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How to explicitly see the ramification over infinity

The ramification over $\infty$ is defined by the value of $\chi$ at $\gamma_\infty$, i.e. $e^{2\pi M/d}$. It is trivial, and hence the cover is unramified over $\infty$ if and only if $d|M$. … The ramification index is equal M\mod d.$ …
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