The modular invariant jqt of quantum tori is defined as a discontinuous, PGL2(double-struckZ)‐invariant multi‐valued map of double-struckR. For θ∈Q it is shown that jqt(θ)=∞. For quadratic irrationalities, experiments conducted with the PARI/GP computer algebra system suggest that jqt(θ) is a finite set. In the case of the golden mean φ, we produce explicit formulas for the experimental supremum and infimum of jqt(φ) involving weighted versions of the Rogers–Ramanujan functions. Finally, we define a universal modular invariant as a continuous and single‐valued map of ultrasolenoids from which jqt as well as the classical modular invariant of elliptic curves may be recovered as subquotients.