# How to work with points in GF(p**2) for a prime p

I'm constructing elliptic curves over fields of size $p^{2}$ for some prime $p$, and I want to do arithmetic with points on these curves. I think my question is best asked with an example: Say I put

```
p=431
k=GF(p**2)
E=EllipticCurve(k,[0,329,0,1,0])
E
```

Sage will say

```
Elliptic Curve defined by y^2 = x^3 + 329*x^2 + x over Finite Field in z2 of size 431^2
```

I understand that `z2`

in this case is the root of an irreducible polynomial used in the construction of `GF(p**2)`

. Say I want to use sage to figure out what $[2] (272*z2 + 405 : 167*z2 + 52 : 1)$ is. This is what I would type into sage:

```
2*E(272*z2 + 405, 167*z2 + 52, 1)
```

But if I do that, I get an error saying that z2 is undefined. I've also gotten the error message

```
unsupported operand parent(s) for +: '<class 'tuple'>' and 'Integer Ring'
```

I'm not sure how to get sage to treat `z2`

as the root of the irreducible polynomial. It seems to think of it as a variable that should have a value assigned to it. Any help would be greatly appreciated.