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.