Let
f
t
(
z
)
=
z
2
+
t
f_t(z)=z^2+t
. For any
z
∈
Q
z\in \mathbb {Q}
, let
S
z
S_z
be the collection of
t
∈
Q
t\in \mathbb {Q}
such that
z
z
is preperiodic for
f
t
f_t
. In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of
S
z
S_z
over
z
∈
Q
z\in \mathbb {Q}
. In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve
C
C
of genus
4
4
defined over
Q
\mathbb {Q}
. We use Chabauty’s method, which requires us to determine the Mordell-Weil rank of the Jacobian
J
J
of
C
C
. We give two proofs that the rank is
1
1
: an analytic proof, which is conditional on the BSD rank conjecture for
J
J
and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree
12
12
and degree
24
24
, respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for
J
J
.