Which modular curves X₁(m,n) have points of unexpectedly small degree?
A point of degree d on X₁(m,n) is an elliptic curve over a number field of degree d whose torsion subgroup contains ℤ/m ⊕ ℤ/n. 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 (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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