Computing Component Groups

Using modular symbols, we can enumerate the optimal quotients $A_f$ of $J_0(N)$ (see, e.g., [1]) and compute the degree $m_A$ (see [12, §3.1]). Suppose $p$ is a prime that exactly divides $N$. As explained in [12], the method of graphs (see [14]) or the ideal theory of quaternion algebras (see [11]) can be used to compute $X=X_{J_0(N),p}$ with its $\mathbf{T}$-action and the monodromy pairing. We can then compute the following three modules:

1. the saturated submodule $\mathcal{L}=\bigcap_{t\in I_f} \ker(t)$ of $X$,
2. the character group degree $m_X=m_\mathcal{L}$, and
3. $\Phi_X=\Phi_\mathcal{L}$.

By Theorem 6.1 we obtain

$$\char93 \Phi_{A,p} = \char93 \Phi_X \cdot \frac{m_A}{m_X}.$$