We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in  to show completeness of the isomorphism relation on many familiar classes in the context of all Σ¹₁ equivalence relations on hyperarithmetical subsets of ω.
"Isomorphism relations on computable structures." J. Symbolic Logic 77 (1) 122 - 132, March 2012. https://doi.org/10.2178/jsl/1327068695