Curve name  $X_{199f}$  
Index  $96$  
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} 1 & 1 \\ 0 & 5 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{199}$  
Curves that $X_{199f}$ minimally covers  
Curves that minimally cover $X_{199f}$  
Curves that minimally cover $X_{199f}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 27t^{16} + 12528t^{14}  79056t^{12}  257472t^{10}  306720t^{8}  1029888t^{6}  1264896t^{4} + 801792t^{2}  6912\] \[B(t) = 54t^{24} + 55728t^{22}  2494800t^{20} + 1100736t^{18} + 4740768t^{16}  120310272t^{14}  404006400t^{12}  481241088t^{10} + 75852288t^{8} + 70447104t^{6}  638668800t^{4} + 57065472t^{2} + 221184\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy = x^3  1644x  30942$, with conductor $102$  
Generic density of odd order reductions  $53/896$ 