Curve name 
$X_{556}$ 
Index 
$96$ 
Level 
$16$ 
Genus 
$3$ 
Does the subgroup contain $I$? 
Yes 
Generating matrices 
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 5 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right]$ 
Images in lower levels 

Meaning/Special name 

Chosen covering 
$X_{297}$ 
Curves that $X_{556}$ minimally covers 
$X_{297}$ 
Curves that minimally cover $X_{556}$ 

Curves that minimally cover $X_{556}$ 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 
$(1 : 0 : 0)$ 
\[78608\]

$(1 : 0 : 1)$ 
\[78608\]

$(0 : 0 : 1)$ 
\[78608\]

$(1 : 0 : 1)$ 
\[78608\]


Comments on finding rational points 
A family of etale double covers maps to genus 2 hyperelliptic curves whose
Jacobians have rank zero or one. 
Elliptic curve whose $2$adic image is the subgroup 
$y^2 = x^3 + x^2  28x + 48$, with conductor $200$ 
Generic density of odd order reductions 
$2195/7168$ 