Let $\mathscr{C}$ be the ColemanMazurBuzzard eigencurve of some fixed tame level $N$. Are there any known examples of a singular point $x\in \mathscr{C}$ which lies in a unique irreducible component of $\mathscr{C}$? Certainly the eigencurve can be singular at points where irreducible components of different flavors (e.g., Eisenstein vs. cuspidal, CM vs. nonCM, distinct inertial types at some prime dividing $N$) cross each other.
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