We study Nivat's conjecture on algebraic subshifts and prove that in some of them every low complexity configuration is periodic. This is the case in the Ledrappier subshift (the 3-dot system) and, more generally, in all two-dimensional algebraic subshifts over $\mathbb{F}_p$ defined by a polynomial without line polynomial factors in more than one direction. We also find an algebraic subshift that is defined by a product of two line polynomials that has this property (the 4-dot system) and another one that does not.