## The modular curve $X_{200d}$

Curve name $X_{200d}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25i}$
Meaning/Special name
Chosen covering $X_{200}$
Curves that $X_{200d}$ minimally covers
Curves that minimally cover $X_{200d}$
Curves that minimally cover $X_{200d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27648t^{16} + 110592t^{14} - 1741824t^{12} + 3290112t^{10} - 2885760t^{8} + 822528t^{6} - 108864t^{4} + 1728t^{2} - 108$ $B(t) = 1769472t^{24} - 10616832t^{22} - 204374016t^{20} + 885620736t^{18} - 2514530304t^{16} + 3459760128t^{14} - 2284498944t^{12} + 864940032t^{10} - 157158144t^{8} + 13837824t^{6} - 798336t^{4} - 10368t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 6657x + 71775$, with conductor $1344$
Generic density of odd order reductions $271/2688$