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


Meaning/Special name  Elliptic curves whose discriminant is twice a square  
Chosen covering  $X_{1}$  
Curves that $X_{5}$ minimally covers  $X_{1}$  
Curves that minimally cover $X_{5}$  $X_{17}$  
Curves that minimally cover $X_{5}$ and have infinitely many rational points.  $X_{17}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{5}) = \mathbb{Q}(f_{5}), f_{1} = 8f_{5}^{2} + 1728\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy + y = x^3  47x  126$, with conductor $1682$  
Generic density of odd order reductions  $3755/7168$ 