Let  be an order of an imaginary quadratic field. The class equation of  is the equation , where  is the extension field minimal polynomial of  over , with  the -invariant of . (If  has generator , then . The degree of  is equal to the class number of the field of fractions  of .

The polynomial  is also called the class equation of  (e.g., Cox 1997, p. 293).

It is also true that

where the product is over representatives  of each ideal class of .

If  has discriminant , then the notation  is used. If  is not divisible by 3, the constant term of  is a perfect cube. The table below lists the first few class equations as well as the corresponding values of , with  being generators of ideals in each ideal class of . In each case, the constant term is written out as a cube times a cubefree part.

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REFERENCES:

Cox, D. A. Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.