For primes p not dividing n, the p-primary Tate-Shafarevich group over a dihedral extension of degree 2n has order equal, modulo fourth powers, to that over the quadratic subextension, or over the product of the three quadratic subextensions when n is even.
Milne On the Arithmetic of Abelian Varieties
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A Note on the Growth of Sha in Dihedral Extensions
For primes p not dividing n, the p-primary Tate-Shafarevich group over a dihedral extension of degree 2n has order equal, modulo fourth powers, to that over the quadratic subextension, or over the product of the three quadratic subextensions when n is even.