Class Invariants

HilbertClassPolynomial(D) : RngIntElt -> RngUPolElt

Given a negative discriminant D, returns the Hilbert class polynomial, defined as the minimal polynomial of j(τ), where Z[τ] is an imaginary quadratic order of discriminant D.

WeberClassPolynomial(D) : RngIntElt -> RngUPolElt

Given a negative discriminant D congruent to 1 modulo 8, returns the Weber class polynomial, defined as the minimal polynomial of f(τ), where Z[τ] is an imaginary quadratic order of discriminant D and f is a particular normalized Weber function generating the same class field as j(τ). A root f(τ) of the Weber class polynomial is an integral unit generating the ring class field related to the corresponding root j(τ) of the Hilbert class polynomial by the expression

j(τ) = ((f(τ)²⁴ - 16)³ /f(τ)²⁴),

where ( GCD)(D, 3) = 1, and

j(τ) = ((f(τ)⁸ - 16)³ /f(τ)⁸),

if 3 divides D. For further details, consult Yui and Zagier [YZ97].