Curve name  $X_{216}$  
Index  $48$  
Level  $16$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 2 \\ 10 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{77}$  
Curves that $X_{216}$ minimally covers  $X_{77}$, $X_{110}$, $X_{112}$  
Curves that minimally cover $X_{216}$  $X_{520}$, $X_{533}$  
Curves that minimally cover $X_{216}$ and have infinitely many rational points.  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{216}) = \mathbb{Q}(f_{216}), f_{77} = \frac{f_{216}^{2}  \frac{1}{8}}{f_{216}}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  x^2  14413x + 1182613$, with conductor $13056$  
Generic density of odd order reductions  $12833/57344$ 