  This paper investigates which integers can appear as 2-Selmer ranks within
the quadratic twist family of an elliptic curve E defined over a number field K
with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined
over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative
integer appears infinitely often as the 2-Selmer rank of a quadratic twist of
E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we
prove the same result for 2-Selmer ranks greater than or equal to r_2, the
number of complex places of K. We also obtain results about the minimum number
of twists of E with rank 0, and subject to standard conjectures, the number of
twists with rank 1, provided E does not have a cyclic 4-isogeny defined over K.
