This paper concerns
elliptic curves defined over quadratic fields and having good reduction at all
primes. All those real fields admitting such curves having a 2-division point
defined over the field and a global minimal model are characterized. The
number of isomorphism classes, over the ground field, of these curves is also
determined. If the number of divisor classes of the field is odd, all the mentioned
curves without a global minimal model are classified and counted as well. It is
shown that there are only eight elliptic curves defined over a quadratic field
having good reduction everywhere and four 2-division points defined over the
field.
