English

Minimal degree of an isogeny between a supersingular elliptic curve and its conjugate

Number Theory 2026-07-16 v1

Abstract

Let EE be a supersingular elliptic curve defined over Fˉp\bar{\mathbb{F}}_p and E(p)E^{(p)} be its conjugate. We give a bound on the minimal degree of an isogeny from EE to E(p)E^{(p)} depending on pp, and show that this bound is both asymptotically optimal as well as sharp in many cases. This bound is obtained by developing a new technique to compute the degree of certain isogenies from a supersingular elliptic curve to its conjugate, and we present extensive computations of the successive minima of the lattice containing these isogenies. Following this, we give several conjectures supported by the data we have obtained, including some on the set of primes pp for which the bound we give in this article is attained.

Keywords

Cite

@article{arxiv.2607.14624,
  title  = {Minimal degree of an isogeny between a supersingular elliptic curve and its conjugate},
  author = {Yves Aubry and Roger Oyono and Christelle Vincent},
  journal= {arXiv preprint arXiv:2607.14624},
  year   = {2026}
}

Comments

accompanying code available at https://github.com/christellevincent/WISDE