Curve name  $X_{17}$  
Index  $6$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves that acquire full $2$torsion over $\mathbb{Q}(\sqrt{2})$  
Chosen covering  $X_{6}$  
Curves that $X_{17}$ minimally covers  $X_{5}$, $X_{6}$  
Curves that minimally cover $X_{17}$  $X_{37}$, $X_{40}$, $X_{41}$, $X_{44}$  
Curves that minimally cover $X_{17}$ and have infinitely many rational points.  $X_{37}$, $X_{40}$, $X_{41}$, $X_{44}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{17}) = \mathbb{Q}(f_{17}), f_{6} = \frac{48f_{17}^{2} + 32}{f_{17}^{2}  2}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy + y = x^3  36x  70$, with conductor $14$  
Generic density of odd order reductions  $5123/21504$ 