The modular curve $X_{566}$

Curve name $X_{566}$
Index $96$
Level $16$
Genus $3$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 14 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 5 \\ 6 & 3 \end{matrix}\right]$
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 $X_{566}$ minimally covers $X_{295}$
Curves that minimally cover $X_{566}$
Curves that minimally cover $X_{566}$ and have infinitely many rational points.
Model $y^2 = x^8 - 4x^7 - 12x^6 + 28x^5 + 38x^4 - 28x^3 - 12x^2 + 4x + 1$
 Rational point Image on the $j$-line $(1 : -1 : 0)$ $\infty$ $(1 : 1 : 0)$ $\frac{16974593}{256}$ $(-1 : -4 : 1)$ $\infty$ $(-1 : 4 : 1)$ $\frac{16974593}{256}$ $(0 : -1 : 1)$ $\infty$ $(0 : 1 : 1)$ $\frac{16974593}{256}$ $(1 : -4 : 1)$ $\infty$ $(1 : 4 : 1)$ $\frac{16974593}{256}$
Comments on finding rational points This curve is isomorphic to $X_{558}$.
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 369014x - 85302700$, with conductor $3362$
Generic density of odd order reductions $6515/21504$