Curve name  $X_{67}$  
Index  $24$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 4 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{24}$  
Curves that $X_{67}$ minimally covers  $X_{24}$, $X_{28}$, $X_{43}$  
Curves that minimally cover $X_{67}$  $X_{255}$, $X_{256}$, $X_{67a}$, $X_{67b}$, $X_{67c}$, $X_{67d}$  
Curves that minimally cover $X_{67}$ and have infinitely many rational points.  $X_{67a}$, $X_{67b}$, $X_{67c}$, $X_{67d}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{67}) = \mathbb{Q}(f_{67}), f_{24} = \frac{2}{f_{67}^{2}}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  63x + 162$, with conductor $360$  
Generic density of odd order reductions  $13/84$ 