The modular invariant j^qt of quantum tori is defined as a discontinuous, PGL(2,Z)-invariant multi-valued map of the reals R. For θ ∈ Q, j^qt(θ) = ∞ and for quadratic irrationalities, experiments conducted with the PARI/GP computer algebra system suggest that j^qt(θ) is a finite set. In the case of the golden mean φ, we produce explicit formulas for the experimental supremum and infimum of j^qt(φ) involving weighted generalizations of the Rogers–Ramanujan functions. Finally, we define a universal modular invariant as a continuous and single-valued map of “ultrasolenoids” (quotients of sheaves of ultrapowers over Stone spaces) from which j^qt as well as the classical modular invariant of elliptic curves may be recovered as subquotients.