5-Torsion in the Shafarevich–Tate Group of a Family of Elliptic Curves, Original Research Article

We compute the φ-Selmer group for a family of elliptic curves, where φ is an isogeny of degree 5, then find a practical formula for the Cassels–Tate pairing on the φ-Selmer groups and use it to show that a particular family of elliptic curves have non-trivial 5-torsion in their Shafarevich–Tate group.