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The class number of a Dedekind ring is the group order of its ideal class group. For every nonzero ideal , there is a nonzero ideal such that

where is a principal ideal. When the class number is finite, a finite list of ideals suffices for all . One representative from each ideal class gives a list whose size is the class number.

Class numbers are usually studied in the context of the orders of number fields. If this order is maximal, then it is the ring of integers of the number field, in which case the class number is equal to the order of the class group of the number field; otherwise it is equal to the order of the Picard group of the nonmaximal order in question.

The ring of integers of a number field has unique factorization iff its class number is 1. Thus the class number measures the failure of unique factorization in this ring.

A formula expressing a class number in terms of other invariants is called a class number formula. Such formulas are available for number fields, including cyclotomic fields. Finding the class number can be computationally difficult.

The Wolfram Language function NumberFieldClassNumber[Sqrt[d]] gives the class number for a fundamental discriminant of a quadratic field. Here is the ordinary ideal class number.

The class number of an order of a quadratic field with discriminant is equal to the number of reduced binary quadratic forms of discriminant that are primitive polynomials and positive definite. Thus the coefficients have greatest common divisor 1, and the forms represent classes under changes of variables with integer coefficients and determinant 1. For example, the class number of the ring of integers of the number field is equal to 3 since there are three such reduced binary quadratic forms of discriminant , namely , , and . Counting these forms gives a direct, though inefficient, way to compute .

For a fundamental discriminant of a quadratic field, the class number is given by the class number formula

where is the Kronecker symbol, is the fundamental unit of the real quadratic field, chosen as the least unit greater than 1, and is the number of roots of unity in the imaginary quadratic field, given by

and the sums include all indicated integers, with the Kronecker symbol equal to 0 when and are not relatively prime (Cohn 1980). For a positive fundamental discriminant , the class number can also be written

where factors with Kronecker symbol 0 contribute 1 to the product.

For a fundamental discriminant of a quadratic field, the class number is related to the Dirichlet L-series by

where is the Dirichlet structure constant.

Oesterlé (1985, pp. 310 and 321) showed that, for a negative fundamental discriminant , the class number satisfies the inequality

where is the floor function, the product is over primes dividing , and the indicates that the greatest prime factor of is omitted from the product. It is also known that if is relatively prime to 5077, then the denominator 7000 in (6) can be replaced by 55.

Gauss's class number problem asks to determine a complete list of fundamental binary quadratic form discriminants such that the class number is given by for a given . Watkins (2004, p. 907) completed the classification for every . Gauss conjectured that the class number of an imaginary quadratic field with binary quadratic form discriminant tends to infinity with , an assertion now known as Gauss's class number conjecture.

The negated discriminants corresponding to imaginary quadratic fields are 3, 4, 7, 8, 11, 15, 19, 20, 23, 24, 31, 35, 39, 40, 43, ... (OEIS A003657), which have corresponding class numbers , 1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 2, 4, 2, 1, ... (OEIS A006641).

Buell (1977) gives the smallest and largest class numbers for fundamental discriminants with , partitioned into even discriminants, discriminants 1 (mod 8), and discriminants 5 (mod 8). Arno et al. (1998) give complete lists of values of fundamental with for odd , 7, 9, ..., 23. Wagner (1996) gives complete lists of values for , 6, and 7. Lists of negative fundamental discriminants corresponding to imaginary quadratic fields having small class numbers are given in the table below. In the table, is the number of negative fundamental discriminants with the given class number. An integer is a fundamental discriminant iff either is squarefree and , or where is squarefree and satisfies or . For example, , but is not a fundamental discriminant because it is divisible by the square number 9 and is odd. Similarly, is not a fundamental discriminant, although . The table lists the absolute values of the discriminants. Its completeness follows from Watkins (2004, p. 907).

The following table lists the absolute values of negative fundamental discriminants having class numbers (Cohen 1993, pp. 229 and 514-515; Cox 1997, p. 271; Watkins 2004, p. 907).

The numbers of negative fundamental discriminants having class number 1, 2, 3, ... are 9, 18, 16, 54, 25, 51, 31, ... (OEIS A046125). The largest absolute values of negative fundamental discriminants having class numbers 1, 2, 3, ... are 163, 427, 907, 1555, 2683, ... (OEIS A038552).

The discriminants corresponding to real quadratic fields are 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, ... (OEIS A003658, omitting its initial value 1), corresponding to class numbers , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, ... (OEIS A003652).

The table below gives lists of the first few positive fundamental discriminants having small class numbers , corresponding to real quadratic fields, augmenting the table of Cohn (1980, pp. 271-274) by including terms divisible by 4 (Cohen 1993, pp. 516-519; Cohen 2000, pp. 534-537).

The smallest such that the real quadratic field with discriminant has class number for , 2, ... are 5, 40, 229, 145, 401, 697, 577, 904, 1129, ... (OEIS A081364).

Kimura (2026) reports an unconditional classification of imaginary quadratic fields with class number at most 2000. The reported total is 16453493, with largest absolute fundamental discriminant 1062350083, occurring at class number 1978. The analytic bounds use a recent OpenAI approach to Landau-Siegel zeros. Kimura (2026) reports Lean and independent-kernel verification of those bounds, but the sieve program and extraction of the final lists are not formally verified. Kimura (2026) describes substantial assistance from Claude in developing the explicit proofs, formalization, and computations.

See also

Binary Quadratic Form Discriminant, Class Field Theory, Class Group, Class Number Formula, Dirichlet L-Series, Dirichlet Structure Constant, Gauss's Class Number Conjecture, Gauss's Class Number Problem, Heegner Number, Ideal, Imaginary Quadratic Field, j-Function, Landau-Siegel Zero, Ring

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References

Arno, S. "The Imaginary Quadratic Fields of Class Number 4." Acta Arith. 60, 321-334, 1992.Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary Quadratic Fields with Small Odd Class Number." Acta Arith. 83, 295-330, 1998. https://doi.org/10.4064/aa-83-4-295-330.Buell, D. A. "Small Class Numbers and Extreme Values of -Functions of Quadratic Fields." Math. Comput. 31, 786-796, 1977. https://doi.org/10.1090/S0025-5718-1977-0439802-X.Cohen, H. "Table of Class Numbers of Complex Quadratic Fields" and "Table of Class Numbers and Units of Real Quadratic Fields." §B.1 and B.2 in A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, pp. 513-519, 1993.Cohen, H. "Hilbert Class Fields of Quadratic Fields." §12.1 in Advanced Topics in Computational Number Theory. New York: Springer-Verlag, pp. 533-542, 2000.Cohn, H. Advanced Number Theory. New York: Dover, pp. 163 and 234, 1980.Cox, D. A. Primes of the Form x²+ny²: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.Davenport, H. "Dirichlet's Class Number Formula." Ch. 6 in Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, pp. 43-53, 1980.Finch, S. R. "Class Number Theory." https://web.archive.org/web/20160419150530/http://www.people.fas.harvard.edu/~sfinch/csolve/clss.pdf.Himmetoglu, S. Berechnung von Klassenzahlen Imaginaer-Quadratischer Zahlkörper. Diplomarbeit. Heidelberg, Germany: University of Heidelberg Faculty for Mathematics, March 1986.Iyanaga, S. and Kawada, Y. (Eds.). "Class Numbers of Algebraic Number Fields." Appendix B, Table 4 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1494-1496, 1980.Kimura, I. "Imaginary Quadratic Fields of Class Number at Most 2000." 15 Oct 2026. https://arxiv.org/abs/2610.21511.Montgomery, H. and Weinberger, P. "Notes on Small Class Numbers." Acta. Arith. 24, 529-542, 1974.Müller, H. "A Calculation of Class-Numbers of Imaginary Quadratic Numberfields." Tamkang J. Math. 9, 121-128, 1978.Oesterlé, J. "Nombres de classes des corps quadratiques imaginaires." Astérisque 121-122, 309-323, 1985. https://numdam.org/item/SB_1983-1984__26__309_0/.Sloane, N. J. A. Sequences A003652/M0051, A003656/M3777, A003657/M2332, A003658/M3776, A006203/M5131, A006641/M0112, A013658, A014602, A014603, A035120, A038552, A046002, A046003, A046125, A048925, A056987, A081364, A094612, A094613, A094614, A094619, and A123563 in "The On-Line Encyclopedia of Integer Sequences."Stark, H. M. "A Complete Determination of the Complex Quadratic Fields of Class Number One." Michigan Math. J. 14, 1-27, 1967.Stark, H. M. "On Complex Quadratic Fields with Class Number Two." Math. Comput. 29, 289-302, 1975.Wagner, C. "Class Number 5, 6, and 7." Math. Comput. 65, 785-800, 1996.Watkins, M. "Class Numbers of Imaginary Quadratic Fields." Math. Comput. 73, 907-938, 2004. https://doi.org/10.1090/S0025-5718-03-01517-5.

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Class Number

Cite this as:

Weisstein, Eric W. "Class Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClassNumber.html

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