About the census

This site keeps a verified list of sporadic points on the modular curves X₁(m,n). It is maintained by Filip Najman (University of Zagreb); the data, the verification code and every certificate live in the GitHub organisation x1mn-sporadic. Contributions of new points are welcome — see submit a point.

Conventions

the curve

X₁(m,n), with m | n, is the modular curve over F = ℚ(ζ_m) whose non-cuspidal points are the isomorphism classes of triples (E, P, Q) with ⟨P, Q⟩ ≅ ℤ/m ⊕ ℤ/n, P of order m, Q of order n, and e_m(P, (n/m)Q) = ζ_m. It is geometrically connected, and over ℂ it is the modular curve of Γ(m) ∩ Γ₁(n). Viewed over ℚ, the same curve parametrizes the triples (E, P, Q) with no condition on the Weil pairing and has φ(m) geometric components; its closed points and their residue fields do not change, which is why degrees can be measured both over ℚ and over F (in the terms of LMFDB's variants of X(N), this is the classical curve over F, equivalently the big curve over ℚ). For m ≥ 3 it is not LMFDB's X_arith,1(m,n), a geometrically connected curve over ℚ whose points are triples (E, Q, C) with C a Galois-stable subgroup of order m rather than a point. The Weil pairing with (n/m)Q identifies C with μ_m, so over F the two curves become isomorphic. Hence they have the same genus, the gonality over ℚ of LMFDB's curve is an upper bound for the gonality of ours over F, and if its Jacobian has positive rank over ℚ, so does ours over F (rank 0 over ℚ does not carry over). For m ≤ 2 the two curves coincide. Each curve page links to the LMFDB page of the curve, and for m ≥ 3 to that of X_arith,1(m,n), whenever the LMFDB has it (all curves except X₁(N) for composite N from 72 to 100). X₁(1,n) = X₁(n). (Derickx–Sutherland write X₁(m, mn) and LMFDB writes X_arith,1(M, MN); here n is always the full cyclic order.)

the degree

As for any curve over F = ℚ(ζ_m), the degree of a closed point x is [F(x) : F]; its absolute degree is [ℚ(x) : ℚ] = φ(m)·[F(x) : F]. For m ≤ 2 (F = ℚ) the two agree. A point of absolute degree d is the same thing as an elliptic curve over a number field of degree d with torsion containing ℤ/m ⊕ ℤ/n, up to the choice of the torsion points and the ambiguity (P,Q) ~ (−P,−Q). This is why the census, like the literature on torsion over number fields, indexes points by absolute degree: the d in a point id m.n.d.x, the degree columns of the curve table and the page of points by degree are absolute degrees. For m ≥ 3 the curve and point pages show both, and say "absolute degree" wherever they mean [ℚ(x) : ℚ]. The verifier always recomputes the residue field of the point, so the degree recorded is the true degree even if the curve was submitted over a larger field.

sporadic

A closed point of degree d is sporadic if X₁(m,n) has only finitely many closed points of degree ≤ d. The notion was introduced by Najman (arXiv:1211.2188, 2012; Math. Res. Lett. 23 (2016)), with a different definition: a point of degree d is sporadic if d < γ, the gonality of the curve. The definition used here first appears in Sutherland's notes Torsion subgroups of elliptic curves over number fields (2012, §§3.3–4): a point on X₁(N) is sporadic if its degree is less than δ(N), the least d for which X₁(N) has infinitely many points of degree d (points of degree < γ are called low degree points there). The weaker property "finitely many points of degree exactly d" — the one studied by van Hoeij's search for low-degree places and by Derickx–van Hoeij, neither of whom uses the word sporadic — is the third column on the curve pages ("infinitely many points of degree d": no).

isolated

Let C = X₁(m,n) over its base field F (degrees here are over F), let φ_d : Sym^d C → Pic^d_C send an effective divisor to its class, and let W^d be its image. Following Bourdon–Ejder–Liu–Odumodu–Viray (arXiv:1808.04520; Adv. Math. 357 (2019), Definition 4.1), a degree-d point x ∈ C is P¹-isolated if there is no other point x′ ∈ (Sym^d C)(F) such that φ_d(x) = φ_d(x′); AV-isolated if there is no positive rank subabelian variety A ⊂ Pic⁰_C such that φ_d(x) + A ⊂ W^d; and isolated if it is P¹-isolated and AV-isolated. Equivalently, x is P¹-isolated if and only if dim L(x) = 1, which is what the census computes; if J₁(m,n) has rank 0 over F, every point is AV-isolated. By their Theorem 4.2(1), C has infinitely many degree-d points if and only if there is a degree-d point on C that is not isolated; in particular, sporadic points are isolated. For a point that is P¹-isolated but not AV-isolated this is the substance of their theorem: earlier arguments give infinitely many points of degree at most d, not of degree exactly d.

the three answers

For each point the census answers sporadic?, isolated? and infinitely many points of degree d? with yes, no or maybe; "maybe" means that no recorded result and no computation decides the question. A point enters the census when it is proven sporadic or isolated (status certified), or when its degree is undecided (verified · undecided). A point of a degree in which the curve provably has infinitely many points is accepted only if it is proven isolated.

one entry per curve

Points are recorded once per elliptic curve over its residue field: two submissions on the same X₁(m,n) whose curves are isomorphic over their (isomorphic) residue fields are the same entry — their level structures differ by a diamond operator, by the choice of the torsion basis, or by Galois conjugation (for X₁(N) this is one entry per diamond orbit, as in van Hoeij's list). The test is an isomorphism computation in Magma, not a comparison of invariants. Cusps and points on genus 0 curves are excluded.

How a point is certified

Every submission is verified from scratch by pipeline/magma/verify_lib.m (Magma 2.29) on Mordell: the field and the curve are constructed, the orders of P and Q and the structure of ⟨P,Q⟩ are checked, the torsion subgroup is bounded by reductions modulo several primes (and computed exactly when the bound is not sharp), and the Tate normal form of (E,Q) gives the residue field of the point and hence its absolute degree d.

Points of a given degree. For every absolute degree e the census decides "finitely many / infinitely many / unknown" from recorded facts, in this order (e′ = e/φ(m) is the degree over the base field, γ = gonality over the base field):

1. Known finiteness. e ≤ 9 and (m,n) ∉ Φ^∞(e), where Φ^∞(e) is the (completely known) set of torsion groups occurring for infinitely many elliptic curves over degree-e fields — Mazur; Kenku–Momose, Kamienny; Jeon–Kim–Schweizer; Jeon–Kim–Park; Derickx–van Hoeij (Theorem 3: X₁(N), e = 5…8); Derickx–Sutherland (m ≥ 2, e = 5, 6, and the degree-7 statements of their introduction); Najman–Varivoda (Theorem 1.1: e = 7, 8, 9). Since these lists are closed under subgroups, infinitely many degree-e points on X₁(m,n) would force (m,n) ∈ Φ^∞(e). Conversely (m,n) ∈ Φ^∞(e) means infinitely many. When the gonality γ is known exactly, the function of degree γ gives infinitely many points of degree γ by Hilbert irreducibility (credited to the source of the gonality).
2. Frey. 2e′ < γ: a curve with infinitely many points of degree ≤ e′ has gonality ≤ 2e′. The gonality is credited to the result it follows from: the Φ^∞ classifications where they determine it, Derickx–van Hoeij (exact for N ≤ 40, upper bounds beyond), the finite-field gonality computations of Najman–Varivoda (Table 1), Abramovich's bound γ ≥ (975/98304)·[PSL₂(ℤ):Γ], or, for a few curves, Najman's unpublished computations (2026).
3. Rank zero. J₁(m,n) has rank 0 over the base field (Derickx–Sutherland, or analytic rank 0 with Kolyvagin–Logachev/Kato) and e′ < γ: then the degree-e′ divisors with infinitely many rational points would form a pencil of degree ≤ e′, contradicting the gonality (Derickx–Sutherland, Proposition 2.3 and Corollary 2.4).

Sporadic is yes when every absolute degree ≤ d is finite, no when some absolute degree ≤ d is infinite, else maybe. Infinitely many points of degree d is the answer for e = d.

For 41 ≤ N ≤ 50, the exact ℚ-gonalities of X₁(N) are now supplied by Varivoda (2026), Theorem D, which proves the earlier upper bounds of Derickx–van Hoeij sharp. For these levels other than N = 43, the Jacobian has rank 0 over ℚ (Derickx–van Hoeij, Lemma 1(3)), so the least degree with infinitely many points equals the gonality: every point below it is sporadic, and every point at or above it is not sporadic. Such a point at a larger degree may still be isolated.

Isolated is yes when absolute degree d is finite, or when the point is P¹-isolated and the Jacobian has rank 0 over the base field. P¹-isolation is decided by pipeline/magma/isolation_lib.m: for a prime q of good reduction (for the curve, the point and the residue field) the point is reduced to a divisor D_q of the same degree as x (that is, d/φ(m)) on Sutherland's model of X₁(m,n) over 𝔽_q (mdmagma), and dim L(D_q) is computed; by upper semicontinuity dim L(x) ≤ dim L(D_q), so dim L(D_q) = 1 for one such prime proves P¹-isolation. When dim L(D_q) ≥ 2 for every prime tried the point probably moves in a pencil, but this is not a proof, so the answer stays maybe (and the point is not accepted if its degree has infinitely many points). When the Jacobian has positive rank, P¹-isolation alone does not decide the question; the census then relies on a published proof of AV-isolation for that specific point, recorded in pipeline/knowledge.py (CURATED_POINTS) with its citation — currently the degree-18 point of X₁(37) (Theorem 48 of the Derickx–van Hoeij appendix to Bourdon–Hashimoto–Keller–Klagsbrun–Lowry-Duda–Morrison–Najman–Shukla).

Each point page credits separately who found the point and who proved it sporadic or isolated: a published proof that preceded this census is named with its year and reference (recorded in the certificate as credits; e.g. the degree-9 point of X₁(28) was found by González-Jiménez–Najman and proved isolated by Bourdon–Gill–Rouse–Watson); otherwise the answer is credited to the cited results it follows from, or to this census's own computation of dim L(x) = 1. Submitters can name the provers on the submission form.

Every fact is credited to its first appearance in print (the paper that proved it), not to the table or survey it was transcribed from; facts that have not appeared in print are credited to "Najman 2026 (unpublished)". The curated facts are in pipeline/knowledge.py; the tables are copied verbatim into data/knowledge/sources/ with their provenance. Genus and index of every curve were computed with Magma (CongruenceSubgroup([n,n,m])). Nothing on this site is asserted from memory: if a fact is not in a cited source or a stored computation, the answer is "maybe".

Sources

Data last built: ….

Workflow and reproducibility

Submissions arrive as GitHub issues (or pull requests). A script on Mordell pulls them, runs the Magma verifier under a time and memory limit, writes one JSON certificate per accepted point to data/points/ together with the Magma log in data/logs/, rebuilds data/curves.json and data/points.json, and pushes to GitHub, where this page is served. Each certificate records the Magma version, the CPU time, the SHA-256 of the verification code, the primes and dimensions of the isolation check, and a Magma snippet that reproduces the point. The whole repository, including rejected submissions with the reason, is public.

How to cite

Please cite the original reference of a point (given on its page) and, for the census itself, the repository x1mn-sporadic/x1mn-sporadic.github.io. The seed data are Mark van Hoeij's low-degree places on X₁(N) (arXiv:1202.4355). The design of this site follows the Elliptic Curve Rank Leaderboard.

Acknowledgements

I thank Abbey Bourdon and Maarten Derickx for helpful suggestions.