Curve name  $X_{12}$  
Index  $6$  
Level  $4$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 3 \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 $X_{12}$ minimally covers  $X_{6}$  
Curves that minimally cover $X_{12}$  $X_{25}$, $X_{30}$, $X_{31}$, $X_{40}$, $X_{51}$, $X_{52}$  
Curves that minimally cover $X_{12}$ and have infinitely many rational points.  $X_{25}$, $X_{30}$, $X_{31}$, $X_{40}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{12}) = \mathbb{Q}(f_{12}), f_{6} = f_{12}^{2}  16\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  x^2  203x  1048$, with conductor $2200$  
Generic density of odd order reductions  $13/42$ 