This paper concerns elliptic
curves defined over complex quadratic fields and having good reduction at all primes.
Those fields are characterized which support such curves having a 2-division
point defined over the field. The number of isomorphism classes, over the
ground field, of these curves is also determined. For curves without a 2-divison
point defined over the field, the possible Galois groups of the 2-division field
over the rationals are determined. Using class field theory, it is shown that
certain complex quadratic fields support no elliptic curves with good reduction
everywhere.
