  We determine all the quadratic points on the genus $13$ modular curve
$X_0(163)$, thus completing the answer to a recent question of Banwait, the
second-named author, and Padurariu. In doing so, we investigate a curious
phenomenon involving a cubic point with complex multiplication on the curve
$X_0(163)$. This cubic point prevents us, due to computational restraints, from
directly applying the state-of-the-art Atkin--Lehner sieve for computing
quadratic points on modular curves $X_0(N)$. To overcome this issue, we
introduce a technique which allows us to work with the Jacobian of curves
modulo primes by directly computing linear equivalence relations between
divisors.
