  The present paper illustrates the utility of Brauer relations, Galois covers
of curves and the theory of regulator constants in the context of studying
isogenies between Jacobians and their relevance to the parity conjecture. This
framework presents a unified approach, enabling the reconstruction of a diverse
array of classical isogenies and the derivation of local expressions for Selmer
rank parities, drawing from an extensive body of existing literature. These
include the local expressions found in the works of Mazur--Rubin (dihedral
extensions), Coates--Fukaya--Kato--Sujatha ($p^g$ isogenies), Kramer--Tunnell
(quadratic twists of elliptic curves), Dokchitser--Maistret (Richelot
isogenies), and Docking (prym construction).
