Generalized class polynomials

Abstract The Hilbert class polynomial has as roots the j -invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute over finite fields with prescribed number points. Since its coefficients are typically rather large, there been continued interest in finding alternative modular functions corresponding polynomials smaller. Best known Weber’s functions, which reduce size by factor 72 for positive density subset discriminants. On other hand, Bröker and Stevenhagen showed that no function will ever do better than 100.83. We introduce generalization polynomials, reduction factors not limited Bröker–Stevenhagen bound. provide examples matching factor. For an infinite family discriminants, their surpass those all previously at least 2.