We study the maximum and minimum occupancy fraction of the antiferromagnetic Ising model in regular graphs. The minimizing problem is known to determine a computational threshold in the complexity of approximately sampling from the Ising model at a given magnetization, and our results determine this threshold for nearly the entire relevant parameter range in the case $Δ=3$. A small part of the parameter range lies outside the reach of our methods, and it seems challenging to extend our techniques to larger $Δ$.