Curve name  $X_{22}$  
Index  $8$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 2 & 3 \\ 1 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 7 \\ 6 & 7 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves whose discriminant is minus twice a square with $a_{p}(E) \equiv 0 \pmod{4}$ for $p \equiv 5 \text{ or } 7 \pmod{8}$  
Chosen covering  $X_{7}$  
Curves that $X_{22}$ minimally covers  $X_{4}$, $X_{7}$  
Curves that minimally cover $X_{22}$  $X_{56}$, $X_{57}$, $X_{83}$, $X_{179}$  
Curves that minimally cover $X_{22}$ and have infinitely many rational points.  $X_{56}$, $X_{57}$, $X_{83}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{22}) = \mathbb{Q}(f_{22}), f_{7} = \frac{\frac{1}{6}f_{22}^{2}  3}{f_{22}^{2} + 12f_{22} + 30}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  x^2  88333x + 10222037$, with conductor $44800$  
Generic density of odd order reductions  $955/1792$ 