More interesting is the question of which fields admit what sorts of elliptic curves having everywhere good reduction. For example, how about quadratic fields? Here are a couple of results.

**Theorem** (Rohrlich [1]) Let $K$ be an imaginary quadratic field and $j$ the invariant of some fixed isomorphism class of elliptic curves with complex multiplication by the ring of integers of $K$. Put $F=\mathbb Q(j)$ and $H=K(j)$. Then there is an elliptic curve with invariant $j$, defined over $F$, which has good reduction at every place of $F$ if and only if the discriminant $−d$ of $K$ is divisible by at least two primes congruent to 3 modulo 4.

**Theorem** (Setzer [2]) There are no elliptic curves having good reduction everywhere over $\mathbb Q(\sqrt{-m})$ for $m=1,2,3,5,6,7,10,13,14,15,17,21,22,\ldots$ .

- Rohrlich, David E., Elliptic curves with good reduction everywhere.
*J. London Math. Soc.* (2) **25** (1982), no. 2, 216–222.
- Setzer, Bennett, Elliptic curves over complex quadratic fields.
*Pacific J. Math.* **74** (1978), no. 1, 235–250.