Working through Ivey and Landsberg's Cartan for Beginners, I am attempting exercise 1.7.3.1, i.e. I am trying to determine special affine invariants for curves ${c}:\mathbb{R} \to \mathbb{R}^2$, that is, differential invariants for curves under the action of $ASL(2,\mathbb{R}^2)$, which is the group of transformations we get by composing a member of $SL(2,\mathbb{R})$ with a translation.
Interpreting $\mathbb{R}^2$ as a homogeneous space $ASL(2,\mathbb{R})/SL(2,\mathbb{R})$ the problem reduces to constructing a sufficiently nice lift $\tilde{c}:\mathbb{R} \to ASL(2,\mathbb{R})$ and then computing the pullback of the Maurer-Cartan form of $ASL(2,\mathbb{R})$ under $\tilde{c}$, and interpreting the non-constant terms as the invariants.
Problem
I am computing $c_1'c_2'' - c_2'c_1''$ and $\frac{c_2''}{c_2'}$ as the invariants. I'm fairly sure the first is correct, but I am expecting some combination of the first and the affine curvature instead of the second.
Work
We can represent a member of $ASL(2,\mathbb{R})$ in $GL(3,\mathbb{R})$ in the obvious way: $$ \rho(t_1,t_2,a,b,c,e) = \begin{bmatrix} 1 & 0 & 0 \\ t_1 & a & b \\ t_2 & c & e \end{bmatrix}, \ \text{with } ae-bc=1. $$
A generic lift of a curve $c$ in $\mathbb{R}^2$ (given by $c(t) = (1,c_1(t),c_2(t)^t)$ is given by $$ \tilde{c}(t) = \begin{bmatrix} 1 & 0 & 0 \\ c_1(t) & A(t) & B(t) \\ c_2(t) & C(t) & E(t) \end{bmatrix}, \ \text{with } AE-BC=1. $$
The Maurer-Cartan form $\omega:=\rho^{-1}d\rho$ is $$ \omega = \begin{bmatrix} 0 & 0 & 0 \\ e \ dt_1 - b \ dt_2 & e \ da - b \ dc & e \ db - b \ de \\ -c \ dt_1 + a \ dt_2 & -c \ da + a \ dc & -c \ db + a \ de \end{bmatrix} $$ and, assuming $A(t) \neq 0$, we have the pullback $$ \tilde{c}^*(\omega) = \begin{bmatrix} 0 & 0 & 0 \\ \frac{1}{A}(1+CB)c_1' - Bc_2' & \frac{1}{A}(1+CB)A' - BC' & \frac{1}{A}(1+CB) B' - BE' \\ -Cc_1' + Ac_2' & -CA' + AC' & -CB' + AE' \end{bmatrix} dt. $$ Now if I set $$ \begin{align} E&=0, \\ B&= -\frac{1}{c_2'}, \\ C&= \frac{c_2'}{c_1'} A, \end{align} $$ we end up with the following matrix $$ \begin{bmatrix} 0 & 0 & 0 \\ 1 & \frac{c_2''}{c_2'} & 0 \\ 0 & c_1'c_2'' - c_2'c_1'' & -\frac{c_2''}{c_2'} \end{bmatrix}. $$ Now, one of these terms gives an invariant, but I am fairly sure the $\frac{c_2''}{c_2'}$ terms do not. I am expecting to get the affine curvature invariant instead here. I expect the problem lies in either setting $E=0$ or in some simple calculation mistakes. Any help here is greatly appreciated.