Elliptic Curves with Everywhere Good Reduction

We consider the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic curves of everywhere good reduction and rational $j$-invariant. Using this, we determine the density of such real and imaginary quadratic fields. If $R(X)$ (respectively $I(X)$) denotes the number of real (respectively imaginary) quadratic fields $K=\mathbb{Q}[\sqrt{m}]$ such that $|\Delta_K| < X$ and for which there exists an elliptic curve $E/K$ with rational $j$-invariant that has everywhere good reduction, then $R(X) \gg \frac{X}{\sqrt{\log(X)}}$. To obtain these estimates we explicitly construct quadratic fields over which we can construct elliptic curves with everywhere good reduction. The estimates then follow from elementary multiplicative number theory. In addition, we obtain infinite families of real and imaginary quadratic fields such that there are no elliptic curves with everywhere good reduction over these fields.