Curve name  $X_{123}$  
Index  $24$  
Level  $16$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 11 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 6 & 3 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{37}$  
Curves that $X_{123}$ minimally covers  $X_{37}$  
Curves that minimally cover $X_{123}$  $X_{239}$, $X_{123a}$, $X_{123b}$, $X_{123c}$, $X_{123d}$, $X_{123e}$, $X_{123f}$  
Curves that minimally cover $X_{123}$ and have infinitely many rational points.  $X_{239}$, $X_{123a}$, $X_{123b}$, $X_{123c}$, $X_{123d}$, $X_{123e}$, $X_{123f}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{123}) = \mathbb{Q}(f_{123}), f_{37} = \frac{f_{123} + 1}{f_{123}^{2}  2}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  x^2  6281x  186919$, with conductor $10880$  
Generic density of odd order reductions  $85091/344064$ 