Conjectural counterexample to Murphy
p=3, 2849p = 8547 = 3*7*11*37
There is an elliptic curve
F:
y^{2} + xy + y = x^{3} + x^{2} - 154x - 478
of conductor 8547, such that
f_{E} = f_{F} (mod 3) and
F has rank 2.
Maybe Sel^{3}(E) = Sel^{3}(F),
which would imply Sha(E) is "explained by a
jump in the rank of the Mordell-Weil"; equivalently,
Sha(E) is visible in J_{0}(8547).
This is only a conjecture.