1 | initial version |
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.