Curve name  $X_{16}$  
Index  $6$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 2 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves with a cyclic $4$isogeny defined over $\mathbb{Q}(\sqrt{2})$  
Chosen covering  $X_{6}$  
Curves that $X_{16}$ minimally covers  $X_{6}$  
Curves that minimally cover $X_{16}$  $X_{29}$, $X_{38}$, $X_{40}$, $X_{42}$  
Curves that minimally cover $X_{16}$ and have infinitely many rational points.  $X_{29}$, $X_{38}$, $X_{40}$, $X_{42}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{16}) = \mathbb{Q}(f_{16}), f_{6} = 2f_{16}^{2} + 48\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy + y = x^3 + 4x  6$, with conductor $14$  
Generic density of odd order reductions  $5123/21504$ 