  We present compact analytic results for the one-loop amplitude for the
process $0 \rightarrow q \bar{q} \ell \bar\ell \ell^\prime \bar{\ell}^\prime
g$, relevant for both the production of a pair of $Z$ and $W$-bosons in
association with a jet. We focus on the gauge-invariant contribution mediated
by a loop of quarks. We explicitly include all effects of the loop-quark mass
$m$, appropriate for the production of a pair of $Z$-bosons. In the limit $m
\to 0$, our results are also applicable to the production of $W$-boson pairs,
mediated by a loop of massless quarks. Implemented in a numerical code, the
results are fast. The calculation uses novel advancements in spinor-helicity
simplification techniques, for the first time applied beyond five-point
massless kinematics. We make use of primary decompositions from
algebraic-geometry, which now involve non-radical ideals, and $p$-adic numbers
from number theory. We show how to infer whether numerator polynomials belong
to symbolic powers of non-radical ideals through numerical evaluations.
