Curve name  $X_{13f}$  
Index  $12$  
Level  $4$  
Genus  $0$  
Does the subgroup contain $I$?  No  
Generating matrices  $ \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  
Chosen covering  $X_{13}$  
Curves that $X_{13f}$ minimally covers  
Curves that minimally cover $X_{13f}$  
Curves that minimally cover $X_{13f}$ and have infinitely many rational points.  
Model  $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 27t^{4} + 432t^{3}  432t^{2}  20736t + 82944\] \[B(t) = 54t^{6} + 1296t^{5}  6480t^{4}  65664t^{3} + 746496t^{2}  1990656t\]  
Info about rational points  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy = x^3 + x^2 + 44x + 55$, with conductor $33$  
Generic density of odd order reductions  $5/42$ 