The modular curve $X_{156}$

Curve name $X_{156}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 12 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 13 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{50}$
Meaning/Special name
Chosen covering $X_{50}$
Curves that $X_{156}$ minimally covers $X_{50}$
Curves that minimally cover $X_{156}$ $X_{291}$, $X_{324}$, $X_{412}$, $X_{413}$, $X_{418}$, $X_{426}$
Curves that minimally cover $X_{156}$ and have infinitely many rational points. $X_{291}$, $X_{324}$
Model $y^2 = x^3 + 8x$
Info about rational points $X_{156}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
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