On the cyclotomic Iwasawa invariants of elliptic curves of rank one

Fix an elliptic curve $E$ over $\mathbb{Q}$ of rank $1$. In this paper, we develop an explicit numerical criterion, comparable to Gold’s criterion, that determines whether the Iwasawa invariants of the elliptic curve at a good (ordinary or supersingular) prime attain their smallest possible value, i.e. whether ${\mu }_p^{\pm }(E)+{\lambda }_p^{\pm }(E)=1$ in the supersingular case or ${\mu }_p(E)+{\lambda }_p(E)=1$ in the ordinary case.