On some open cases of a conjecture of Conrad, Edixhoven and Stein

Let [Formula: see text] be a prime. In 2003, Conrad et al. conjectured that the rational torsion subgroup of the modular Jacobian [Formula: see text] coincides with the rational cuspidal divisor class group. Using explicit computations in Magma, the open case [Formula: see text] has been proven by [M. Derickx, S. Kamienny, W. Stein and M. Stoll, Torsion points on elliptic curves over number fields of small degree, Algebra Number Theory 17(2) (2023) 267–308, doi:10.2140/ant.2023.17.267]. We extend these results to primes [Formula: see text] and [Formula: see text]. In addition, we provide a list of the groups [Formula: see text] for every prime up to [Formula: see text]. However, our method is general and can be applied to larger primes.