Curve name 
$X_{563}$ 
Index 
$96$ 
Level 
$16$ 
Genus 
$3$ 
Does the subgroup contain $I$? 
Yes 
Generating matrices 
$
\left[ \begin{matrix} 7 & 14 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 5 \\ 6 & 11 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$ 
Images in lower levels 

Meaning/Special name 

Chosen covering 
$X_{295}$ 
Curves that $X_{563}$ minimally covers 
$X_{295}$ 
Curves that minimally cover $X_{563}$ 

Curves that minimally cover $X_{563}$ and have infinitely many rational
points. 

Model 
\[y^2 = x^7 + 4x^6  7x^5  8x^4 + 7x^3 + 4x^2  x\] 
Info about rational points 
Rational point  Image on the $j$line 
$(1 : 0 : 0)$ 
\[2048\]

$(1 : 0 : 1)$ 
\[2048\]

$(0 : 0 : 1)$ 
\[2048\]

$(1 : 0 : 1)$ 
\[2048\]


Comments on finding rational points 
This curve is isomorphic to $X_{556}$. 
Elliptic curve whose $2$adic image is the subgroup 
$y^2 = x^3 + x^2  3x  2$, with conductor $200$ 
Generic density of odd order reductions 
$2195/7168$ 