Curve name  $X_{75g}$  
Index  $48$  
Level  $8$  
Genus  $0$  
Does the subgroup contain $I$?  No  
Generating matrices  $ \left[ \begin{matrix} 5 & 5 \\ 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 $X_{75g}$ minimally covers  
Curves that minimally cover $X_{75g}$  
Curves that minimally cover $X_{75g}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 442368t^{8} + 3317760t^{6}  926208t^{4} + 51840t^{2}  108\] \[B(t) = 113246208t^{12} + 1783627776t^{10}  1842020352t^{8} + 390168576t^{6}  28781568t^{4} + 435456t^{2} + 432\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3  x^2 + 55743x  1756863$, with conductor $14784$  
Generic density of odd order reductions  $307/2688$ 