TITLE: Distribution of Elliptic Curve Generators over Q
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: I gathered some statistics using the Cremona tables. There are 565803 elliptic curves with rank 1, trivial torsion group, positive discriminant (which means that the group of real points has two connected components) and conductor $\leq 4 \times 10^5$ (the current limit of the tables). Among them 239106 curves have the generator of $E(\mathbf{Q})$ located on the identity component.
Here is the histogram representing the location of the generator $P$ on the identity component for these curves. More precisely, this is the distribution of the (unique) real number $z_P \in (0,1/2)$ such that the Mordell-Weil group $E(\mathbf{Q})$ is generated by the class of $z_P \Omega_E^+$, where $\Omega_E^+$ is the real period of $E$. We can note the behaviour near $z=0,1/4,1/3,1/2$ which might be explained by the fact that the torsion points somehow repel the point of infinite order.