Curve name  $X_{9}$  
Index  $6$  
Level  $4$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 2 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves that acquire a cyclic $4$isogeny over $\mathbb{Q}(i)$  
Chosen covering  $X_{6}$  
Curves that $X_{9}$ minimally covers  $X_{6}$  
Curves that minimally cover $X_{9}$  $X_{24}$, $X_{37}$, $X_{9a}$, $X_{9b}$, $X_{9c}$, $X_{9d}$  
Curves that minimally cover $X_{9}$ and have infinitely many rational points.  $X_{24}$, $X_{37}$, $X_{9a}$, $X_{9b}$, $X_{9c}$, $X_{9d}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{9}) = \mathbb{Q}(f_{9}), f_{6} = f_{9}^{2} + 48\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  12x  11$, with conductor $180$  
Generic density of odd order reductions  $83/336$ 