For prime-power level, rational isolated j-invariants are exactly 15 values on X_1 (13 CM plus two non-CM) and 19 values on X_0 (13 CM plus six non-CM).
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Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$
For prime-power level, rational isolated j-invariants are exactly 15 values on X_1 (13 CM plus two non-CM) and 19 values on X_0 (13 CM plus six non-CM).