The modular curve $X_{167}$

Curve name $X_{167}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 2 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{12}$ $8$ $12$ $X_{30}$
Meaning/Special name
Chosen covering $X_{30}$
Curves that $X_{167}$ minimally covers $X_{30}$
Curves that minimally cover $X_{167}$ $X_{295}$, $X_{297}$, $X_{419}$, $X_{423}$
Curves that minimally cover $X_{167}$ and have infinitely many rational points. $X_{295}$, $X_{297}$
Model $y^2 = x^3 + x^2 + x + 1$
Info about rational points $X_{167}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup None. All the rational points lift to covering modular curves.
Generic density of odd order reductions N/A