Murmuring elliptic curves
The average p-th Dirichlet coefficients of elliptic curves (p prime) are found to oscillate dramatically in a fixed conductor range with a given rank.
Murmurations of Elliptic Curves
We investigate the average value of the pth Dirichlet coefficients of elliptic curves for a prime p in a fixed conductor range with given rank. Plotting this average yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we perform various data-scientific experiments with the goal of classifying elliptic curves according to their ranks.