Curve name  $X_{34}$  
Index  $12$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 5 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{13}$  
Curves that $X_{34}$ minimally covers  $X_{13}$  
Curves that minimally cover $X_{34}$  $X_{75}$, $X_{79}$, $X_{94}$, $X_{100}$, $X_{34a}$, $X_{34b}$, $X_{34c}$, $X_{34d}$, $X_{34e}$, $X_{34f}$, $X_{34g}$, $X_{34h}$  
Curves that minimally cover $X_{34}$ and have infinitely many rational points.  $X_{75}$, $X_{79}$, $X_{94}$, $X_{100}$, $X_{34a}$, $X_{34b}$, $X_{34c}$, $X_{34d}$, $X_{34e}$, $X_{34f}$, $X_{34g}$, $X_{34h}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{34}) = \mathbb{Q}(f_{34}), f_{13} = 8f_{34}^{2}  8\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy = x^3  x^2 + 58x  284$, with conductor $350$  
Generic density of odd order reductions  $513/3584$ 