Curve name  $X_{19}$  
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 & 3 \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 $X_{19}$ minimally covers  $X_{4}$, $X_{6}$  
Curves that minimally cover $X_{19}$  $X_{29}$, $X_{48}$  
Curves that minimally cover $X_{19}$ and have infinitely many rational points.  $X_{29}$, $X_{48}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{19}) = \mathbb{Q}(f_{19}), f_{6} = \frac{48f_{19}^{2}  32}{f_{19}^{2} + 2}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy + y = x^3 + 4x + 20$, with conductor $66$  
Generic density of odd order reductions  $5123/21504$ 