextuple).

My question to the group is this, given the Mordell-Weil basis (which 
seems quite easy to find for E), how can I determine a cut-off value of 
elliptic curve point heights for creating a pool of 10K points that are 
on the curve?

I am using a gp-pari command called "ellpool(E,p,h)" where E is the 
curve, p is the Mordell-Weil basis and h is the maximum height of the 
points stored in the pool (or lattice or vector) of rational points. By 
carefully adding points from the MW basis, we can slowly build a pool of 
points on the curve <= h.

I need to determine h such that the count of the pool is around 10K or 
so. Determining h seems somewhat hard to figure out, although I was 
using the regulator^(1/rank) * F to set the height h (where F is around 
5 to 12 or so) and that generally works, but not always. Of  course the 
torsion group for the curve E plays a huge part too in finding the pool 
of points, as the torsion points automatically increase the rational 
points found.

Has someone done this work before? Can any papers be cited?

Right now I am setting this ad-hoc, but wish that a better scheme could 
be found.

Example:

% find([5/4, 5/36, 32/9])

rank = 2,

curve = [0, 6625/1296, 0, 2225/729, 2500/6561]

MW_basis = [[124/9, 4879/81], [1600/9, 194750/81]]

triple: [5/4, 5/36, 32/9]

And playing around with h in the ellpool(e,p,h) command eventually finds 
h = 942 gives 9979 points, 943 = 9987 points, and 944 = 10027 points so 
943.5 gives 10003 points which is close enough.

The regulator of this curve is 12.62021889005498 and the rank = 2, so 
taking reg^(1/rank) gives F = 265.588 above in order to come out to 943.5

Any ideas on a better way to find h? The curve is a Z2xZ2 curve.

Randall