Abstract: For a positive integer $N$, let $\Gamma_0(N)$ denote the subgroup of $SL_2(\mathbb Z)$ consisting of matrices whose lower left entry is divisible by $N$ and let $X_0(N)$ denote the quotient of the extended upper half plane by the action of $\Gamma_0(N)$. $X_0(N)$ can be viewed as an algebraic curve over $\mathbb Q$. Let $J_0(N)$ denote the Jacobian variety of $X_0(N)$. The famous Taniyama-Shimura conjecture, now a theorem of Wiles et al, states that any "non-CM" elliptic curve over $\mathbb Q$ is isogenous to a $\mathbb Q$-simple factor of $J_0(N)$ for a suitable $N$. Can we find an upper bound for $N$ such that all $\mathbb Q$-simple factors of $J_0(N)$ are elliptic curves, or more generally, abelian varieties of bounded dimension? We attack this geometric question "effectively" (pun intended) by analytic techniques from equidistribution theory.