If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:

(1.) $\operatorname{rank}E(K(l))=\operatorname{rank}E(\mathbb{Q}(l))$

(2.) $l$ is inert in $K/\mathbb{Q}$

where $\operatorname{rank}$ denotes the algebraic rank and $K(l)$ and $\mathbb{Q}(l)$ denote the ray class fields of conductor $l.$ Assume for simplicity that $K$ has class number $1$.

If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:

(1.) $\operatorname{rank}E(K(l))=\operatorname{rank}E(K)$

(2.) $l$ is inert in $K/\mathbb{Q}$

where $\operatorname{rank}$ denotes the algebraic rank and $K(l)$ is the ray class fields of conductor $l.$ Assume for simplicity that $K$ has class number $1$.