Negative of elliptic curve points

Question

F₈ is defined as:

 0    000
 1    100
 β1   010
 β2   001
 β3   110
 β4   011
 β5   111
 β6   101

What type of elliptic curve model is meant by y² + xy = x³ + β²x² + β⁶ over F₈?

What are the negatives of the following points: P = (β³, β⁶), Q = (0, β³), R = (β, 0) ?

Answer 1

This is generalized Weierstrass form, with the restriction that you have specified. There’s no name for it, far as I know. In all cases, you get the negative of a point $P$ in the finite plane by finding the other point with the same $x$-coordinate.