First, let me formulate the question in the language of elliptic curves. Almost every elliptic curve has a Weierstrass normal form $y^2=x^3+px+1$, but this form is bad from the point of view of invariant theory, and I have heard that there is another normal form written in terms of symmetric polynomials.
Second, from the point of view of invariant theory, the group $\operatorname{SL}_3$ acts standardly on $S^3\mathbb C^3$, there is an open invariant subset $W \subset S^3\mathbb C^3$ and some subset $$U=\{ y^2z-x^3-pxz^2-qz^3 \} \subset S^3\mathbb C^3,$$ such that any $\operatorname{SL}_3$-orbit in $W$ intersects $U \cap W$ exactly once. Namely, $W$ consists of polynomials that define smooth irreducible curve in $\mathbb{CP}^3$, the intersection $U \cap W$ is defined by non-vanishing of the discriminant of $x^3+px+q$, and its elements are called Weierstrass normal forms.
Then the invariants $p$ and $q$ can be defined via some complicated contractions of tensors involving many indices, but it gives little insight. Could you help me with an anologous normal form, but involving symmetric polynomials, and its invariants? I am in particular interesting in the case of $\operatorname{SL}_3$, not just $\operatorname{GL}_3$. It is probably very classical subject, but I can not find the answer, so any reference is welcome!