How to construct an isogeny [i] such that [i]^2= -1?

asked 2022-01-31 15:22:54 +0100

DrewC
57 ●2 ●7

Let $E: y^2 = x^3 + x$ be an elliptic curve over a field $K$ of characteristic $p\neq 2, 3$. It is well known that the map $[i]$ defined as below is an endomorphism on $E$ and $[i]^2=-1$

$[i]:(x,y)\mapsto(-x, iy)$

I'm wondering how to construct this isogeny on sage when $K$ is a finite field. If I can define an isogeny by giving its rational maps then I can simply compute a fourth root of unity $i$ in $K$ and let $\phi = (-x, iy)$, but as I know of sage doesn't let you define an isogeny from rational maps?

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answered 2022-02-01 12:43:29 +0100

Luca
890 ●6 ●21

You can use the .automorphisms() method to get all the automorphisms of $E$.

sage: E = EllipticCurve(GF(13), [1, 0])
sage: Aut = E.automorphisms()
sage: Aut
[Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 13
   Via:  (u,r,s,t) = (1, 0, 0, 0),
 Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 13
   Via:  (u,r,s,t) = (5, 0, 0, 0),
 Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 13
   Via:  (u,r,s,t) = (8, 0, 0, 0),
 Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 13
   Via:  (u,r,s,t) = (12, 0, 0, 0)]

Your map $[i]$ would be the second or the third map above, depending on your choice for $i ∈ 𝔽_p$. Automorphisms in Sage (currently version 9.4) do not inherit from isogenies, so they lack some of their methods (e.g., degree), and you cannot compose them with other isogenies, so it's a bit annoying to work with them. However the isogeny package is evolving quickly these days, so it's possible that future versions will handle these things better.