Which complexity class does this number theory problem belong to? 'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by=c$' is $\mathsf{NP}$-complete.

Which complexity class does 'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by^2=c$' belong to?
Added later: As noted in the comments, the NP upper bound is trivial if a, b, and c are positive, as was asked.

Theorem 1.2 in this paper shows that deciding if a given diophantine equation in two variables has a solution is in NP.