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 the 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 four (or five) images. Higher order imaging is brought about by deviations of the isophotes from pure ellipses, such as pointedness caused by embedded discs 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 ε and the fourth-order Fourier coefficients ( a ₄ and b ₄) used by observers to parametrize the isophote shapes. The six or eight 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 per cent of all multiply imaged quasars may be sextuplet systems or higher. Forthcoming satellites such as the Global Astrometric Interferometer for Astrophysics ( GAIA ) will provide data sets of ～4000 multiply imaged systems, and so ～40 will show sextuple imaging or higher.