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


Meaning/Special name  
Chosen covering  $X_{34}$  
Curves that $X_{34b}$ minimally covers  
Curves that minimally cover $X_{34b}$  
Curves that minimally cover $X_{34b}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 108t^{8} + 648t^{6}  1323t^{4} + 972t^{2}  108\] \[B(t) = 432t^{12}  3888t^{10} + 13770t^{8}  23814t^{6} + 19764t^{4}  5832t^{2}  432\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3 + 196x  5488$, with conductor $3136$  
Generic density of odd order reductions  $289/1792$ 