I have computed the torsion subgroup and component group of hundreds of optimal quotients at prime level. In every example, the same pattern emerges: the torsion subgroup is generated by the image of 0-oo and has the same order as the component group; furthermore, the product of the orders of the component groups equals the numerator of (p-1)/12. This indicates that a refinement of the conjecture of Ogg and other theorems proved by Mazur may be true.

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