Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields
Abstract
For any primes p ≠ ℓ, ℓ-isogeny graphs of ordinary elliptic curves defined over 𝔽p2 have a typical structure called ℓ-volcanoes, and the structure is the core of Sutherland's supersingularity testing algorithm for elliptic curves. In this paper, by exploiting the properties of $3$-isogenies between twisted Hessian curves, we show that when p ≡ 2 pmod3 and ℓ = 3, every ordinary twisted Hessian curve defined over 𝔽p lies on the surface of the $3$-volcano. As an application, we give an improved version of Sutherland's supersingularity testing algorithm specialized to twisted Hessian curves defined over 𝔽p with p ≡ 2 pmod3. We also give a generalization of the known fact that any supersingular j-invariant is a cube in 𝔽p2; we show that for any twisted Hessian curve H(a,d) defined over 𝔽p2, its j-invariant is not a cube in 𝔽p2 if and only if H(a,d) is ordinary and lies on the floor of a $3$-volcano.
Related
- Non-critical p-adic twisted adjoint L-function for ordinary families on definite unitary groups
- Supersingularity and Superspeciality Verification of Abelian Surfaces
- Recovering curves from isogeny classes of Prym varieties
- Faltings' Isogeny Theorem via Equidistribution
- Isogeny rigidity under algebraic group actions and finite translation orbits
BibTeX
@misc{hashimoto2026ordinary,
title = {{Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields}},
author = {Yuji Hashimoto and Koji Nuida},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11037},
eprint = {2609.11037},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11037},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields
AU - Yuji Hashimoto
AU - Koji Nuida
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11037
UR - https://arxiv.org/abs/2609.11037
AB - For any primes p ≠ ℓ, ℓ-isogeny graphs of ordinary elliptic curves defined over 𝔽_p^2 have a typical structure called ℓ-volcanoes, and the structure is the core of Sutherland's supersingularity testing algorithm for elliptic curves. In this paper, by exploiting the properties of $3$-isogenies between twisted Hessian curves, we show that when p ≡ 2 pmod3 and ℓ = 3, every ordinary twisted Hessian curve defined over 𝔽_p lies on the surface of the $3$-volcano. As an application, we give an improved version of Sutherland's supersingularity testing algorithm specialized to twisted Hessian curves defined over 𝔽_p with p ≡ 2 pmod3. We also give a generalization of the known fact that any supersingular j-invariant is a cube in 𝔽_p^2; we show that for any twisted Hessian curve H(a,d) defined over 𝔽_p^2, its j-invariant is not a cube in 𝔽_p^2 if and only if H(a,d) is ordinary and lies on the floor of a $3$-volcano.
ER -