Curve name 
$X_{364}$ 
Index 
$48$ 
Level 
$16$ 
Genus 
$2$ 
Does the subgroup contain $I$? 
Yes 
Generating matrices 
$
\left[ \begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 12 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 10 & 15 \end{matrix}\right]$ 
Images in lower levels 

Meaning/Special name 

Chosen covering 
$X_{81}$ 
Curves that $X_{364}$ minimally covers 
$X_{81}$, $X_{150}$, $X_{153}$ 
Curves that minimally cover $X_{364}$ 

Curves that minimally cover $X_{364}$ and have infinitely many rational
points. 

Model 
\[y^2 = x^6 + 10x^4  20x^2  8\] 
Info about rational points 
Rational point  Image on the $j$line 
$(1 : 1 : 0)$ 
\[1728 \,\,(\text{CM by }4)\]

$(1 : 1 : 0)$ 
\[1728 \,\,(\text{CM by }4)\]


Comments on finding rational points 
The rank of the Jacobian is 2. This curve admits a family of etale double
covers that map to rank zero elliptic curves. 
Elliptic curve whose $2$adic image is the subgroup 
None 
Generic density of odd order reductions 
N/A 