  We prove an optimal systolic inequality for nonpositively curved Dyck's
surfaces. The extremal surface is flat with eight conical singularities, six of
angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta)
in Q(sqrt{19}). Relying on some delicate capacity estimates, we also show that
the extremal surface is not conformally equivalent to the hyperbolic surface
with maximal systole, yielding a first example of systolic extremality with
this behavior.
