Class Equation

Let O be an order of an imaginary quadratic field. The class equation of O is the equation H_O=0, where H_O is the extension field minimal polynomial of j(O) over Q, with j(O) the j-invariant of O. (If O has generator tau, then j(O)=j(tau)). The degree of H_O is equal to the class number of the field of fractions K of O.

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

It is also true that


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

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


See also

Algebraic Number Minimal Polynomial, Class Group, Class Number, Discriminant, Ideal, Ideal Class, j-Function, j-Invariant, Number Field Order

This entry contributed by David Terr

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

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Class Equation

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Terr, David. "Class Equation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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