Class invariants from a new kind of Weber-like modular equation

A technique is described for explicitly evaluating quotients of the Dedekind eta function at quadratic integers. These evaluations do not make use of complex approximations but are found by an entirely ‘algebraic’ method. They are obtained by means of specialising certain modular equations related to Weber’s modular equations of ‘irrational type’. The technique works for certain eta quotients evaluated at points in an imaginary quadratic field with discriminant \(d \equiv 1 \pmod {8}\).