Question about some term in Sage while using GF(9)

Question

I tried to define an elliptic curve over $GF(9)$ in Sage, and some term $z2$ appeared, see below (click on the image if the font is too small):

I know that it has something to do with the definition of $GF(9)$ - probably it describes how it works as a $GF(3)$-space.

However, I do not know how to access the necessary information about $z2$ (computing its minimal polynomial does not work as well). Could you please explain me what $z2$ is? Thank you!

Answer 1

You can specify the name for the root of an irreducible polynomial to be used in construction of $GF(9)$, e.g.:

sage: G = GF(9,'a')

You can also get the polynomial used:

sage: G.modulus()
x^2 + 2*x + 2

And then

sage: E2 = EllipticCurve(G, [-1,0]) 
sage: E2
Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field in a of size 3^2
sage: E2.rational_points()
[(0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 1), (2 : 0 : 1), (a : a : 1), (a : 2*a : 1), (a + 1 : a : 1), (a + 1 : 2*a : 1), (a + 2 : a : 1), (a + 2 : 2*a : 1), (2*a : a + 2 : 1), (2*a : 2*a + 1 : 1), (2*a + 1 : a + 2 : 1), (2*a + 1 : 2*a + 1 : 1), (2*a + 2 : a + 2 : 1), (2*a + 2 : 2*a + 1 : 1)]
sage: E2.cardinality()
16