Are there sextuplet and octuplet image systems?

We study gravitational lensing by the family of scale-free galaxies with flat rotation curves. The models are defined by a shape function, which prescribes the radius of the isophote as a function of position angle from the major axis. The critical curves are analytic, while the caustic network is reducible to a simple quadrature. The cusps are always located at the turning points of the shape function. We show that the models with exactly elliptic isophotes never admit butterfly or swallowtail cusps and so have at most 4 (or 5) images. Higher order imaging is brought about by deviations of the isophotes from pure ellipses -- such as pointedness caused by embedded disks or boxiness caused by recent merging. The criteria for the onset of sextuple and octuple imaging can be calculated analytically in terms of the ellipticity and the fourth-order Fourier coefficients (a_4 and b_4) used by observers to parametrise the isophote shapes. The 6 or 8 images are arranged roughly in a circle, which appears as an incomplete Einstein ring if inadequately resolved. Using data on the shapes of elliptical galaxies and merger remnants, we estimate that 1% of all multiply imaged quasars may be sextuplet systems or higher. Forthcoming satellites like the Global Astrometric Interferometer for Astrophysics (GAIA) will provide datasets of roughly 4000 multiply imaged systems, and so about 40 will show sextuple imaging or higher.