Visibility theorems imply nontrivial ℓ-torsion in Sha of quadratic twists of elliptic curves with additive reduction at ℓ; for ℓ=3 this yields pairs of curves with identical BSD data and Kodaira symbols but isomorphic Sha groups containing 3-torsion.
Sutherland, A local-global principle for rational isogenies of prime degree, Journal de théorie des nombres de Bordeaux 24 (2012), no
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Nontrivial torsion in the Tate--Shafarevich group of elliptic curves via visibility and twists
Visibility theorems imply nontrivial ℓ-torsion in Sha of quadratic twists of elliptic curves with additive reduction at ℓ; for ℓ=3 this yields pairs of curves with identical BSD data and Kodaira symbols but isomorphic Sha groups containing 3-torsion.