Papadimitriou in https://lara.epfl.ch/w/_media/papadimitriou81complexityintegerprogramming.pdf shows ILP is fixed parameter tractable in number of constraints and Lenstra in https://people.csail.mit.edu/rrw/presentations/Lenstra81.pdf shows ILP is fixed parameter tractable in number of variables. It is shown in https://dl.acm.org/doi/10.5555/284943.284957 that two variable integer programming is equivalent to GCD but the program has an objective function.
What about the case of deciding
$$\exists x\in\mathbb Z^d$$ $$Ax\leq b$$
when number of variables and number of constraints are both fixed?
Is this in $NC$?