Let $X$ be a smooth genus one curve over $k$. I don't it elliptic curve because it will have no rational points.

By index of $X$ be mean the smallest degree of a closed point on $X$; equivalently by Riemann-Roch that's the same as the smallest positive degree of a divisor, or the greatest common divisor of all degrees of closed points.

Of course, the index of a curve equals one if the field is algebraically closed, so we consider non-closed fields here. For example, a smooth cubic curve in $\mathbb{P^2}$ can have index one or three.

Here is the question: can one characterize fields $k$ which admit genus one curves of index $d$?

I am especially interested in the $d = 5$ case; then the model of such curve over algebraic closure is a linear section of the Grassmannian $\mathrm{Gr}(2,5)$. For which fields do we have genus one curves of index $5$? Is there a way to figure this out without writing explicit equations?