2 problems

Let $E_m$ denote the cubic twist considered in the paper, and let $m$ be prime. For primes $m\equiv 4,7\pmod 9$, Elkies proved that the rank of $E_m$ is exactly $1$. Rank conjectur…

Let $f(k,X)$ denote the number of cube-free integers $d\leq X$ satisfying $(*)$ and such that $|\text{Ш}(E_d)|=k^2$. Here $E_d$ is the cubic twist of $X_0(27)$ associated with $d$.…