Imaginary quadratic fields whose class groups have large ℓ-rank

Exhibit imaginary quadratic fields whose class groups have record ℓ-rank, such as 3-rank 9 or 5-rank 6, certified by independent forms.

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State the problem, from FrontierMath: Open Problems

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	1	+# Imaginary quadratic fields whose class groups have large ℓ-rank
	2	+
	3	+The class group of a number field measures how far its ring of integers is from unique factorisation. Heuristics predict that for every prime ℓ and every r, some imaginary quadratic fields have class groups containing r independent elements of order ℓ, but explicit examples run out quickly: for ℓ = 3 the record is rank 8. The task is to beat the records.
	4	+
	5	+## Problem
	6	+
	7	+For a positive integer $D$ such that $-D$ is a fundamental discriminant, let $\mathrm{Cl}(K)$ be the class group of $K = \mathbb{Q}(\sqrt{-D})$, identified with the group of equivalence classes of positive definite binary quadratic forms $ax^2 + bxy + cy^2$ of discriminant $b^2 - 4ac = -D$ under composition. For an odd prime $\ell$, the ℓ-rank of $\mathrm{Cl}(K)$ is the integer $r$ with $\mathrm{Cl}(K)[\ell] \cong (\mathbb{Z}/\ell\mathbb{Z})^r$.
	8	+
	9	+Task. For as many pairs $(\ell, r)$ as possible from the challenge set
	10	+
	11	+- $\ell = 3$ and $r = 9, \dots, 16$;
	12	+- $\ell = 5$ and $r = 6, \dots, 12$;
	13	+- $\ell = 7$ and $r = 5, \dots, 10$;
	14	+- $\ell = 11$ or $\ell = 13$, and $r = 4, \dots, 8$;
	15	+
	16	+find $D$ together with $r$ forms of order exactly $\ell$ in $\mathrm{Cl}(\mathbb{Q}(\sqrt{-D}))$ that generate a subgroup isomorphic to $(\mathbb{Z}/\ell\mathbb{Z})^r$. This certifies that the ℓ-rank is at least $r$. The larger goal is a method that produces such fields for any $(\ell, r)$.
	17	+
	18	+## Background
	19	+
	20	+Cohen and Lenstra's heuristics (1984) predict that for each odd prime $\ell$ and each $r$, a positive proportion of imaginary quadratic fields have ℓ-rank exactly $r$ [1]; this is not proved for any $\ell$. Producing fields of large rank on demand therefore tests how well class groups can be steered by construction rather than found by search. For $\ell = 3$ there is a link with elliptic curves: descent by 3-isogeny on a Mordell curve $y^2 = x^3 + k$ of large rank forces large 3-ranks in the class groups of related quadratic fields [2].
	21	+
	22	+## What is known
	23	+
	24	+- The record 3-rank for a quadratic field is 8, for $\mathbb{Q}(\sqrt{-3D})$ with $D = 72513834653847828539450325493$. Elkies obtained it from a Mordell curve of rank 16 and computed the class group unconditionally, using the Burgess bounds (2025) [2].
	25	+- For every ℓ, families with ℓ-rank at least 3 are known, but for all but a few ℓ no example of rank 4 or more is known (Epoch's summary).
	26	+- Known constructions give $D$ growing exponentially with ℓ even for rank 1, whereas the heuristics suggest polynomial growth (Epoch's summary).
	27	+- Every pair in the challenge set is one for which no field is currently known.
	28	+
	29	+## What would count as progress
	30	+
	31	+- For a pair $(\ell, r)$: a discriminant $-D$ and $r$ forms $(a_i, b_i, c_i)$ with $b_i^2 - 4a_ic_i = -D$, each of order $\ell$ and jointly independent. Composition and reduction of forms make this quick to check, even for very large $D$.
	32	+- Epoch scores submissions with its own verifier: one certificate per line in the form `ℓ\|r\|D\|a,b,c\|...`, with credit for each certified pair of the challenge set.
	33	+- A general construction taking $\ell$ and $r$ and returning $D$ would be the real prize, and might shed light on the Cohen–Lenstra conjecture itself.
	34	+
	35	+## References
	36	+
	37	+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)
	38	+2. N. D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, Research in Number Theory 11 (2025), article 70. [doi:10.1007/s40993-024-00601-x](https://doi.org/10.1007/s40993-024-00601-x)
	39	+
	40	+## Source
	41	+
	42	+Imported from [FrontierMath: Open Problems, class-group-rank](https://epoch.ai/frontiermath/open-problems/class-group-rank), 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.