Isolated and sporadic points on X₁(m,n)

Submit a point

Found an elliptic curve with a point of order n (and full m-torsion) over a number field of unusually small degree? Submit it here. Every submission is re-verified with Magma on the Mordell workstation; if the point is genuine and a recorded theorem or a Riemann–Roch computation shows that it is sporadic or isolated, it is added to the census with its certificate and logs.

1. Fill in the form

Submissions are GitHub issues created from a form (a free GitHub account is all you need):

Open the submission form ↗

You give the pair (m, n), the number field, the elliptic curve, and (recommended) the torsion points. Use Magma-style expressions: the field is defined by a polynomial in x, and elements of the field are written as polynomials in a, a root of that polynomial, e.g. (3*a^2 - 2*a + 1)/7. Only digits, a/x/y, + - * / ^ ( ) and spaces are allowed — nothing else is ever evaluated.

The curve can be given in one of three ways:

a-invariants [a1, a2, a3, a4, a6] over the field, ideally with the coordinates of Q (order n) and, for m > 1, of P (order m). Without the points, Magma has to compute the full torsion subgroup, which can take a long time for fields of degree ≳ 8.
Tate normal form b, c: the curve y² + (1 − c)xy − by = x³ − bx² with Q = (0,0) of order n; give P for m > 1.
van Hoeij's format: eqx (the field polynomial) and eqxy, as in his LowDegreePlaces file (m = 1 only).

Please say who discovered the point and when (these are shown in the tables), and add a reference (paper, arXiv number, or "new"). If someone has already proved that the point is sporadic or isolated, name them too ("names (year), reference" in the fields Proved sporadic by / Proved isolated by): the point page credits the discovery and these proofs separately. Left empty, the proofs are credited to the cited results or to the census's own computation.

2. What happens next

The issue is labelled submission and waits in the queue (the home page shows how many are waiting).
Periodically the queue is pulled to Mordell. For each submission Magma checks that the field and curve are well defined, that Q has exact order n and P exact order m with ⟨P,Q⟩ ≅ ℤ/m ⊕ ℤ/n, bounds the torsion subgroup by reductions, computes the Tate normal form of (E, Q), and determines the residue field ℚ(b, c, x(P), y(P)) of the point on X₁(m,n) — its degree is the degree of the point, whatever field the curve was submitted over.
A second Magma job decides whether the point is P¹-isolated (a Riemann–Roch computation on a model of the curve modulo a good prime; m ≤ 2), and the three questions — sporadic? isolated? infinitely many points of this degree? — are answered yes / no / maybe from the curated table of what is known about X₁(m,n) (gonality, rank of the Jacobian, the sets Φ^∞(d), Frey's theorem; see about). The outcome is certified (sporadic or isolated proven), verified · undecided (genuine point, degree undecided; listed with that label), or rejected (the data do not define such a point, it is a duplicate, or it lies in a degree with infinitely many points without being proven isolated).
The result is posted as a comment on the issue, the issue is closed, and accepted points appear on this site with their certificate and Magma log. If a submission is rejected because of a data problem, just fix it and submit again.

Limits of the automatic verification

Magma time on the server is limited, so a submission is verified automatically only when it is cheap: the number field must have degree at most 30; when the degree exceeds 8 the torsion points Q (and P) must be given, since otherwise Magma would have to compute the full torsion subgroup; the isolation check is run only for curves of genus at most 60; each Magma job runs under a time limit (10 minutes for the verification, 15 for the isolation check); and each verification cycle processes at most 8 submissions and about 45 minutes of Magma time, the rest waiting for the next cycle. A submission beyond these limits is not lost: the issue is labelled manual and the maintainer runs it by hand.

Bulk submissions

For many points at once, open a pull request adding files to submissions/inbox/ of the repository, one JSON file per point:

{
  "schema": "x1mn-sporadic/submission/1",
  "m": 2, "n": 18,
  "field": "x^5 - 2*x^4 + 2*x^3 - 3*x^2 - x + 4",
  "curve": {"ainvs": ["0", "a^2 - 1", "0", "(-3*a + 2)/5", "1"]},
  "points": {"P": ["0", "0"], "Q": ["a", "2*a - 1"]},
  "degree": 5,
  "submitter": "Your Name", "affiliation": "Your institution", "github": "your-handle",
  "discoverer": "J. Doe", "year": 2026,
  "reference": "arXiv:XXXX.XXXXX, Theorem 3",
  "credits": {"isolated": {"by": "A. Author, B. Author", "year": 2025, "reference": "arXiv:YYYY.YYYYY, Theorem 1"}},
  "notes": "anything the curators should know"
}

"credits" (optional) names who proved the point sporadic and/or isolated, when that was done before this census.

Instead of "ainvs" you may give "curve": {"tate": {"b": "...", "c": "..."}} or "curve": {"vanhoeij": {"eqxy": "..."}}. The example values above are placeholders, not a real point.

Which curves?

The census currently lists X₁(n) for n ≤ 100, X₁(2,n) for n ≤ 60, and X₁(m,n) for 3 ≤ m ≤ 10 and the n for which Derickx–Sutherland's models exist (see the home page). If your point lies on a curve outside this list, open an issue and it will be added.