It is conjectured that there
are do not exist elliptic curves over $\mathbb Q$ of arbitrarily high rank. I was wondering wether someone made a similar conjecture if one restricts to a fixed $j$-invariant. If there are specific values of $j$ known such that the following question has a negative answer would also be nice to know.
Let $j \in \mathbb Q$ be fixed. Do there exist elliptic curves $E/\mathbb Q$ with $j(E)=j$ of arbitrary high rank over $\mathbb Q$?
I'm specifically interested in the value $j=0$.
Edit I recently found an article by Bjorn Poonen called "Heuristics for the arithmetic of elliptic curves". And in that article he proves (Thm 3.7) that if elliptic curves over $\mathbb Q$ follow his heuristics then 21 is the switching point between finitely and infinitely many elliptic curves of that rank and in particular there will be only finitely many elliptic curves of rank > 21. This of course implies the boundedness of the rank of all elliptic curves over $\mathbb Q$, so the same should hold if one restricts to just one $j$-invariant.