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Let $p$ be a prime number and $n$ be positive integer. Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve.

LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$. This is the biggest size of $Sha(E_{p,n})$ I have observed ever. Largest Tate-Shafarevich group with j-invariant $1728$ in LMFDB is this curve.

My question is, is it possible to find more large $Sha(E_{p,n})$?

I searched more than 100 primes with the following code of Magma, but the largest one was above.

A:=EllipticCurve([0,0,0,p^n,0]); MordellWeilShaInformation(A: ShaInfo:=true);

Thank you for your assistance. Even a single example without any background would be also appreciated. I believe the Tate-Shafarevich group for this family might not be well-documented, but if you're aware of any references for this curve's family, I'd greatly appreciate it.

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    $\begingroup$ One suggestion is as follows. Curves with $j$-invariant equal to 1728 tend to have very large 2-Selmer groups, and no bias when it comes to 3-Selmer. Comparing the two will give you good estimates for Sha. $\endgroup$
    – Stanley Yao Xiao
    Commented Sep 20, 2023 at 18:50

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