Is the converse of the Reduction Theorem for Torsion Points true?

Question

Let $E/K$ be an elliptic curve defined over a number field $K$.

It's a well known result that, for a prime $v$ of $K$ such that the reduction of the elliptic curve $\tilde{E}/k$ is non-singular (here $k$ denotes the corresponding residue field of the completion $K_v$), the reduction map $E(K)[m] \to \tilde{E}(k)$ is injective (for $m$ coprime to $p=char(k)$).

This allows one to quickly find a method for finding the torsion subgroup of an elliptic curve. For instance, for the elliptic curve $E$ defined by $y^2=x^3+3$ with discriminant $\Delta=-3^5 2^4$, we can check that $\#\tilde{E}({\mathbb{F}_5})=6$ and $\#\tilde{E}({\mathbb{F}_7})=13$, so obviously the torsion of $E$ is trivial. My question concerns the reciprocal of this result, namely:

1) Is it true that $\#E(K)_{\text{torsion}} = \gcd\{ \#\tilde{E}({k}) | E \text{ has good reduction mod v}\}$?

2) If not, are there some (nice) conditions one can ask on $E$ that make this result true? For instance, is $E$ isogenous to a curve for which 1) is true? (see the answer by the user tracing)

Answer 1

Well, in the meantime I found a counter-example to my question 1):

$E/\mathbb{Q}$ defined by $y^2=x^3+x$ with discriminant $\Delta=-2^5$. We have $E(\mathbb{Q})_{\text{torsion}}=\{O,(0,0)\}$, but one can check that $4|\# E(\mathbb{F_p})$ for every $p \geq 3$ (this is Silverman AEC, exercise 5.12).

Answer 2

One point to note is that $a_p,$ and hence #$E(\mathbb F_p)$, is an isogeny invariant, while #$E(\mathbb Q)_{\mathrm{torsion}}$ is not.

E.g. the curve $y^2 = x^3 + x$, which is curve 64.a4 on the LMFBD tables, is isogenous to some other curves which do have 4 torsion points (see the table for the isogeny class 64.a).