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Class Equation

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.

 0

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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References

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

Class Equation

Cite this as:

Terr, David. "Class Equation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ClassEquation.html