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.