Frey-Mazur's conjecture.

In his rational isogenies paper, Mazur asks if there are any pairs of elliptic curves E and F and prime l>5 such that E[l] \isom F[l] as Galois module. This question lead to the conjecture that no such pair of elliptic curves exist (I'm not sure where this was first published as a conjecture). Of course, with more computation power, we can actually find such pairs for l=7, 11, and even 13 just by looking at small Cremona's table. Dave Brown pointed out to me that the moduli space of such (ordered) pairs of elliptic curves for l=7 is in birational to P^2, and Sander Dahman and I spent a few weeks trying to come up with an explicit birational map for this moduli space. Similarly the moduli space of unordered pairs for l=11 and 13 have infinite number of rational points for similarly good geometric reasons. The natural question is what happens for 17? Doing a slow exhaustive search on Cremona's table doesn't seem to find any example for level less than 25000. This search is slowing down considerably, so I might need to come up with faster way of searching it right now. Right now, I go through all elliptic curves in the table, check their discriminant, see if we can do level lowering modulo 17 or so, if so, I look at all possible lower levels, and check if the two modular forms are congruent modulo 17. I guess the last step can be sped up by just checking if they are congruent, vs calculating the congruence number. I will start doing that soon.

J(11)[2],

In Poonen, Schaefer, and Schtoel, they find all rational points of $X_E(7)$ for various $E$'s by first finding ranks of $J_E(7)$ and applying Chabauty to find the rational points on $X_E(7)$ (and more stuff as well). They find the ranks of $J_E(7)$ by doing two descent, specially since they have an explicit generators for
$J_E(7)[2]$ in terms of the cusps of $X_E(7)$.
Now, it is natural to ask, does $J(p)[2]$ also have such an explicit set of generators? It seems unlikely, since, for example, $J(11)$ is 26 dimensional, and hence you would need 52 generators for it, by there are only 60 generators. This doesn't rule it out explicitly, but doesn't seem to be very likely, specially when one digs a bit deeper on how the cusps of $X_E(7)$ generated the two torsion points.