Curve name  $X_{38b}$  
Index  $24$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  No  
Generating matrices  $ \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 6 & 7 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{38}$  
Curves that $X_{38b}$ minimally covers  
Curves that minimally cover $X_{38b}$  
Curves that minimally cover $X_{38b}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 108t^{8} + 648t^{6}  1728t^{4} + 2592t^{2}  1728\] \[B(t) = 432t^{12}  3888t^{10} + 10368t^{8}  41472t^{4} + 62208t^{2}  27648\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  3724x  82320$, with conductor $3136$  
Generic density of odd order reductions  $289/1792$ 