Curve name  $X_{4}$  
Index  $2$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 1 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves whose discriminant is minus twice a square  
Chosen covering  $X_{1}$  
Curves that $X_{4}$ minimally covers  $X_{1}$  
Curves that minimally cover $X_{4}$  $X_{19}$, $X_{22}$  
Curves that minimally cover $X_{4}$ and have infinitely many rational points.  $X_{19}$, $X_{22}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{4}) = \mathbb{Q}(f_{4}), f_{1} = 2f_{4}^{2} + 1728\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy + y = x^3  x  2$, with conductor $50$  
Generic density of odd order reductions  $3755/7168$ 