  A point of a Veech surface is periodic if it has a finite orbit under the
surface's affine automorphism group. We show that the periodic points of Prym
eigenforms in the minimal strata of translation surfaces in genera 2, 3 and 4
are the fixed points of the Prym involution. This answers a question of
Apisa--Wright and gives a geometric proof of M\"oller's classification of
periodic points of Veech surfaces in the minimal stratum in genus 2.
