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Torsion of elliptic curves over number fields of degree 66, 77, 88 and 99

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Filip Najman

Version 2 / Nov 04, 2026 / CC BY 4.0

Abstract

We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree dd for d=6d=6, 77, 88 and 99. Apart from the groups that occur infinitely often, exactly 44 groups occur in degree 66, 33 in degree 77, 88 in degree 88 and 99 in degree 99, and among them the groups Z/2Z×Z/24Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} over sextic fields, Z/3Z×Z/15Z\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/15\mathbb{Z} over octic fields and Z/2Z×Z/26Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/26\mathbb{Z} over nonic fields were not known to occur. For 1313 of these 2424 pairs of a degree and a group we also determine all the elliptic curves that realize them. The proofs combine the global method of Derickx, Hecke sieves, formal immersion criteria at the prime 22 with Hecke operators, and Riemann-Roch computations on modular curves and their quotients over finite fields.

Author comments: Fixed ten references to equations (6) and (10), which appeared as "??" in v1 (compiled with TeX Live 2025). No mathematical changes.

Provenance statement

The mathematical content of this paper was produced by AI models. OpenAI's GPT-6.1 Sol produced almost all of the mathematics of the first versions of the proofs, separately for degrees 6, 7, 8 and 9, and carried out most of the computations, starting in mid-October 2026. Anthropic's Claude Opus 5.5 reviewed these versions, independently recomputed some of the computations and the explicit examples, merged the proofs into one paper, wrote the manuscript and organized the code repositories. The merged paper was also reviewed by independent AI agents (Claude Opus 5.5, ChatGPT-6.1 Sol and ChatGPT-6 Astra). I posed the questions, guided the work and made the editorial decisions, and I worked to keep the presentation understandable and well organized. I have carefully checked, understood and rewritten the degree 6 part; that version will appear separately on arXiv. Mathematically, the present paper should be regarded as raw data: I have not yet fully checked the parts on degrees 7, 8 and 9, and its degree 6 part differs in places from the checked version.

Tools used

OpenAI
ChatGPTVersion 6.1 Sol
Anthropic
Claude OpusVersion 5.5
Magma
MagmaVersion c2.30-1

References

  1. Abramovich, Dan. A linear lower bound on the gonality of modular curves. Internat. Math. Res. Notices, no. 20, pp. 1005–1011. 1996DOI
  2. Bosma, Wieb; Cannon, John; Playoust, Catherine. The Magma algebra system. I. The user language. J. Symbolic Comput., vol. 24, no. 3-4, pp. 235–265. 1997. Computational algebra and number theory (London, 1993)DOI
  3. Bars, Francesc; Kontogeorgis, Aristides; Xarles, Xavier. Bielliptic and hyperelliptic modular curves X(N)X(N) and the group Aut(X(N)){\rm Aut}(X(N)). Acta Arith., vol. 161, no. 3, pp. 283–299. 2013DOI
  4. Peter Bruin; Filip Najman. A criterion to rule out torsion groups for elliptic curves over number fields. Res. Number Theory, vol. 2, pp. Paper No. 3. 2016. 13 pp.DOI
  5. Box, Josha. Quadratic points on modular curves with infinite Mordell–Weil group. Math. Comp., vol. 90, no. 327, pp. 321–343. 2021DOI
  6. Bourdon, Abbey; Pollack, Paul. Torsion subgroups of CM elliptic curves over odd degree number fields. Int. Math. Res. Not. IMRN, no. 16, pp. 4923–4961. 2017
  7. Bruin, Nils; Stoll, Michael. The Mordell–Weil sieve: proving non-existence of rational points on curves. LMS J. Comput. Math., vol. 13, pp. 272–306. 2010DOI
  8. Clark, Pete L.; Corn, Patrick; Rice, Alex; Stankewicz, James. Computation on elliptic curves with complex multiplication. LMS J. Comput. Math., vol. 17, no. 1, pp. 509–535. 2014DOI
  9. Clark, Pete L.; Genao, Tyler; Pollack, Paul; Saia, Frederick. The least degree of a CM point on a modular curve. J. Lond. Math. Soc. (2), vol. 105, no. 2, pp. 825–883. 2022DOI
  10. Derickx, Maarten; Etropolski, Anastassia; van Hoeij, Mark; Morrow, Jackson S.; Zureick-Brown, David. Sporadic cubic torsion. Algebra Number Theory, vol. 15, no. 7, pp. 1837–1864. 2021DOI
Show all 58 referencesShow fewer references
  1. Daniels, Harris B.; González-Jiménez, Enrique. On the torsion of rational elliptic curves over sextic fields. Math. Comp., vol. 89, no. 321, pp. 411–435. 2020DOI
  2. Diamond, Fred; Im, John. Modular forms and modular curves. Seminar on Fermat's Last Theorem (Toronto, ON, 1993–1994), CMS Conf. Proc., vol. 17, pp. 39–133. 1995
  3. Derickx, Maarten; Kamienny, Sheldon; Stein, William; Stoll, Michael. Torsion points on elliptic curves over number fields of small degree. Algebra Number Theory, vol. 17, no. 2, pp. 267–308. 2023DOI
  4. Derickx, Maarten; Najman, Filip. Classification of torsion of elliptic curves over quartic fields. J. Reine Angew. Math., vol. 829, pp. 123–156. 2025DOI
  5. Deligne, P.; Rapoport, M. Les schémas de modules de courbes elliptiques. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), vol. 349, pp. 143–316. 1973
  6. Drinfel'd, V. G. Two theorems on modular curves. Funktsional. Anal. i Prilozhen., vol. 7, no. 2, pp. 83–84. 1973
  7. Derickx, Maarten; Sutherland, Andrew V. Explicit models of X1(m,mn)X_1(m,mn). 2016. https://math.mit.edu/~drew/X1mn.htmllink
  8. Derickx, Maarten; Sutherland, Andrew V. Torsion subgroups of elliptic curves over quintic and sextic number fields. Proc. Amer. Math. Soc., vol. 145, no. 10, pp. 4233–4245. 2017DOI
  9. Derickx, Maarten; Stoll, Michael. Prime order torsion on elliptic curves over number fields. Part I: Asymptotics. 2025. Preprint, arXiv:2505.14109v1
  10. — , Personal communication, 2026.
  11. Derickx, Maarten; van Hoeij, Mark. Gonality of the modular curve X1(N)X_1(N). J. Algebra, vol. 417, pp. 52–71. 2014DOI
  12. Faltings, Gerd. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math., vol. 73, no. 3, pp. 349–366. 1983DOI
  13. Frey, Gerhard. Curves with infinitely many points of fixed degree. Israel J. Math., vol. 85, no. 1-3, pp. 79–83. 1994DOI
  14. González-Jiménez, Enrique; Najman, Filip. An algorithm for determining torsion growth of elliptic curves. Exp. Math., vol. 32, no. 1, pp. 70–81. 2023DOI
  15. Hartshorne, Robin. Algebraic Geometry. Springer-Verlag, New York-Heidelberg, vol. 52. 1977
  16. Jeon, Daeyeol; Kim, Chang Heon; Park, Euisung. On the torsion of elliptic curves over quartic number fields. J. London Math. Soc. (2), vol. 74, no. 1, pp. 1–12. 2006DOI
  17. Jeon, Daeyeol; Kim, Chang Heon; Schweizer, Andreas. On the torsion of elliptic curves over cubic number fields. Acta Arith., vol. 113, no. 3, pp. 291–301. 2004DOI
  18. Kamienny, Sheldon. Torsion points on elliptic curves and qq-coefficients of modular forms. Invent. Math., vol. 109, no. 2, pp. 221–229. 1992DOI
  19. Kamienny, Sheldon. Torsion points on elliptic curves over fields of higher degree. Internat. Math. Res. Notices, no. 6, pp. 129–133. 1992DOI
  20. Katz, Nicholas M. Galois properties of torsion points on abelian varieties. Invent. Math., vol. 62, no. 3, pp. 481–502. 1981DOI
  21. Kato, Kazuya. pp-adic Hodge theory and values of zeta functions of modular forms. Astérisque, no. 295, pp. ix, 117–290. 2004. Cohomologies pp-adiques et applications arithmétiques. III
  22. Kenku, M. A. The modular curves X0(65)X_0(65) and X0(91)X_0(91) and rational isogeny. Math. Proc. Cambridge Philos. Soc., vol. 87, no. 1, pp. 15–20. 1980DOI
  23. Kenku, M. A. On the number of Q{\bf Q}-isomorphism classes of elliptic curves in each Q{\bf Q}-isogeny class. J. Number Theory, vol. 15, no. 2, pp. 199–202. 1982
  24. Khawaja, Maleeha. Torsion primes for elliptic curves over degree 88 number fields. Res. Number Theory, vol. 10, no. 2, pp. Paper No. 48. 2024
  25. Kim, Henry H. Functoriality for the exterior square of GL4{\rm GL}_4 and the symmetric fourth of GL2{\rm GL}_2. J. Amer. Math. Soc., vol. 16, no. 1, pp. 139–183. 2003. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Henry H. Kim and Peter SarnakDOI
  26. Kenku, M. A.; Momose, Fumiyuki. Torsion points on elliptic curves defined over quadratic fields. Nagoya Math. J., vol. 109, pp. 125–149. 1988DOI
  27. Kubert, Daniel Sion. Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. (3), vol. 33, no. 2, pp. 193–237. 1976DOI
  28. Liu, Qing; Lorenzini, Dino. Models of curves and finite covers. Compositio Math., vol. 118, pp. 61–102. 1999
  29. The L-functions and modular forms database. 2026. The LMFDB Collaboration; online, accessed September 2026. https://www.lmfdb.orglink
  30. Manin, Ju. I. Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR Ser. Mat., vol. 36, pp. 19–66. 1972
  31. Mazur, Barry. Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math., no. 47, pp. 33–186 (1978). 1977. With an appendix by Mazur and M. Rapoport
  32. Mazur, B. Rational isogenies of prime degree. Invent. Math., vol. 44, no. 2, pp. 129–162. 1978
  33. mdmagma. M. Derickx and A. V. Sutherland, \textttMagma package. https://github.com/nt-lib/mdmagmalink
  34. Merel, Loïc. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math., vol. 124, no. 1-3, pp. 437–449. 1996DOI
  35. Najman, Filip. Torsion of rational elliptic curves over cubic fields and sporadic points on X1(n)X_1(n). Math. Res. Lett., vol. 23, no. 1, pp. 245–272. 2016DOI
  36. Najman, Filip. Classification of torsion of elliptic curves over quintic fields. 2026. Preprint, https://arxiv.org/abs/2609.12846link
  37. Najman, Filip. Isolated and sporadic points on X1(m,n)X_1(m,n). 2026. https://x1mn-sporadic.github.iolink
  38. Najman, Filip. Points of prime order on elliptic curves over number fields of degree 99 to 2222. 2026. Preprint, https://github.com/F-Najman/Sdlink
  39. Najman, Filip; Varivoda, Marin. Torsion groups of elliptic curves that appear infinitely often over septic, octic and nonic fields. 2026. Preprint, https://arxiv.org/abs/2602.03513link
  40. Oda, Tadao. The first de Rham cohomology group and Dieudonné modules. Ann. Sci. École Norm. Sup. (4), vol. 2, pp. 63–135. 1969
  41. Parent, Pierre. Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. Reine Angew. Math., vol. 506, pp. 85–116. 1999DOI
  42. Parent, Pierre. Torsion des courbes elliptiques sur les corps cubiques. Ann. Inst. Fourier (Grenoble), vol. 50, no. 3, pp. 723–749. 2000DOI
  43. Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer, Dordrecht, vol. 106. 2009
  44. Sutherland, Andrew V. Magma intrinsics for subgroups of GL2(Z/NZ)\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z}) and modular curves. \textttMagma package; point counting in \textttgl2points.m. https://github.com/AndrewVSutherland/Magmalink
  45. Sutherland, Andrew V. Constructing elliptic curves over finite fields with prescribed torsion. Math. Comp., vol. 81, no. 278, pp. 1131–1147. 2012DOI
  46. Sutherland, Andrew V. Optimized equations for X1(N)X_1(N). 2012. https://math.mit.edu/~drew/X1_altcurves.htmllink
  47. van Hoeij, Mark. Low degree places on the modular curve X1(N)X_1(N). 2014. Preprint, https://arxiv.org/abs/1202.4355; data at https://www.math.fsu.edu/~hoeij/files/X1N/link
  48. Waterhouse, William C. Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4), vol. 2, pp. 521–560. 1969DOI

Version history

  1. v1 · 99 pagesSubmitted by Filip NajmanInitial depositSupersededNov 03, 2026
  2. v2 · 99 pagesSubmitted by Filip NajmanFixed ten references to equations (6) and (10), which appeared as "??" in v1 (compiled with TeX Live 2025). No mathematical changes.CurrentNov 05, 2026