Databases of elliptic curves ordered by height and distributions of   Selmer groups and ranks
  Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava-Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$ ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$ ordered by height in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon observed in these databases is that the average rank eventually decreases as height increases. 
Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William   Stein, James Weigandt
math.NT math.AG