The $\mathbb{Q}$-rational cuspidal group of $J_{1}(2p)

Let p be a prime not equal to 2 or 3. In this paper we study the \mathbb{Q}-rational cuspidal group \mathcal{C}_{\mathbb{Q}} of the jacobian J_{1}(2p) of the modular curve X_{1}(2p). We prove that the group \mathcal{C}_{\mathbb{Q}} is generated by the \mathbb{Q}-rational cusps. We determine the order of \mathcal{C}_{\mathbb{Q}}, and give numerical tables for all p\leq127. These tables give also other cuspidal class numbers for the modular curves X_{1}(2p) and X_{1}(p). We give a basis of the group of the principal divisors supported on the \mathbb{Q}-rational cusps, and using this we determine the explicit structure of \mathcal{C}_{\mathbb{Q}} for all p\leq127. We determine the structure of the Sylow p-subgroup of \mathcal{C}_{\mathbb{Q}}, and the explicit structure for all p\leq4001.