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.
Shiga, Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants, arXiv:2507.18574 (2025)
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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.