The monodromy pairing is a nondegenerate Hecke equivariant integer valued pairing on the module of supersingular points. (Note that it is not perfect, in general.) This pairing is simple to describe in terms of the basis for the supersingular module given by the enhanced supersingular elliptic curves. If E and F are two supersingular elliptic curve in characteristic p equipped with level N structure, then E and F pair to 0 unless E isomorphic to F, in which case they pair to half the number of automorphisms of E.
The monodromy pairing of elements P and Q of a supersingular module.
The diagonal entries that define the monodromy pairing on the ambient space.
> M := SupersingularModule(11);
> MonodromyWeights(M);
[ 2, 3 ]
> P := Basis(CuspidalSubspace(M))[1]; P;
(1, 1) - (0, 0)
> Q := Basis(EisensteinSubspace(M))[1]; Q;
3*(1, 1) + 2*(0, 0)
> Basis(M);
[
(1, 1),
(0, 0)
]
> MonodromyPairing(P,Q);
0
> MonodromyPairing(P,P);
5