The standard arithmetic operations + and - and left scalar multiplication are defined for elements of supersingular modules. Also one can add and intersect two submodules of an ambient supersingular module.
The submodule generated by all sums of elements in the supersingular modules M1 and M2.
The intersection of the supersingular modules M1 and M2.
> M := SupersingularModule(11); > P := M.1; P; (1, 1) > Q := M.2; Q; (0, 0) > P + Q; (1, 1) + (0, 0) > P - Q; (1, 1) - (0, 0) > 3*P; 3*(1, 1)Next we illustrate some arithmetic on submodules.
> E := EisensteinSubspace(M);
> S := CuspidalSubspace(M);
> V := E + S;
> V;
Supersingular module associated to X_0(1)/GF(11) of dimension 2
> Basis(V);
[
(1, 1) + 4*(0, 0),
5*(0, 0)
]
The index of E + S in M is of interest since it is related
to congruences between Eisenstein series and cusp forms.
Upon converting each of E and S to an RSpace,
we find that the index is 5.
> RSpace(M)/RSpace(V); Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 5 ]The intersection of E and S is the zero module.
> W := E meet S; W; Supersingular module associated to X_0(1)/GF(11) of dimension 0 > Basis(W); []