Quadratic Fields
Imaginary Quadratic Class Numbers by Counting Reduced Forms
Computing class numbers and group structure for imaginary quadratic fields by enumerating reduced forms, and where the method runs out.
Engineering / MathematicsQuadratic Fields2 min readKV-MATH-0604
For imaginary quadratic fields the class number is the number of reduced forms of the discriminant. This makes small discriminants entirely elementary and gives a reliable check on more sophisticated methods.
The method
Enumerate all reduced positive definite forms of the given discriminant. Because each class contains exactly one, the count is the class number.
Recovering the group structure
Class group structure from reduced forms
- EnumerateList all reduced forms; the count is the class number.
- Choose candidate generatorsTake forms with small first coefficient.
- Compute ordersBy repeated composition until the principal form is reached.
- AssembleCombine generators until their orders multiply to the class number; extract invariant factors.
Cost and limits
| Discriminant size | Feasibility |
|---|---|
| Up to about 10 digits | Immediate |
| 10 to 14 digits | Feasible but slow |
| Beyond | Impractical; use sub-exponential methods |
Value as a check
Class number one
There are exactly nine imaginary quadratic fields of class number one, a celebrated result. The corresponding discriminants appear repeatedly in complex multiplication and in elliptic curve primality proving.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 5.3.1. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.