The fundamental distinction that creates murmuration patterns
The rank of an elliptic curve is the number of independent generators of its group of rational points. Rank 0 means only finitely many (torsion) points; rank 1+ means infinitely many.
When researchers averaged a_p values separately for rank 0 and rank 1 curves, they discovered distinct oscillating patterns—the murmurations. This separation was key to the discovery!
This famous Millennium Prize problem predicts that rank equals the order of vanishing of the L-function at s=1. Murmurations may provide new insights.