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

2026/07/16 by Yves Aubry, Roger Oyono, Christelle Vincent
#math.NT

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Abstract

Let E be a supersingular elliptic curve defined over \mathbbF_p and E^(p) be its conjugate. We give a bound on the minimal degree of an isogeny from E to E^(p) depending on p, 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 p for which the bound we give in this article is attained.

Related

• Computing isogenies between supersingular elliptic curves over F_p
• Gross lattices of supersingular elliptic curves
• Isogenies of elliptic curves over function fields
• Higher-degree supersingular group actions
• Computing Isogenies at Singular Points of the Modular Polynomial