  It is shown that $c=-29/16$ is the unique rational number of smallest
denominator, and the unique rational number of smallest numerator, for which
the map $f_c(x) = x^2+c$ has a rational periodic point of period $3$. Several
arithmetic conditions on the set of all such rational numbers $c$ and the
rational orbits of $f_c(x)$ are proved. A graph on the numerators of the
rational $3$-periodic points of maps $f_c$ is considered which reflects
connections between solutions of norm equations from the cubic field of
discriminant $-23$.
