We study numerically frictionless ellipse packings versus the aspect ratio alpha, and find that the jamming transition is fundamentally different from that for spherical particles. The normal mode spectra possess two gaps and three distinct branches over a range of alpha. The energy from deformations along modes in the lowest-energy branch increases quartically, not quadratically. The quartic modes cause novel power-law scaling of the static shear modulus and their number matches the deviation from isostaticity. These results point to a new critical point at alpha>1 that controls jamming of aspherical particles.