Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants

We prove that there exist infinitely many pairs of non-isomorphic elliptic curves over $\mathbb{Q}$ sharing the same BSD invariants – including their Mordell–Weil groups, Tate–Shafarevich groups, Tamagawa numbers, regulators, and real periods – and their Kodaira symbols and minimal discriminants, while having distinct $j$-invariants.