I know the question is old, but is possible to give an exact characterization to $d$, at least if it's a prime number:
It is known that considering a prime $p \equiv 1\mod 4$, there is always a solution to $x^2 - py^2 = -1$ in integers, the proof is from Mordell "Diophantine Equations" pages 55-56 (thanks to @Will Jagy for pointing that out).
Furthermore Dirichlet proved that if the prime $p \equiv 1 \mod 4$ and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, the equation $x^2 - 2py^2 = -1$ has still integer solutions (reference: https://www.forgottenbooks.com/en/download/ThePellEquation_10024828.pdf pag. 80)
So you can say that if $p$ is a prime and $p \equiv 5 \mod 8$, or if $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, both NPEₚ NPE₂ₚ have integer solutions
