Curve name  $X_{75j}$  
Index  $48$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  No  
Generating matrices  $ \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 5 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{75}$  
Curves that $X_{75j}$ minimally covers  
Curves that minimally cover $X_{75j}$  
Curves that minimally cover $X_{75j}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 110592t^{8} + 829440t^{6}  231552t^{4} + 12960t^{2}  27\] \[B(t) = 14155776t^{12}  222953472t^{10} + 230252544t^{8}  48771072t^{6} + 3597696t^{4}  54432t^{2}  54\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  x^2 + 13936x + 212640$, with conductor $3696$  
Generic density of odd order reductions  $65/896$ 