ag.algebraic geometry – Tate-Shafarevich groups of high-rank elliptic curves over $mathbb Q$


Assume the BSD conjecture. By checking various examples, it seems that the Tate-Shafarevich groups of elliptic curves over $mathbb Q$ satisfies the following propositions:

  • If an elliptic curve E over ℚ has rank ≥ 2, then Ш(E)=1 or Ш(E)=4.
  • If an elliptic curve E over ℚ has rank ≥ 3, then Ш(E)=1.

EDIT: The second proposition is false. The elliptic curve $E:y^2 = x^3 + 1916840x$ has rank 3 and Ш(E)=4, by the following SageMath computation:

A=EllipticCurve((0,0,0,1916840,0))
A.rank()                   #=3
A.sha().an_numerical()     #=4.0000000000

Question: Are there references, heuristics, counterexamples, etc. to the first proposition above?