Considering elliptic curves over Q with positive discriminant and rank>0, are there any results or proposed heuristics regarding the fraction of generators that are located on the identity vs non-identity component? How does this vary with rank?
The family of curves I've been examining (associated with $a/(b+c) + b/(a+c) + c/(a+b) = N$) have the following distributions for N even:
Rank 1: 313 have generator on identity, 601 on non-identity Rank 2: 107 have 2 on identity, 153 have 1, 16 have 0 Rank 3: 3 have 3 on identity, 16 have 2, 3 have 1, 0 have 0
For N odd, all of the generators appear on the identity component.