Curve name 
$X_{52}$ 
Index 
$12$ 
Level 
$8$ 
Genus 
$1$ 
Does the subgroup contain $I$? 
Yes 
Generating matrices 
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$ 
Images in lower levels 

Meaning/Special name 

Chosen covering 
$X_{12}$ 
Curves that $X_{52}$ minimally covers 
$X_{12}$ 
Curves that minimally cover $X_{52}$ 
$X_{126}$, $X_{129}$ 
Curves that minimally cover $X_{52}$ and have infinitely many rational
points. 

Model 
\[y^2 = x^3 + x\] 
Info about rational points 
Rational point  Image on the $j$line 
$(0 : 1 : 0)$ 
\[ \infty \]

$(0 : 0 : 1)$ 
\[ \infty \]


Comments on finding rational points 
None 
Elliptic curve whose $2$adic image is the subgroup 
None 
Generic density of odd order reductions 
N/A 