On cubic factors of j-invariants of quadratic Q-curves of prime degree

For a rational prime p, the j-invariants of quadratic Q-curves of degree p are nearly cubes. In order to understand these cubic factors, we study factorizations of the form j=r3⋅R, where j is the modular elliptic function, r is a function on the modular curve X0(p) and the divisor of the function R satisfies certain conditions. We prove the existence of such factorizations and show how to compute them.