Points by degree
The sporadic and isolated points of the census grouped by absolute degree [ℚ(x) : ℚ], the degree of the number field over which the elliptic curve and its torsion are defined: all points of absolute degree 3, then 4, and so on. For m ≤ 2 this is the degree of the point; for m ≥ 3 the degree of the point is its degree over ℚ(ζm), the absolute degree divided by φ(m), and is given on the curve and point pages. Within a degree the points are ordered by curve. The three answers are explained on the about page; hover over an answer for the reason.
Completeness of the sporadic points in degrees ≤ 10
Let Φ(d) be the set of torsion groups E(K)tors of elliptic curves over number fields of degree d, and Φ∞(d) ⊆ Φ(d) the groups that occur for infinitely many ℚ̄-isomorphism classes of curves. A group in Φ(d) \ Φ∞(d) is a sporadic torsion group in degree d. This is a property of a group, not of a point, but the two are related. If x is a non-cuspidal sporadic point of absolute degree d on X₁(m,n), with elliptic curve E over the residue field K = ℚ(x), then E(K)tors ⊇ ℤ/m ⊕ ℤ/n is a sporadic torsion group in degree d: otherwise the infinitely many curves with this torsion group would give infinitely many points of degree at most that of x on X₁(m,n). For the same reason, a group in Φ∞(e) for some e < d gives no sporadic points in degree d. So where all the elliptic curves with a sporadic torsion group are known, all the sporadic points they give are known, and the census contains every one of them wherever the table says "all known". The table records how far the classifications of Φ(d) get; the sets Φ∞(d) for d ≤ 9 are due to Jeon–Kim–Schweizer, Jeon–Kim–Park, Derickx–Sutherland and Najman–Varivoda. Nothing here concerns isolated points that are not sporadic.
| degree | classification | sporadic torsion groups | sporadic points of this absolute degree |
|---|---|---|---|
| 1 | Mazur 1977 | none | none |
| 2 | Kenku–Momose, Kamienny | none | none |
| 3 | DEvHMZB 2021 | ℤ/21, realized only by 162.c3 over ℚ(ζ₉)⁺ (Najman 2016) | all known: 1.21.3.a |
| 4 | Derickx–Najman 2025 | none | none |
| 5 | Najman 2026 (5) | ℤ/28, ℤ/30, ℤ/2 ⊕ ℤ/18; all curves known (1, 2 and 1) | all known: 1.28.5.a, 1.30.5.a, 1.30.5.b. The ℤ/2 ⊕ ℤ/18 point 2.18.5.a is isolated but not sporadic, since X₁(2,18) has infinitely many points of degree 4 |
| 6 | Najman 2026 (6–9) | ℤ/25, ℤ/37, ℤ/4 ⊕ ℤ/12, ℤ/2 ⊕ ℤ/24; all curves known for ℤ/37 and ℤ/2 ⊕ ℤ/24 | known except on X₁(4,12): 1.37.6.a, 2.24.6.a, 4.12.6.a; further sextic curves with torsion ℤ/4 ⊕ ℤ/12 are not excluded. ℤ/25 gives no sporadic points in degree 6 (X₁(25) has infinitely many points of degree 5) |
| 7 | Najman 2026 (6–9) | ℤ/25, ℤ/33, ℤ/36; all curves known (1, 1 and 2) | all known: 1.33.7.a, 1.36.7.a, 1.36.7.b. The ℤ/25 point 1.25.7.a is isolated but not sporadic |
| 8 | Najman 2026 (6–9) | ℤ/33, ℤ/34, ℤ/35, ℤ/39, ℤ/40, ℤ/42, ℤ/3 ⊕ ℤ/15, ℤ/5 ⊕ ℤ/10; all curves known except for ℤ/42 and ℤ/5 ⊕ ℤ/10 | all known on X₁(33), X₁(34), X₁(35), X₁(39), X₁(40) and X₁(3,15); on X₁(42) and X₁(5,10) there may be further ones besides 1.42.8.a–c and 5.10.8.a |
| 9 | Najman 2026 (6–9) | ℤ/29, ℤ/31, ℤ/32, ℤ/33, ℤ/34, ℤ/39, ℤ/40, ℤ/42, ℤ/2 ⊕ ℤ/26; all curves known for ℤ/33 and ℤ/2 ⊕ ℤ/26 | all known on X₁(33) and X₁(2,26); there may be further ones on X₁(29), X₁(31), X₁(34), X₁(39), X₁(40) and X₁(42). ℤ/32 gives no sporadic points in degree 9 (X₁(32) has infinitely many points of degree 8) |
| 10 | Najman 2026 (10); Φ∞(10) is not known | ℤ/2 ⊕ ℤ/26, ℤ/3 ⊕ ℤ/15, ℤ/3 ⊕ ℤ/18, ℤ/6 ⊕ ℤ/12, with all curves known; for 15 further groups of Φ(10) it is not known whether they occur infinitely often: ℤ/28, ℤ/30, ℤ/2 ⊕ ℤ/18, ℤ/2 ⊕ ℤ/28, ℤ/2 ⊕ ℤ/30 and ℤ/N for N = 29, 31, 35, 37, 38, 39, 40, 42, 45, 50 | all known on X₁(2,26), X₁(3,15), X₁(3,18) and X₁(6,12); there may be further ones on X₁(2,28), X₁(2,30) and X₁(N) for N = 29, 31, 35, 37, 38, 39, 40, 42, 45, 50. ℤ/28, ℤ/30 and ℤ/2 ⊕ ℤ/18 give no sporadic points in degree 10 (they lie in Φ∞(6), Φ∞(6) and Φ∞(4)) |