Curve name  $X_{77}$  
Index  $24$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{29}$  
Curves that $X_{77}$ minimally covers  $X_{29}$, $X_{43}$, $X_{45}$  
Curves that minimally cover $X_{77}$  $X_{216}$, $X_{257}$, $X_{258}$, $X_{275}$, $X_{276}$, $X_{373}$, $X_{391}$  
Curves that minimally cover $X_{77}$ and have infinitely many rational points.  $X_{216}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{77}) = \mathbb{Q}(f_{77}), f_{29} = \frac{f_{77}^{2}  \frac{1}{2}}{f_{77}}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  x^2 + 307x  4587$, with conductor $5376$  
Generic density of odd order reductions  $401/1792$ 