Some remarks on the torsion of elliptic curves

We prove the following.
Theorem. {\it Let $k$ be a number field, and $J(n)$ the Jacobian of the curve parametrizing the elliptic curves with distinguished cyclic subgroups of order $n$. If the number $N$ is written as $n\cdot a,$ where $J(a)$ contains a $k$-simple abelian subvariety $A$ such that
$$ \tau(n)\times\operatorname{rk}\operatorname{End}_k(A)>\operatorname{rk}A_k, $$
then the set of $k$-isomorphism classes of elliptic curves over the field $k$ possessing $k$-points of order $N$ is finite}.
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