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We study the rationality of the intersection points of certain lines andsmooth plane quartics C defined over F_q. For q \geq 127, we prove theexistence of a line such that the intersection points with C are all rational.Using another approach, we further prove the existence of a tangent line withthe same property as soon as the characteristic of F_q is different from 2 andq \geq 66^2+1. Finally, we study the probability of the existence of a rationalflex on C and exhibit a curious behavior when the characteristic of F_q isequal to 3.

On rationality of the intersection points of a line with a plane quartic