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?