Reduction of elliptic curves over certain real quadratic number fields

The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic field has a -rational point under certain hypotheses (primarily on class numbers of related fields). It extends the earlier case in which no ramification at is allowed. Small fields satisfying the hypotheses are then found, and in four cases the non-existence of such elliptic curves can be shown, while in three others all such curves have been classified.