Curve name  $X_{166}$  
Index  $24$  
Level  $16$  
Genus  $1$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 5 & 15 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 15 & 13 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 11 & 11 \\ 2 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{39}$  
Curves that $X_{166}$ minimally covers  $X_{39}$  
Curves that minimally cover $X_{166}$  $X_{281}$, $X_{289}$, $X_{302}$, $X_{304}$, $X_{366}$, $X_{374}$, $X_{379}$, $X_{389}$, $X_{395}$, $X_{402}$  
Curves that minimally cover $X_{166}$ and have infinitely many rational points.  $X_{281}$, $X_{289}$, $X_{302}$, $X_{304}$  
Model  \[y^2 = x^3 + x^2  3x + 1\]  
Info about rational points  $X_{166}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$  
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 