American Citizen on Sat, 21 Jun 2025 06:07:14 +0200


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question on the height setting for qfminim on rank 2 elliptic curves


Suppose we have the following GP-Pari script:

---- start script below this line

default(realprecision,16);
read("elltorsion.gp");

ht=80;

e=[0,0,0,5015293065225,0]
E=ellinit(e);
p=[[5184,161243352],[2992900/289,1119687287050/4913]]
P=Set(p);
[13.52420283807603,13.52757028189808]
h=ellheightmatrix(E,p);
M=qfminim(h,ht,,2)[3];
sz=matsize(M);
for(i=1,sz[2],\
  Q=[0];\
  for(j=1,sz[1],\
    Q=elladd(E,Q,ellmul(E,p[j],M[j,i]));\
  );\
  R=ellneg(E,Q);\
  P=setunion(P,[Q]); P=setunion(P,[R]);\
);
print("ht:",ht," #P:",#P)
\\P

ht=60;

print();
e=[0,0,0,2394144900,0]
E=ellinit(e);
p=[[11650,5428900],[1978765727665766400/51862365611809,4526613785427224046216710400/373489574658819939377]]
P=Set(p);
[2.641519528683110,39.38323971555240]
h=ellheightmatrix(E,p);
M=qfminim(h,ht,,2)[3];
sz=matsize(M);
for(i=1,sz[2],\
  Q=[0];\
  for(j=1,sz[1],\
    Q=elladd(E,Q,ellmul(E,p[j],M[j,i]));\
  );\
  R=ellneg(E,Q);\
  P=setunion(P,[Q]); P=setunion(P,[R]);\
);
print("ht:",ht," #P:",#P)
\\P

---------- end script

The output is

%4 = [0, 0, 0, 5015293065225, 0]
%6 = [[5184, 161243352], [2992900/289, 1119687287050/4913]]
%8 = [13.52420283807603, 13.52757028189808]
ht:80 #P:20

%16 = [0, 0, 0, 2394144900, 0]
%18 = [[11650, 5428900], [1978765727665766400/51862365611809, 4526613785427224046216710400/373489574658819939377]]
%20 = [2.641519528683110, 39.38323971555240]
ht:60 #P:20
Question:

I had to play around with the script file trying different values for ht in the qfminim command until the number of points were equal.

How can one adjust the height value in qfminim for rank 2 elliptic curves (when the Mordell-Weil basis is known)  such that the same number of points can be found irregardless of the ratio of the 2 heights of the points on the curve?

Right now, whenever I get two points whose ratio is divergent from about 0.9 or so, qfminim is returning much different values. The problem with not enough points occurs when the heights are almost equal, which is why I choose these 2 curves to give an example.

Thanks for any input on balancing the output of qfminim

Randall