The motivation of this question can be found in
Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order?
Given the elliptic curve: $$C:y²=x³+ax+b$$
for $a,b∈ℤ$.
We know that $C(ℚ)≠∅$, so the rank is $r≥0$. From the current literature we do know about the case $r≥2$ except some special cases. My question is then:
Given an arbitrary rank $r≥2$, is it possible to say that there is a curve $C$ such that its rank exactly $r$?