One interesting feature of the heuristics of Garton, Park, Poonen, Wood, Voight, discussed here previously: they predict there are fewer elliptic curves of rank 3 than there are of rank 2. Is this what we believe? On one hand, you might believe that having three independent points should be “harder” than having only two. But there’s the parity issue. All right-thinking people believe that there are equally many rank 0 and rank 1 elliptic curves, because 100% of curves with even parity have rank 0, and 100% of curves with odd parity have rank 1. If a curve has even parity, all that has to happen to force it to have rank 2 is to have a non-torsion point. And if a curve has odd parity, all that has to happen to force it to have rank 3 is to have one more non-torsion point you don’t know about it. So in that sense, it seems “equally hard” to have rank 2 or rank 3, given that parity should be even half the time and odd half the time.
So my intuition about this question is very weak. What’s yours? Should rank 3 be less common than rank 2? The same? More common?