**Conjecture:** If I have an elliptic curve with j-invariant 0 of the form $y^2 = x^3 + b$ over some prime-order field $\mathbb{F}_p$ (where $p$ is not 2 or 3), and the group of rational curvepoints has prime order $q$ (which is not 2 or 3), then there is some curve of the form $y^2 = x^3 + b'$ over $\mathbb{F}_q$ with prime order $p$.

I'm not sure about the characteristic 2-or-3 case, I guess I could check that exhaustively, but it's less interesting to me and I haven't taken the time to do it yet. For larger primes I've found a few dozen cases with small primes, a couple with large primes, and no counterexamples.

I have 3 questions:

- Is this conjecture true?
- What does it mean? This seems to give an equivalence relation between ground fields of different characteristic, which seems really weird to me... what am I looking at here? (Sorry this is so vague.)
- What is the natural way to generalize this to curves whose groups of rational points has composite order, and/or fields with non-prime order?

Exper. Math.20(3)(2011), 329-357. $\endgroup$ – Joe Silverman Sep 15 '16 at 21:49