Isolated points on $$X_1(\ell ^n)$$ with rational j-invariant

Abstract

Let \(\ell \) be a prime and let \(n\ge 1\). In this note we show that if there is a non-cuspidal, non-CM isolated point x with a rational j-invariant on the modular curve \(X_1(\ell ^n)\), then \(\ell =37\) and the j-invariant of x is either \(7\cdot 11^3\) or \(-7.137^3\cdot 2083^3\). The reverse implication holds for the first j-invariant but it is currently unknown whether or not it holds for the second.