Rank Distribution of 130 Million Elliptic Curves

William Stein

October 17, 2004

This is a graph of the distribution of the ranks of the more than 130 million elliptic curves in the Stein-Watkin database. These are all curves with |Δ| ≤ 10^{12} and c_{4} ≤ 1.44 · 10^{12} and N < 10^{8}. The top graph is rank 1, the second graph from the top is rank 0, the third graph is rank 2, and the fourth is rank ≥ 3. Notice that there are a surprisingly large number of elliptic curves of rank 2 in this data. (Acknowledgement: Baur Bektemirov)

0.00 0.51 0.25 100000  100000000 50050000