The action of a subgroup of the torsion group of elliptic curves on integral points?

Question

Let $E$ be an elliptic curve given in long Weierstraß form with all coefficients $a_1,a_2,a_3,a_4,a_6 in mathbb{Z}$. It is known that the rational points $E(mathbb{Q})$ form a group which has a finite torsion subgroup $T$.

Question:

What is known about the action of the following subgroup $hat{T} le T$ on the integral points $E(mathbb{Z})$:
$$hat{T} := { t in T | t + E(mathbb{Z}) subset E(mathbb{Z}) }$$
?
I have found one elliptic curve where this $hat{T}$ is not the trivial group:
https://www.lmfdb.org/EllipticCurve/Q/210/e/6
For this curve:
$$T := left[left(0 : 1 : 0right), left(4 : 58 : 1right), left(64 : 418 : 1right), left(-26 : 148 : 1right), left(28 : -14 : 1right), left(-26 : -122 : 1right), left(64 : -482 : 1right), left(4 : -62 : 1right), left(-36 : 18 : 1right), left(34 : -122 : 1right), left(-8 : -122 : 1right), left(244 : -3902 : 1right), left(frac{31}{4} : -frac{31}{8} : 1right), left(244 : 3658 : 1right), left(-8 : 130 : 1right), left(34 : 88 : 1right)right]
$$
and
$$hat{T}:= left[left(0 : 1 : 0right), left(frac{31}{4} : -frac{31}{8} : 1right)right]$$
I must admit, that in most cases where I looked at numerical examples, we had $hat{T} = 1$.

Question: Are there examples of elliptic curves with $1 < hat{T} = T$?

Thanks for your help.

user6671 · Answer

I will try to answer question 1. on my own as good as I can:
Consider two cases:
1.) $hat{T} cap E(mathbb{Z}) neq emptyset$
Let $Q$ be an element in this intersection. Then $-Q in hat{T}$ and adding this to $Q in E(mathbb{Z})$ we get by hypothesis on $hat{T}$ a point in $E(mathbb{Z})$, so
$$ O = (-Q) + Q in E(mathbb{Z})$$
which is a contradiction, to SAGEMATHs definition of integral points $E(mathbb{Z})$.
So this case can not happen.
2.) $hat{T} cap E(mathbb{Z}) = emptyset$
Then by Nagel-Lutz Theorem, either $hat{T}=1$ or each $Q in hat{T}$ is of the form $Q=(frac{m}{4},frac{n}{8},1)$ and has $operatorname{ord}(Q)=2$. This means in the latter case that $hat{T} = C_2 times ldots times C_2$ which leaves by a theorem of Mazur only the cases $C_2,C_2 times C_2$ open.
All in all, we get three possible cases:
$$hat{T} = 1, C_2, C_2 times C_2$$
of which I have seen examples for the first two. Is there an example for $hat{T} = C_2 times C_2$?

The action of a subgroup of the torsion group of elliptic curves on integral points?

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