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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