We investigate the relation between the dynamical features of a supercooled liquid and those of its potential energy landscape, focusing on a model liquid with density anomalies. We consider, at fixed temperature, pairs of state points with different density but the same diffusion constant, and find that surprisingly they have identical dynamical features at all length and time scales. This is shown by the collapse of their mean square displacements and of their self--intermediate scattering functions at different wavevectors. We then investigate how the features of the energy landscape change with density, and establish that state points with equal diffusion constant have different landscapes. In particular, we find a correlation between the fraction of instantaneous normal modes connecting different energy minima and the diffusion constant, but unlike in other systems these two quantities are not in one--to--one correspondence with each other, showing that additional landscape features must be relevant in determining the diffusion constant.