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ArticlePublished 7 Aug 20262 min readBy Kevin Joginimaginary quadraticclass numberreduced formsenumeration
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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.

Key point

Uniqueness of the reduced form is doing all the work. Without it the enumeration would count classes with multiplicity, and no simple counting argument would be available.

Recovering the group structure

Class group structure from reduced forms

  1. EnumerateList all reduced forms; the count is the class number.
  2. Choose candidate generatorsTake forms with small first coefficient.
  3. Compute ordersBy repeated composition until the principal form is reached.
  4. AssembleCombine generators until their orders multiply to the class number; extract invariant factors.

Note

Order computation in the class group requires the class number to be known first, since the order divides it. This is why the enumeration is done before the structure is determined.

Cost and limits

Enumeration cost ~ sqrt(|D|)The bound on the first coefficient.
Practical range of the enumeration method
Discriminant sizeFeasibility
Up to about 10 digitsImmediate
10 to 14 digitsFeasible but slow
BeyondImpractical; use sub-exponential methods

Cost

Because the class number itself grows roughly like the square root of the discriminant, the enumeration cost is proportional to the size of the answer. No enumeration method can do better, which is why sub-exponential methods are needed — see sub-exponential quadratic class groups.

Value as a check

Key point

For discriminants where both are feasible, the enumeration gives an unconditional class number against which a GRH-conditional sub-exponential result can be verified. This is the standard way to validate an implementation of the harder method.

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.

Related pages

  • The Ideal Class Group
  • Complex Multiplication and Class Numbers
  • Composition of Binary Quadratic Forms
  • Class Numbers from Analytic Class Number Formulas

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