Curve name  $X_{20}$  
Index  $8$  
Level  $4$  
Genus  $0$  
Does the subgroup contain $I$?  Yes  
Generating matrices  $ \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 3 & 0 \end{matrix}\right]$  
Images in lower levels 


Meaning/Special name  Elliptic curves whose discriminant is minus a square with $a_{p}(E) \equiv 0 \pmod{4}$ for $p \equiv 3 \pmod{4}$  
Chosen covering  $X_{7}$  
Curves that $X_{20}$ minimally covers  $X_{3}$, $X_{7}$  
Curves that minimally cover $X_{20}$  $X_{60}$, $X_{180}$, $X_{20a}$, $X_{20b}$  
Curves that minimally cover $X_{20}$ and have infinitely many rational points.  $X_{60}$, $X_{20a}$, $X_{20b}$  
Model  \[\mathbb{P}^{1}, \mathbb{Q}(X_{20}) = \mathbb{Q}(f_{20}), f_{7} = \frac{f_{20} + 1}{f_{20}^{2}  3}\]  
Info about rational points  None  
Comments on finding rational points  None  
Elliptic curve whose $2$adic image is the subgroup  $y^2 + xy = x^3  x^2  6x + 8$, with conductor $162$  
Generic density of odd order reductions  $37/84$ 