  Using Semaev's summation polynomials, we derive a new equation for the
$\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve
defined over $\mathbb{F}_q$. Using this equation, we produce an optimal-size
representation for such points. Our representation is compatible with scalar
multiplication. We give a point compression algorithm to compute the
representation and a decompression algorithm to recover the original point (up
to some small ambiguity). The algorithms are efficient for trace zero varieties
coming from small degree extension fields. We give explicit equations and
discuss in detail the practically relevant cases of cubic and quintic field
extensions.
