Which modular curves X₁(m,n) have points of unexpectedly small degree?
A point on X₁(m,n) whose residue field has degree d over ℚ is an elliptic curve over a number field of degree d whose torsion subgroup contains ℤ/m ⊕ ℤ/n (for m ≥ 3 this is the absolute degree; the curve is defined over ℚ(ζm), over which the point has degree d/φ(m)). The point is sporadic when the curve has only finitely many points of degree ≤ d, and isolated when it belongs to no positive-dimensional family of degree-d points. This census collects sporadic and isolated points, re-verifies every submission with Magma, and records the result that decides each question. Submit a point →
The curves
Prefer to browse by degree? All sporadic and isolated points grouped by degree →
Sorted by m, then n; X₁(1,n) = X₁(n). Click a curve for its points and for what is known about it. Finite and infinite list the degrees in which the curve is known to have finitely, respectively infinitely, many points (absolute degrees for m ≥ 3; hover for the source; see about for the conventions).
| curve | genus | gonality | rk J₁ | finite in degree | infinite in degree | points | in degrees |
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