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We theoretically investigate the properties of magnetically levitated superconducting elements confined in anti-Helmholtz traps, for application in magnetomechanical experiments. We study both the translational modes and the librational mode. The librational mode gives an additional degree of freedom that levitated spheres do not have access to. We compare levitated particles of different shapes: ellipsoids (with analytical and numerical treatment), cylinders, and rectangular cuboids (numerical treatment). We find that the stable orientations of the particles depend on their aspect ratios.