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Let $f_t(z)=z^2+t$. For any $z\in\mathbb{Q}$, let $S_z$ be the collection of$t\in\mathbb{Q}$ such that $z$ is preperiodic for $f_t$. In this article,assuming a well-known conjecture of Flynn, Poonen, and Schaefer, we prove auniform result regarding the size of $S_z$ over $z\in\mathbb{Q}$. In order toprove it, we need to determine the set of rational points on a specificnon-hyperelliptic curve $C$ of genus $4$ defined over $\mathbb{Q}$. We useChabauty's method, which requires us to determine the Mordell-Weil rank of theJacobian $J$ of $C$. We give two proofs that the rank is $1$: an analyticproof, which is conditional on the BSD rank conjecture for $J$ and somestandard conjectures on L-series, and an algebraic proof, which isunconditional, but relies on the computation of the class groups of two numberfields of degree $12$ and degree $24$, respectively. We finally combine theinformation obtained from both proofs to provide a numerical verification ofthe strong BSD conjecture for $J$.

Dynamics of quadratic polynomials and rational points on a curve of genus $4$