Curve name  $X_{7}$  
Index  $4$  
Level  $4$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 2 & 1 \\ 3 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves with $a_{p}(E) \equiv 0 \pmod{4}$ if $p$ is inert in $\mathbb{Q}(\sqrt{\Delta})$  
Chosen covering  $X_{1}$  
Curves that $X_{7}$ minimally covers  $X_{1}$  
Curves that minimally cover $X_{7}$  $X_{20}$, $X_{21}$, $X_{22}$, $X_{26}$, $X_{55}$  
Curves that minimally cover $X_{7}$ and have infinitely many rational points.  $X_{20}$, $X_{22}$, $X_{26}$, $X_{55}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{7}) = \mathbb{Q}(f_{7}), f_{1} = \frac{32f_{7}  4}{f_{7}^{4}}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  12x + 20$, with conductor $216$  
Generic density of odd order reductions  $179/336$ 