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](https://arxiv.org/abs/1711.10112v2) 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.



The boundedness of the rank of twists of a fixed curve
This is a very interesting question, and there was renewed interest in it lately. As Chris Wuthrich says in the comments, the situation used to be that most people believed in unboundedness of ranks for elliptic curves with a given $j$-invariant, although there was no particular evidence in favour of this. In fact, there was an old conjecture of Honda


T. Honda, Isogenies, rational points and section points of group varieties, Japan. J. Math.,
30 (1960), 84-101


that asserted boundedness, but people did not really believe it. However, recently, there was work by Andrew Granville, Mark Watkins and others with some serious heuristics pointing towards boundedness as well. Mark's slides for his recent talk at the Warwick conference about this are [here](http://www2.warwick.ac.uk/fac/sci/maths/research/events/2012-2013/numbertheory/selmergroups/talks). In particular, the heuristics suggest that for the family 
$$
 X^3+Y^3=A\qquad (j=0)
$$ 
the correct answer could be that the ranks are bounded by $9$, except for finitely many curves in the family. Mark also notes, though, that all these heuristics are quite shaky, and are also very difficult to verify, so at the moment it is really not clear what to expect.


Finally, people working in random matrix theory, notably Nina Snaith, Jon Keating and their collaborators are also looking into higher ranks in the family of quadratic twists. Nina spoke about it  [here](http://rational.epfl.ch/workshop2.php) just two weeks ago, and these are her  [slides](http://rational.epfl.ch/Nina_Snaith.pdf). They are working on a precise heuristic for the number of rank 2 twists, with a hope that the approach would eventually lead to a conjectural formula for the number of higher rank twists as well.


In any case, the current status is that the precise answer is not known for a single $j$-invariant.


