Curve name  $X_{34f}$  
Index  $24$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  No  
Generating matrices  $ \left[ \begin{matrix} 3 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{34}$  
Curves that $X_{34f}$ minimally covers  
Curves that minimally cover $X_{34f}$  
Curves that minimally cover $X_{34f}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 432t^{4} + 864t^{2}  108\] \[B(t) = 3456t^{6}  10368t^{4} + 6480t^{2} + 432\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3 + x + 2$, with conductor $56$  
Generic density of odd order reductions  $643/5376$ 