Curve name  $X_{11}$  
Index  $6$  
Level  $4$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves with discriminant $\Delta$ whose $2$isogenous curve has discriminant in the square class of $\Delta$  
Chosen covering  $X_{6}$  
Curves that $X_{11}$ minimally covers  $X_{6}$  
Curves that minimally cover $X_{11}$  $X_{23}$, $X_{24}$, $X_{26}$, $X_{27}$, $X_{28}$, $X_{29}$, $X_{35}$, $X_{39}$, $X_{41}$, $X_{43}$, $X_{45}$, $X_{47}$, $X_{49}$, $X_{50}$, $X_{53}$, $X_{54}$  
Curves that minimally cover $X_{11}$ and have infinitely many rational points.  $X_{23}$, $X_{24}$, $X_{26}$, $X_{27}$, $X_{28}$, $X_{29}$, $X_{35}$, $X_{39}$, $X_{41}$, $X_{43}$, $X_{45}$, $X_{47}$, $X_{49}$, $X_{50}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{11}) = \mathbb{Q}(f_{11}), f_{6} = f_{11}^{2}  16\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3 + 72x + 485$, with conductor $2772$  
Generic density of odd order reductions  $83/336$ 