Curve name  $X_{85j}$  
Index  $48$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  No  
Generating matrices  $ \left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{85}$  
Curves that $X_{85j}$ minimally covers  
Curves that minimally cover $X_{85j}$  
Curves that minimally cover $X_{85j}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 108t^{8} + 864t^{6} + 4320t^{4}  24192t^{2}  1728\] \[B(t) = 432t^{12} + 5184t^{10}  77760t^{8} + 483840t^{6}  684288t^{4}  912384t^{2} + 27648\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  x^2  2177x  108927$, with conductor $1344$  
Generic density of odd order reductions  $635/5376$ 