  Let K be a field of characteristic different from 2 and C an elliptic curve
over K given by a Weierstrass equation. To divide an element of the group C by
2, one must solve a certain quartic equation. We characterise the quartics
arising from this procedure and find how far the quartic determines the curve
and the point. We find the quartics coming from 2-division of 2- and 3-torsion
points, and generalise this correspondence to singular plane cubics. We use
these results to study the question of which degree 4 maps of curves can be
realised as duplication of a multisection on an elliptic surface.
