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

Curve name $X_{200g}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{200}$
Curves that $X_{200g}$ minimally covers
Curves that minimally cover $X_{200g}$
Curves that minimally cover $X_{200g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -442368t^{24} + 4423680t^{22} - 42688512t^{20} + 237330432t^{18} - 629462016t^{16} + 832204800t^{14} - 600044544t^{12} + 208051200t^{10} - 39341376t^{8} + 3708288t^{6} - 166752t^{4} + 4320t^{2} - 108$ $B(t) = -113246208t^{36} + 1698693120t^{34} + 3822059520t^{32} - 151976411136t^{30} + 1011826556928t^{28} - 3649896972288t^{26} + 8480605077504t^{24} - 12985617088512t^{22} + 12688948297728t^{20} - 7860634288128t^{18} + 3172237074432t^{16} - 811601068032t^{14} + 132509454336t^{12} - 14257410048t^{10} + 988111872t^{8} - 37103616t^{6} + 233280t^{4} + 25920t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 326209x + 25271231$, with conductor $9408$
Generic density of odd order reductions $271/2688$