The one-level density measures the statistical distribution of zeros across a family of $L$-functions near the central point. In this talk, I will present several results concerning the one-level density for families of $L$-functions associated to CM elliptic curves of the form $y^2=x^3-dx$. The talk will moreover discuss applications to bounding the average rank of such curves (joint work with Chantal David, Lucile Devin, and Alessandro Fazzari).