Let $E/\mathbb{Q}$ be an elliptic curve, and let $Sha(E/\mathbb{Q})$ denote the Tate-Shafarevich group of $E/\mathbb{Q}$. It is known that the 2-primary component $Sha(E/\mathbb{Q})[2]$ can become arbitrarily large as we vary $E$.

For example, Kramer found an elliptic curve over $\mathbb{Q}$ with discriminant $m(16m+1)$ and $\#Sha(E/\mathbb{Q})[2] \geq 2^{2k-2}$, where $k$ is the number of prime factors of $16m+1$ ("A family of semistable elliptic curves with large Tate-Shafarevich groups," Proceedings of the AMS, 89 (1983)).

This elliptic curve has a discriminant $\Delta_E$ with a very large number of prime factors.

However, is it possible to enlarge the 2-part of the Tate-Shafarevich group while keeping the number of prime factors of the discriminant small? Specifically, can we increase $\dim_2Sha(E/\mathbb{Q})[2] - \omega(\Delta_E)$ arbitrarily?

The largest value of $\dim_2Sha(E/\mathbb{Q})[2] - \omega(\Delta_E)$ that I am aware of is 12. If anyone knows of an example with a larger $\dim_2Sha(E/\mathbb{Q})[2] - \omega(\Delta_E)$, I would greatly appreciate the information.

Let $\omega(\Delta_E)$ denote the number of distinct prime factors of $\Delta_E$. The following two elliptic curves have a small number of prime factors of their discriminants, $\omega(\Delta_E) = 2$, and $\dim_2Sha(E/\mathbb{Q})[2] - \omega(\Delta_E) = 4$:

1. $E_d: y^2 = x^3 + dx$ when we let $d$ be $p = 2^{2^k} + 1$, then $Sha(E_{65537}) \cong (\mathbb{Z}/8\mathbb{Z})^2$. For more information, refer to the discussion on MathOverflow [here](https://mathoverflow.net/questions/151396/the-exponent-of-Ш-of-y2-x3-px-where-p-is-a-fermat-prime).

2. Let $E_{p,n}: y^2 = x^3 + p^nx$ be an elliptic curve. According to LMFDB, for $(p,n) = (73,3)$, $\#Sha(E_{p,n}) = 64$. This was mentioned in a MathOverflow discussion available [here](https://mathoverflow.net/questions/454965/large-tate-shafarevich-group-of-an-elliptic-curve-with-the-form-e-p-ny2-x3).

In Table 2 of the document available at [Sza.pdf](https://math.berkeley.edu/~wodzicki/prace/Sza.pdf), the case of $(n,p) = (16,48)$ is cited as an example where $Sha(E/\mathbb{Q}) - \omega(\Delta_E) = 18 - 6 = 12$.

This record of $\dim_2Sha(E/\mathbb{Q})[2] - \omega(\Delta_E) = 12$ is the maximal one I have ever seen.