Curve name  $X_{75h}$  
Index  $48$  
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 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{75}$  
Curves that $X_{75h}$ minimally covers  
Curves that minimally cover $X_{75h}$  
Curves that minimally cover $X_{75h}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 7077888t^{12} + 54853632t^{10}  28200960t^{8} + 5363712t^{6}  440640t^{4} + 13392t^{2}  27\] \[B(t) = 7247757312t^{18}  111434268672t^{16} + 160356630528t^{14}  74516004864t^{12} + 16955080704t^{10}  2119385088t^{8} + 145539072t^{6}  4893696t^{4} + 53136t^{2} + 54\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3 + x^2 + 682848x  74301228$, with conductor $25872$  
Generic density of odd order reductions  $193/1792$ 