Isolated and sporadic points on X₁(m,n)

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 whose non-cuspidal points are the isomorphism classes of triples (E, P, Q) with ⟨P, Q⟩ ≅ ℤ/m ⊕ ℤ/n, P of order m and Q of order n. It is defined over ℚ(ζ_m) and geometrically connected there. X₁(1,n) = X₁(n). (Derickx–Sutherland write the same curve as X₁(m, mn); here n is always the full cyclic order.)
the degree: The degree of a closed point x is its absolute degree [ℚ(x) : ℚ]. For m ≥ 3 this is φ(m) times the degree over the base field ℚ(ζ_m); both are shown. A point of 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). 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 (Bourdon–Ejder–Liu–Odumodu–Viray). 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: A closed point x of degree d is P¹-isolated if dim L(x) = 1, i.e. it does not move in a pencil of degree-d divisors, and AV-isolated if no positive-rank abelian subvariety of the Jacobian moves it (automatic when J₁(m,n) has rank 0 over the base field); isolated means both. A point that is not isolated lies in a positive-dimensional family of degree-d points, so finitely many points of degree d implies isolated, and sporadic implies isolated.
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 degree d.

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

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

Isolated is yes when 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 degree d 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.