Curve name  $X_{68}$  
Index  $24$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 7 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{23}$  
Curves that $X_{68}$ minimally covers  $X_{23}$, $X_{28}$, $X_{47}$  
Curves that minimally cover $X_{68}$  $X_{255}$, $X_{264}$  
Curves that minimally cover $X_{68}$ and have infinitely many rational points.  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{68}) = \mathbb{Q}(f_{68}), f_{23} = \frac{f_{68}}{f_{68}^{2} + \frac{1}{4}}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy = x^3  x^2  805514x  264095580$, with conductor $16810$  
Generic density of odd order reductions  $109/448$ 