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For a fixed prime $p$ congruent to $1$ modulo $4$ we may define the modularcurve $X_{H}\left( p \right)$ associated to the subgroup of non-zero squaresmodulo $p$. This curve has four cusps and we consider the subgroup of theJacobian $J_{H}\left( p \right)$ of $X_{H}\left( p \right)$ generated by thesepoints, which we will call the cuspidal subgroup of $J_{H}\left( p \right)$.This is a finite subgroup by the results of Manin and Drinfeld, and lies insidethe $\mathbb{Q} \left( \sqrt{ p} \right)$-rational torsion subgroup. In thispaper we compute the cuspidal subgroup for all such curves of genus $g$, $2\leq g \leq 10$, namely those with $p \in \{ 29, 37, 41, 53, 61, 73 \}$, andcompare this with $J_{H}\left( \mathbb{Q} \left( \sqrt{p} \right)\right)_{\text{tors}}$.

Computing the Cuspidal Subgroup of the Modular Jacobian $J_{H}\left( p \right)$