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
![H_O(X)=product[X-j(a)],](/images/equations/ClassEquation/NumberedEquation1.svg) |
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.
 |  |  |  |
 |  |  | 0 |
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  | ||
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |
