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


Meaning/Special name  Elliptic curves whose discriminant is minus a square  
Chosen covering  $X_{1}$  
Curves that $X_{3}$ minimally covers  $X_{1}$  
Curves that minimally cover $X_{3}$  $X_{10}$, $X_{20}$, $X_{3a}$, $X_{3b}$  
Curves that minimally cover $X_{3}$ and have infinitely many rational points.  $X_{10}$, $X_{20}$, $X_{3a}$, $X_{3b}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{3}) = \mathbb{Q}(f_{3}), f_{1} = f_{3}^{2} + 1728\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy + y = x^3 + x^2  30x  76$, with conductor $121$  
Generic density of odd order reductions  $59/112$ 