Curve name  $X_{10}$  
Index  $6$  
Level  $4$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves that acquire full $2$torsion over $\mathbb{Q}(i)$  
Chosen covering  $X_{6}$  
Curves that $X_{10}$ minimally covers  $X_{3}$, $X_{6}$  
Curves that minimally cover $X_{10}$  $X_{27}$, $X_{42}$, $X_{10a}$, $X_{10b}$, $X_{10c}$, $X_{10d}$  
Curves that minimally cover $X_{10}$ and have infinitely many rational points.  $X_{27}$, $X_{42}$, $X_{10a}$, $X_{10b}$, $X_{10c}$, $X_{10d}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{10}) = \mathbb{Q}(f_{10}), f_{6} = \frac{48f_{10}^{2}  16}{f_{10}^{2} + 1}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 = x^3 + 33x  74$, with conductor $180$  
Generic density of odd order reductions  $83/336$ 