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 


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 