ag.algebraic geometry - Square of an elliptic curve and projective plane

Let's assume one takes $E = mathbb{C}^* / langle p rangle$ an elliptic (Tate) curve over the complex field ($p = e^{2 pi i tau}$ where $1, tau$ are the 2 periods in additive notation; $Im tau > 0$). On this take points $u_1, u_2, u_3$ such that $u_1 u_2 u_3 = 1$ and then mod out by the action of the symmetric group $S_3$. So we essentially have a hypersurface in $E^3$ - a copy of $E^2$ with coordinates $(u_1, u_2)$ and we mod out by permuting $u_1, u_2$ and $1/u_1 u_2$ (the $u_i$'s are zeros and their reciprocals poles of an elliptic function - essentially the only one up to constant with these zeros and poles).

The question: is this quotient space $mathbb{P}^2$? I believe the answer is yes, but I can't see a way of using theta functions or other gadgets to explicitly give the isomorphism (whereby a theta function I mean $$theta_p(x) = prod_{l ge 0}(1-p^l x)(1-p^{l+1}/x)$$ which reduces to the Jacobi theta via the triple product identity).

Finally, does this work over other fields (reals, finite fields, other reasonable fields)?