The modular curve $X_{563}$

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
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{12}$ $8$ $24$ $X_{93}$
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 $y^2 = x^7 + 4x^6 - 7x^5 - 8x^4 + 7x^3 + 4x^2 - x$
 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 $2195/7168$