A new 'level of a fiber' invariant is defined for points on X1(l^n), and a Lang-Trotter-style theorem shows that maximal degree at level k+1 forces maximal degree for all higher lifts.
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Computing the Level of a Fiber for Points on Modular Curves
A new 'level of a fiber' invariant is defined for points on X1(l^n), and a Lang-Trotter-style theorem shows that maximal degree at level k+1 forces maximal degree for all higher lifts.