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Specific Tables

Component Groups of J0(N)(R) and J1(N)(R)

• URL: http://wstein.org/Tables/real_tamagawa/

• New Code:

This function computes the J_0(N) real component groups.

def f(N):
    M = ModularSymbols(N).cuspidal_subspace()
    d = M.dimension()//2
    S = matrix(GF(2),2*d,2*d, M.star_involution().matrix().list()) - 1
    return 2^(S.nullity()-d)

For J_1(N) it is:

def f(N):
    M = ModularSymbols(Gamma1(N)).cuspidal_subspace()
    d = M.dimension()//2
    S = matrix(GF(2),2*d,2*d, M.star_involution().matrix().list()) - 1
    return 2^(S.nullity()-d)

Future extension: one could replace Gamma1(N) by GammaH(N,...). One could also do the new subspace.

Cuspidal Subgroup

Computing the structure of the cuspidal subgroup of J0(N) and J1(N) (say).

• URL: http://wstein.org/Tables/cuspgroup/ (the displayed formula is backwards at the top)

• New Sage code:

def cuspidal_subgroup_J0(N):
    J = J0(N)
    I = C.cuspidal_subgroup().invariants()
    # maybe pickle J
    return I

def cuspidal_subgroup_J0(N):
    J = J1(N)
    I = C.cuspidal_subgroup().invariants()
    # maybe pickle J
    return I