We have to find an elliptic curve $E:= y^2=x^3+Ax+B $ (A, B are integers) which has points $P, Q$ with rational coordinates and satisfy $P=[n]Q, n>1$. So,
Input: Rational coordinates = $P$.
Output: Integers $A, B, n$ (when $n>1$), and rational coordinates $Q,$ $$OR,$$
$$Return \; 0.$$
What is the best algorithm or what kind approach should be taken to get the output? is there anything in the literature related to the above problem?
Previously I thought that it is not needed to assume something about the rational point. For example, if rational point $(x,y)=(p/q,r/2)$ where $q$ and $r$ odd, then we consider such $A$ and $B$ that makes $x^3 + Ax + B - y^2 = (p^3+Apq^2+Bq^3)/q^3 - r^2/4$ equal $0$, in this case, $(p^3+Apq^2+Bq^3)/q^3 - r^2/4=0 \implies Apq^2+Bq^3 = p^3-r^2\times q^3/4 \cdots$, but it is not true, in that case, please put necessary and sufficient condition on $P$, if $P$ does not satisfy that we return $0$.
It might not be possible to have such Integers $A, B, n>1, Q$ for a given $P$ satisfying above condition, in that case we return $0.$
I am not considering Torsion Group or points of finite order only, $P, Q$ could be points of infinite order.