A prescribed class-group structure for imaginary quadratic fields

Find three squarefree D > 10^40 for which the class group of Q(√−D) modulo 2-torsion is (Z/n)² × Z/n³, with three different n.

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State the problem, from FrontierMath: Open Problems

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	1	+# A prescribed class-group structure for imaginary quadratic fields
	2	+
	3	+Class groups of imaginary quadratic fields are well understood on average but very hard to control one field at a time. This problem asks for three huge fields, with discriminants beyond $10^{40}$, whose class groups modulo 2-torsion all have the unusual shape $(\mathbb{Z}/n)^2 \times \mathbb{Z}/n^3$, for three different values of $n$.
	4	+
	5	+## Problem
	6	+
	7	+For a squarefree integer $D > 0$, let $G = \mathrm{Cl}(\mathbb{Q}(\sqrt{-D}))$ be the class group and $C(D) = G / G[2]$ its quotient by the 2-torsion subgroup.
	8	+
	9	+Task. Find three distinct squarefree integers $D_1, D_2, D_3 > 10^{40}$ such that
	10	+$$C(D_i) \cong (\mathbb{Z}/n_i\mathbb{Z})^2 \times \mathbb{Z}/n_i^3\mathbb{Z} \qquad (i = 1, 2, 3)$$
	11	+for integers $n_1, n_2, n_3 > 1$ that are pairwise distinct.
	12	+
	13	+## Background
	14	+
	15	+Gauss's genus theory determines $G[2]$ from the prime factors of $D$, so it is natural to study $C(D)$, whose structure is far more mysterious. The Cohen–Lenstra heuristics [1] predict that, among groups of a given size, a group $H$ occurs with frequency proportional to $1/\|\mathrm{Aut}(H)\|$; work of Holmin, Jones, Kurlberg, McLeman and Petersen [2] and of Soundararajan [3] studies how often individual groups occur. Combining this with the Brauer–Siegel size $\|G\| = D^{1/2 + o(1)}$, the problem's authors estimate that a group of this shape (order $n^5$, automorphism group of order about $n^{11}$) should appear a constant number of times on average in each dyadic range of $D$, hence infinitely often. Yet no method is known for constructing infinitely many quadratic fields with a prescribed whole class group; even Gauss's conjecture that infinitely many real quadratic fields have class number 1 is open.
	16	+
	17	+## What is known
	18	+
	19	+- Up to $10^9$ there are 65 such $D$: 19 with $n = 2$, 35 with $n = 3$, 9 with $n = 4$ and 2 with $n = 5$, spread fairly evenly over dyadic ranges (Epoch, by brute-force search).
	20	+- Some abelian groups never occur as class groups of imaginary quadratic fields [2], so heuristics must be used with care.
	21	+- Brute force becomes impractical far beyond $10^9$, where suitable $D$ should become sparse; the problem's authors know no better method than search.
	22	+
	23	+## What would count as progress
	24	+
	25	+- Three integers $D_1, D_2, D_3$. Each is checked by computing a class group of discriminant size about $10^{40}$, which is routine with standard computer algebra when one assumes the generalized Riemann hypothesis, and by reading off $C(D_i)$.
	26	+- Epoch scores submissions with its own verifier, given the three integers.
	27	+- A construction that explains why the fields have this class group would be far more valuable than a lucky search, and might extend to other prescribed groups.
	28	+
	29	+## References
	30	+
	31	+1. H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, in Number Theory Noordwijkerhout 1983, Lecture Notes in Math. 1068 (1984), 33–62. [doi:10.1007/BFb0099440](https://doi.org/10.1007/BFb0099440)
	32	+2. S. Holmin, N. Jones, P. Kurlberg, C. McLeman and K. L. Petersen, Missing class groups and class number statistics for imaginary quadratic fields, Experiment. Math. 28 (2019), 233–254. [doi:10.1080/10586458.2017.1383952](https://doi.org/10.1080/10586458.2017.1383952)
	33	+3. K. Soundararajan, The number of imaginary quadratic fields with a given class number, 2007. [arXiv:0707.0237](https://arxiv.org/abs/0707.0237)
	34	+
	35	+## Source
	36	+
	37	+Imported from [FrontierMath: Open Problems, square-class-groups](https://epoch.ai/frontiermath/open-problems/square-class-groups), curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.