On Elliptic Curves with Everywhere Good Reduction

We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quad-ratic fields m for. These results of computations give best-possible data including structures of Mor-dell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method